some mathematical model s dynamics of biological...

Dynamics of Biological Species:

Some Mathematical Models

THESIS

Submitted in partial fulfilment

of the requirements for the degree of

DOCTOR OF PHILOSOPHY

By

ATASI PATRA

Under the Supervision of

PROF. BALRAM DUBEY

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI

PILANI CAMPUS, RAJASTHAN-333031

December, 2013

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE

PILANI (RAJASTHAN)

CERTIFICATE

This is to certify that the thesis entitled ''Dynamics of Biological Species :

Some Mathematical Models'' submitted by Atasi Patra, ID No.

2009PHXF419P for award of Ph.D. Degree of the Institute embodies original

work done by her under my supervision.

_____________

Signature of the Supervisor

Name : BALRAM DUBEY

Date / / 2013. Designation : PROFESSOR

Dedicated To

My Beloved Children

Oli and Roni

Acknowledgements

It is my great pleasure and privilege to express my deep sense of gratitude and

everlasting indebtedness to my research supervisor, Prof. Balram Dubey for his patience,

continuous encouragement and guidance throughout my research work. Without his active

guidance it would not have been possible for me to complete the work.

I would like to express my sincere thanks to Vice Chancellor, Director, Deputy

Directors, BITS Pilani, Pilani Campus for providing me the opportunity to pursue my

doctoral studies. I express my gratitude to Prof. S. K. Verma, Dean, ARD, Prof. H. Jadav,

Dr. C. B. Das, Prof. S. Kumar for their official support and encouragement.

I owe my sincere gratitude to Prof. P. H. Keskar, HOD, Mathematics who provided me

with requisite facilities to carry out my doctoral research. I acknowledge my deep sense of

gratitude to others faculty members of our Department, specially Prof. Ram Awtar, Dr. Rakhi

and Dr. A. K. Verma.

I am thankful to my DAC members Prof. R. Kumar and Dr. U. S. Dubey who spared

their valuable time to review my draft thesis and provided me with constructive and valuable

suggestion, which helped in improving the quality of the thesis.

I express my sincere thanks to DST, New Delhi for providing me the financial support

under Woman Scientist Scheme (WOS-A) throughout my doctoral research work.

I am grateful to Dr. S. K. Sahani (SAU- New Delhi), Dr. P. K. Srivastava (IIT-Patna),

Prof. R. K. Upadhyay, and Prof. N. K. Thakur (ISM-Dhanbad) who helped me to understand

the subject clearly.

I would like to thank all the co-scholars, specially Mr. Rajesh who helped throughout my

research work. I also express my thanks to the official staff of SRCD and ARD, specially,

Jadav ji, and our attendant Mahendar ji for their official support.

Most importantly, my family deserve credit. I would like to express my sincere gratitude

to my parents and members of father-in-law's house for their unquestioning support. I also

express my thanks to my maid Saraswati di for proper caring of my children. Special thanks

to Saumi di, Babi di, Deepa di, Yogita di, Madhu and Pijus.

My heartfelt thanks are due to my husband Dilip who gave me the opportunity to

complete this work. He rejoiced along with me in my successes and consoled me in my

frustrations. It was a long and difficult journey and I could not completed it without his

continual support, understanding, motivation and love.

At last, but by no means the least, I express my deepest love to my dear Children: Oli

and Roni whom I may have neglected way too much during my research work.

Dated ATASI PATRA

Abstract

The depletion of renewable and non-renewable resources at an alarming rate is causing a

great concern in the developing countries. These resources are being depleted due to rapid

industrialization, fast urbanization and increasing population. The utilization of resources is

an essential part for the development of any country. But for the ecological balance it is also

very important to keep these resources at an optimal level.

It is well known that the resource carrying capacity of our planet is limited. The

environmental crisis which society faces today can be controlled by:

i) restoring the productivity of the ecosystem and by conserving genetic resources,

ii) insisting and assessing the environmental impact of developmental projects to ensure

harmonization of environment with developments,

iii) incorporating environmental safeguards and beneficial analysis for development of

industrial projects,

iv) affecting better land use, and

v) promoting environmental education awareness among people.

Since the beginning of life in the earth the natural renewable food resources such as fish,

whale, deer, forest etc. constitute an increasingly important class of economic resources for

human welfare on the planet earth. The common characteristic of these resources are that they

can be directly consumed by humans and they can reproduce themselves with a specific

spread of renewal in a given a specific environment.

Manmade projects such as the expansion of agricultural land for food production,

growing of mining and wood based industries, development of housing complexes, timber

trade, and cutting of trees for fuel and fodder, pollutants in the form of sulphur dioxide,

nitrogen oxide, carbon monoxide, hydrocarbons, fluorines, fly ash, pollulated water, etc. are

responsible for the ecological degradation.

Human population have exhibited wide fluctuations throughout time and their growth

may still be responding to environmental changes in modern times. It may be affected

dramatically by advances in housing, agricultural practices, health care, and so on. In

retrospect, it is not surprising to see dramatic changes in the carrying capacity of the earth

Abstract

vi

over time, because these changes are driven by strong environmental shifts and events that

include the public health transformation experienced over the last five decades via the

widespread use of antibiotics and the implementation of large-scale vaccination policies.

It may be pointed out here that consumers of the resources wish to maximize the yield in

the long run and to maximize the net economic benefit in the long run. However, it is also the

prime duty of consumers to reduce the risk of extinction of the resources in the long run. In

the case when fish is taken as a natural resource then this can be achieved by

i) restriction on quotas,

ii) restriction on fish age, size and gender,

iii) restriction on net type, its number and size,

iv) restriction on type of fishing boats,

v) restriction on number of licences,

vi) restriction on harvesting time,

vii) restriction on space (creating reserve zones),

viii) imposing taxes, etc.

It is difficult task to form a perfect models that will serve our multiple objectives. The

validation that any model truly represent the real world system is even more difficult. A

useful and pragmatic approach to validation of models is to confront model output with

measurement data as well as expert judgment and sensitivity analysis. Janssen and Heuberger

(1995) discern three major aspects and relevant questions pertaining to each:

i) Assessment of the ability of the model to reproduce the real world system behaviour:

Does the model output resemble observations and measurement data?

ii) Assessment of suitability of the model for the intended use :

Does the model answer the research question ? and

iii) Assessment of the robustness or sensitivity of the model :

Does the assessment of the problem and its various solution differ greatly when

parameter have extreme but acceptable values ?

The advantages of modelling are:

i) multi-disciplinary knowledge is structured consistently,

Abstract

vii

ii) it helps to describe the logical consequence of assumptions, data and the described

knowledge of processes, and

iii) it can explore the behaviour of complex system more reliably than mental model,

provided the assumptions are well-understood.

Keeping the above aspects in mind, in this thesis we propose and analyze some

mathematical models to study the dynamics of some biological species. Various aspect of

interaction of biological populations with crowding effects, various types of nonlinear

harvesting rate of renewable resources and different types of nonlinear treatment rates of

infected populations are considered in the formulation of the models. These models have been

analyzed using the stability theory of Ordinary Differential Equations and finally computer

simulations have been carried out to validate the qualitative analysis.

The whole work is discussed in seven separate chapters whose abstracts are given below.

In Chapter 1, a brief introduction with literature has been presented to provide a

background required for the upcoming chapters. Further an overview of mathematical tools

for establishing local and global stability of the steady states of dynamical models, which has

been used in the sub sequent chapters, is also given.

In Chapter 2, a dynamical model has been proposed and analyzed to study the effect of

population on resource biomass by taking into account the crowing effect. The resource

biomass and population, both are growing logistically. The population partially depends on

the resource and utilizes the resource for its own growth and development. The population

grow both linearly and non-linearly with the presence of the resource. The resource biomass,

which has commercial importance, is harvested according to catch-per-unit effort hypothesis

and the harvesting effort is a control variable.

The existence of equilibria and stability analysis have been discussed with the help of

stability theory of ordinary differential equations. It has been shown that the positive

equilibrium (whenever it exists) is always locally asymptotically stable. But it is not always

globally asymptotically stable. However, we are able to find a sufficient condition under

which the positive equilibrium point is globally asymptotically stable. This condition gives a

threshold value of the specific growth rate of resource biomass. If specific growth rate of

resource biomass is larger than this threshold value, then positive equilibrium point is

globally asymptotically stable. Using Bendixon-Dulac criteria, it has been observed that the

model system has no limit cycle in the interior of the positive quadrant.

Abstract

viii

The bionomical equilibrium of the model has also been discussed. It has been observed

that the bionomical equilibrium of the resource biomass does not depend upon the growth rate

and carrying capacity of population. An analysis for sustainable yield and maximum

sustainable yield has been carried out. It has been shown that if sustainable yield is greater

than maximum sustainable yield then the resource biomass will tend to zero and if sustainable

yield is less than maximum sustainable yield then the resource biomass and population may

be maintained at desired level.

The global asymptotic stability behaviour of the positive equilibrium has also been

studied through output feedback control method. We have constructed an appropriate

Hamiltonian function and then using Pontryagin’s Maximum Principle, the optimal

harvesting policy has been discussed. An optimal equilibrium solution has been obtained.

The results are validated based on the numerical simulation. Here we observed that the

depletion rate of the resource due to its crowding, the growth rate of the population through

the crowding of resource and harvesting effort are important parameters governing the

dynamics of the system. It has been found that if depletion rate of the resource due to

crowding increases, then the resource biomass and the population both decrease. However, if

growth rate of the population through the crowding of resource increases, then the population

density increases and the resource biomass density decreases. A threshold value for the

optimal harvesting effort has been found theoretically as well as numerically. It has been

shown that the harvesting effort should be always kept less than optimal harvesting effort to

maintain the resource and the population at an optimal equilibrium level.

In Chapter 3, the model studied in Chapter-2 is extended. In Chapter-2, the harvesting

effort has been considered to be a control variable, but here the harvesting effort has been

assumed to be a dynamic variable and taxation has been used as a control variable. The

harvesting rate has been assumed to follow the catch-per-unit effort (CPUE) hypothesis.

The existence of equilibrium points has been discussed and stability analysis has been

carried out by eigenvalue method, Routh-Hurwitz criteria and Liapunov direct method. A

threshold level of the intrinsic growth rate of resource biomass has been found and it has been

shown that if the intrinsic growth rate of the resource biomass is larger than a threshold value

and price of the per unit harvested biomass is larger than the tax imposed on it, then the non-

zero equilibrium point exists. And whenever this point exists, it is always locally and globally

asymptotically stable. The bionomical equilibrium of the model has been found and it has

Abstract

ix

been shown that the bionomical equilibrium of the resource biomass does not depend upon

the growth rate and carrying capacity of population utilizing the resource biomass. The

maximum sustainable yield (MSY) of the model has been obtained. The optimal harvesting

policy has been discussed using Pontryagin’s Maximum Principle. Constructing an

appropriate Hamiltonian function the optimal tax policy has been found.

A computer simulation has been performed to illustrate all theoretical results. We

observe that the depletion of the resource due to crowding and the growth rate of the resource

due to crowding are important parameters governing the dynamics of the model. If the

depletion of the resource due to its crowding increases, then resource density and population

density both initially decreases with respect to time, but after a threshold value of the

depletion rate, the density of resource biomass and population decreases as depletion rate

decreases. We note that the densities of the resource biomass and population increase as tax

increases, but the density of effort decreases as tax increases. For an optimal level of the tax

imposed on per unit of harvested biomass, the resource biomass, the population and the effort

settle down at their respective optimal level. It has been shown that if the regulatory agency

imposes a tax on the per unit harvested biomass at an optimal level, then the densities of the

resource biomass and population can be maintained at an appropriate level.

In the Chapter 4, we proposed a model of resource biomass and population in which

both are growing logistically. The resource biomass, which has commercial importance, is

harvested according to a realistic non-linear catch-rate function. The population utilizes the

resource for its own growth and development and grow linearly and non-linearly in the

presence of the resource. The harvesting effort has been assumed to be a dynamical variable.

Tax on per unit harvested resource biomass has been used as a tool to control exploitation of

the resource.

Here we have discussed the existence of equilibria, local stability by Eigen value method

and Routh-Hurwitz criteria and global stability using Liapunov direct method. The non-

negative equilibrium point exists if tax on the harvested biomass has been less than a

threshold value. This threshold value depend on the selling price per unit biomass and the

fixed cost of harvesting per unit of effort. The non-zero equilibrium point is locally and

globally asymptotically stable under certain conditions. When the population does not utilized

the resource directly, but the resource biomass is harvested according to same catch-rate

function, then we obtained the range of the tax on per unit harvested biomass which can be

used by the regulatory agencies. But, when the population utilized the resource biomass for

Abstract

x

its growth and development and the resource is harvested, then the range of the tax is again

obtained. This is same as previous one, but in some modified form. The maximum sustainable

yield (MSY) has been computed for our model system. Then bionomic equilibrium has been

obtained and we observed that for this model bionomic equilibrium point exists under certain

conditions. Choosing an appropriate Hamiltonian function and using the Pontryagin's

Maximum Principle, we have analysed optimal harvesting policy.

Finally, a numerical simulation experiments have been carried out to verify our

theoretical results. It has been observed that depletion rate of the resource and growth rate of

the population due to crowding are very sensitive parameters in comparison to depletion rate

of resource and growth rate of the population. We note that density of resource, population

and harvesting effort with respect to time all decrease as depletion of resource due to

crowding increases. Again it is observed that if growth rate of population due to crowding

increases then density of population also increases very quickly with respect to time and goes

to the peak after that it decreases very quickly and settle down at its equilibrium level. Again,

density of resource and harvesting effort decrease as growth rate of population due to

crowding increases and then obtain its respective equilibrium levels. It is also observed here

that dynamics of the system is highly sensitive with respect to the growth rate of population

due to crowding. An optimal level of tax to be imposed by the regulatory agency has been

suggested. It has been shown that if tax increases, then densities of resource biomass and

population also increase with respect to time, but if tax increases then harvesting effort

decreases. If tax is greater than optimal tax, then harvesting effort decreases and goes to the

zero level but resource biomass and population density increases with respect to time t. Thus

the regulatory agency should keep tax less than optimal tax, so that one can maintain resource

and population at an optimal level.

In Chapter 5, we have considered a 2D spatial Rosenzweig-MacArthur type model with

Holling type IV functional response for zooplankton-phytoplankton-fish interaction with self

and cross-diffusion. We assumed that nutrient level of the system is constant. Phytoplankton

is a food for Zooplankton. The phytoplankton equation is based on the logistic growth

formulation, with a Monod type of nutrient limitation. We consider the growth limitations by

different nutrients are not treated separately and there is an overall carrying capacity which is

a function of the nutrient level of the system, and the phytoplankton do not deplete the

nutrient level. Holling type functional response-II, III and IV all appear in this model.

Abstract

xi

Using the stability theory of ordinary differential equations, we analyzed the local

stability analysis of a non-spatial model. It has been observed that the positive equilibrium

point is locally asymptotically stable under certain conditions. Choosing an appropriate

Liapunov function the global stability has been also analysed and the positive equilibrium

point proved to be globally asymptotically stable under some conditions.

Then we have considered the model of one dimensional case and constant diffusivity

with initial and zero flux boundary conditions, and two dimensional case with constant

diffusivity. The effect of the critical wave length which can drive a system to instability has

been investigated. We have observed that cross-diffusion coefficient can be quite significant,

even for small values of off-diagonal terms in the diffusion matrix. It has been observed that

the rate of convergence towards its equilibrium in two-dimension is faster in comparison to

one-dimension case.

The condition of stability of positive equilibrium point in one dimensional case and two

dimensional case are different. So, if positive equilibrium point is stable in one-dimensional

case, it need not be stable in two-dimensional case. We also notice that if the positive

equilibrium point of the model equation is unstable, then it can be made stable by increasing

self-diffusion coefficient to sufficiently large values.

In the numerical simulation, we have observed the effect of the ratio of the predator’s

immunity from or tolerance of the prey to the half-saturation constant in the absence of any

inhibitory effect, non-dimensional half- saturation constant and non-dimensional form of fish

predation rate on zooplankton respectively and observed the following:

(i) For the increasing values of the ratio of the predator’s immunity from or tolerance of

the prey to the half-saturation constant, in the absence of any inhibitory effect and

non-dimensional half- saturation constant, we observed the spot and spot-strip

mixture in the whole domain.

(ii) For the increasing values of fish predation, we observed the strip, spot-strip mixture

and spot patterns in the whole domain for both phytoplankton and zooplankton

population .

(iii) As we increase the time steps from t =200 to t =1000, we observed the destruction

of regular spiral pattern to patchy irregular patterns which spread in the whole

domain. This phenomena also persists for the longer period. After changing the

Abstract

xii

initial condition into a constant–gradient zooplankton distribution we observed the

similar dynamics.

In Chapter 6, a new mathematical model has been proposed and analyzed to study the

interaction of phytoplankton- zooplankton-fish population in an aquatic environment with

Holloing’s type II, III and IV functional responses. It is assumed that the growth rate of

phytoplankton with a Monod type, depends upon the constant level of nutrient. Phytoplankton

fully depends on zooplankton for food and fish population grows logistically and partially

depends on phytoplankton for food and the fish population is harvested according to CPUE

(catch per unit effort) hypothesis. The existence of equilibrium points and their stability

analyses have been discussed with the help of stability theory of ordinary differential

equations. The positive equilibrium point is locally and globally asymptotically stable under a

fixed region of attraction when certain conditions are satisfied. We have observed that model

system has a chaotic solution for the chosen set of parameter values. For a control parameter,

measuring the ratio of the predator’s immunity from or tolerance of the prey to the half

saturation constant in the absence of any inhibitory effect, the system exhibits bifurcation

phenomena. The bifurcation and blow-up bifurcation diagrams exhibits the transition from

chaos to order through a sequence of period having bifurcation. The blow-up bifurcation

diagram shows that the model system possesses a rich variety of dynamical behaviour. The

chaotic behaviour of the system is not continuing further, as the unstable period-3 orbits

which originate at the time of saddle-node bifurcation do not allow it to move further.

We have discussed the bionomical equilibrium of the model and found the sustainable

yield (h) and maximum sustainable yield (hMSY). It has been shown that if h > hMSY, then the

over-exploitation of fish population takes place and if h < hMSY, then the fish population is

under exploitation. By constructing an appropriate Hamiltonian function and using the

Pontryagin’s Maximum Principal, the optimal harvesting policy has been discussed. We also

found an optimal equilibrium solution. The idea contained in the chapter provides a better

understanding of the relative role of different factors; e.g., different predation rate of

phytoplankton and zooplankton by fish population and intensity of interference among

individual of predator.

In Chapter 7, an SEIR epidemic model has been proposed for treatment of infectives

considering the development of acquired immunity in recovered individuals. This model

consists of four variables, namely, susceptible, exposed, infectious and recovered individuals.

Abstract

xiii

This work is, in fact, the generalization of the work (Zhang and Suo (2010)), by introducing

exposed class in the model and by taking three different types of treatment rates. Zhang and

Suo (2010) studied an SIR model by taking the treatment rate in the form of Holling type-II.

We have considered two different types of treatment rates for the infected individuals by

taking in the form of Holling type-III and IV. Stability analysis for disease free as well as

endemic equilibria is performed. In both the types of treatment functions, the basic

reproduction number 0 1 and R R are reported. It is observed that the existence of unique

endemic equilibrium depends on the basic reproductive number 0R as well as on treatment

rate. In both the cases, the model has been analysed using stability theory of ordinary

differential equations. Criteria for the disease free equilibrium point and the endemic

equilibrium point to be locally and globally asymptotically stable are obtained. It has been

shown that in the case of Holling type-III recovery the disease free equilibrium point is

locally asymptotically stable if the basic reproduction number 0 1R and unstable if

0 1.R

Similar results are obtained in the case of Holling type-IV recovery function. It has also been

found that under certain conditions, the endemic equilibrium can also be made locally as well

as globally asymptotically stable. The model of Zhang and Suo (2010) is further generalized

in the case of Holling type-II treatment function by taking into account the exposed class.

Numerical simulation has also been performed to investigate the dynamics of interacting

subpopulations. It has been observed that the number of infective as well as susceptible in all

three cases (Holling type-II, III and IV) depend upon the treatment function but do so in a

variable manner. In Holling type-II, the treatment slowly increases, then attains its peak and

finally settles down at its saturation value. This case is applicable when availability of

treatment is poor including newly emergent diseases. In such a case a large proportion of

population may get infected as in the case of highly infectious diseases such as HIV.

In Holling type-III, the treatment initially increases fast, then increases slowly, attains its

peak and is finally stabilized. For different initial values, the infected population shows

different patterns. This indicates that the dynamics of infected population depends on the

parameters as well as initial values. Here the quantum of infection (prevalence of disease as

well as the new infection) depends upon the maximum treatment capacity in the community.

The infectivity is high when the treatment is low and vice-versa. Our numerical experiment

also shows that in case of Holling type-III treatment rate function, the susceptible individuals

first decrease and attain its minimum value and then starts increasing to enter the steady state.

Abstract

xiv

In the case of Holling type-IV, the treatment increases slowly, attains its peak and then

decreases to zero. Here the infected population initially increases and then shows transient

oscillations and finally it is stabilized at a much lower equilibrium level. This pertains to the

situation when the disease has been brought under control by an effective treatment. In case

of Holling type-IV treatment rate function, the susceptible individuals first show transient

oscillations and then settles down at its equilibrium level.

Thus in this thesis multiple aspects of the dynamics of Biological species have been

studies by Mathematical modeling and Computational analysis.

Contents

List of Figures......................................................................................................... xix-xxiii

List of Abbreviations/Symbols................................................................................. xxiv

1. General Introduction 1-23

1.1 Introduction ................................................................................................. 1

1.2 Objective of the Thesis................................................................................ 3

1.3 Depletion of Resource Biomass and its Conservation................................. 4

1.3.1 Harvesting Model .......................................................................... 4

1.3.2 Maximum Sustainable Yield (MSY)................................................ 5

1.3.3 Output Feedback Control............................................................... 5

1.3.4 Optimal Harvesting Policy.............................................................. 6

1.3.5 Taxation........................................................................................... 6

1.4 Effects of Diffusion...................................................................................... 7

1.4.1. Pattern Formation and Turing Instability...................................... 8

1.5 Different Treatment rate of Infected Population........................................... 10

1.5.1 Basic Reproductive Number............................................................ 12

1.6 Basic Definition and Tools........................................................................... 13

1.6.1 Malthusian Growth.......................................................................... 13

1.6.2 Logistic Growth.............................................................................. 13

1.6.3 Basic Prey-Predator Model............................................................ 13

1.6.4 Ecological Tools: Holling Type Functional Response.................. 14

1.6.5 Stability for Ordinary Differential Equation.................................. 15

1.6.5.1 Linear Stability................................................................. 15

1.6.5.2 Non-linear Stability.......................................................... 16

1.6.6 Limit Cycle...................................................................................... 17

1.6.7 Hopf-bifurcation............................................................................. 17

1.6.8 Pontryagin's Maximum Principle................................................... 19

Contents

xvi

1.6.9 Bendixon-Dulac Thoreem.............................................................. 20

1.7 Mathematical Techniques Used in the Thesis............................................... 20

1.7.1 The Method of Characteristic Roots............................................... 20

1.7.1.1 Routh-Hurwitz's Criteria ................................................. 20

1.7.1.2 Descartes' Rule of Sign..................................................... 21

1.7.2 Liapunov’s Direct Method ............................................................. 22

1.7.2.1 Sylvester Criterion............................................................ 22

1.7.3 Numerical Simulations................................................................... 23

2. A Mathematical Model for Optimal Management and Utilization

of a Renewable Resource by Population 24-45

2.1 Introduction.................................................................................................. 24

2.2 Mathematical Model..................................................................................... 26

2.3 Stability Analysis.......................................................................................... 28

2.4 Bionomic Equilibrium.................................................................................. 32

2.5 The Maximum Sustainable Yield................................................................. 33

2.6 Output Feedback Control.............................................................................. 34

2.7 Optimal Harvesting Policy............................................................................ 35

2.8 Numerical Simulations................................................................................. 39

2.9 Conclusions................................................................................................... 44

3. Optimal Management of a Renewable Resource Utilized

by a Population with Taxation as a Control Variable 46-65

3.1 Introduction................................................................................................... 46



3.4 Bionomic Equilibrium.................................................................................. 55


3.6 Optimal Harvesting Policy ........................................................................... 56


Contents

xvii

3.8 Conclusions.................................................................................................. 64

4. Modeling the Dynamics of a Renewable Resource under Harvesting

with Taxation as a Control Variable 66-98

4.1 Introduction................................................................................................... 66




4.5 Bionomic Equilibrium ................................................................................. 79

4.6 Optimal Harvesting Policy .......................................................................... 80

4.7 Hopf-bifurcation........................................................................................... 83


4.9 Conclusions.................................................................................................. 97

5. A Predator-Prey Interaction Model with Self and Cross-Diffusion

in an Aquatic System 99-124

5.1 Introduction ................................................................................................... 99

5.2 Mathematical Model...................................................................................... 101

5.3 Stability Analysis of a Non-Spatial Model................................................... 103

5.3.1 Stability Analysis............................................................................ 106

5.4. Stability Analysis with Diffusion................................................................ 108

5.4.1 One Dimensional Case................................................................... 108

5.5 Two Dimensional Case................................................................................. 114


5.7 Conclusions................................................................................................... 123

6. Dynamics of Phytoplankton, Zooplankton and Fishery Resource Model 125-151

6.1 Introduction................................................................................................... 125



6.4 Numerical Simulations.................................................................................. 140

6.5 Bionomic Equilibrium................................................................................... 144

6.6 The Maximum Sustainable Yield.................................................................. 146

Contents

xviii

6.7 Optimal Harvesting Policy .......................................................................... 146

6.8 Conclusions................................................................................................... 150

7. Modeling and Analysis of an SEIR model with Different Types of

Non-linear Treatment Rates 152-181

7.1 Introduction................................................................................................... 152



7.4 Numerical Simulations................................................................................... 170

7.5 Conclusions.................................................................................................... 179

Conclusions............................................................................................................... 182-183

Specific Contributions............................................................................................... 184-186

Future scope of Work................................................................................................ 187-188

References................................................................................................................. 189-204

List of Publications and Presentations...................................................................... 205

Brief Biography of the Supervisor........................................................................... 206

Brief Biography of the Candidate............................................................................ 207

List of Figures

Fig. No. Caption Page No.

2.1 Plot of B and N verses t of the model system (2.3)-(2.4) for the values

of parmeters

1 2 1 2

1.6, 1.2, 100, 100, 100, 0.1,

0.01, 0.001, 0.01, 0.01, 0.1.

r s K L E p

q

........................................ 40

2.2 Global stability of the model system (2.3)-(2.4) converge to the point

*(17.4311, 117.0579)P for the values of parameters

1 2 1 2

1.6, 1.2, 100, 100, 100,

0.1, 0.01, 0.001, 0.0001, 0.01, 0.0001.

r s K L E

p q

................ 40

2.3 Plot of B and N with respect to time t for different values of 2 , and

other values of parameters are same as figure (2.2)......................................... 41

2.4 Plot of B and N with respect to time t for different values of 2 with

2 0.0001 and other values of parameters are same as in

figure (2.2)......................................................................................................... 42

2.5 Plot of B vs t for different values of E, for the values of parameters

1.6, 1.2, 100, 100, 100, 0.1, 0.01,r s K L E p q

1 2 1 20.001, 0.0001, 0.01, 0.0001 , with

0.001 and 5.c ............................................................................................ 43

2.6 Plot of N vs t for different values of E and others values are same as

figure (2.5).......................................................................................................... 43

3.1 Plot of B, N and E verses t of the model system (3.3)-(3.5) for the

values of parameters

0

1 2 1 2

1.6, 1.2, 100, 100, 0.5, 0.01, 0.1,

0.001, 0.0001, 0.01, 0.0001, 0.001, 0.1.

r s K L p q

c

............... 60

3.2 Global stability of the model system (3.3)-(3.5) for the set of

parameters same as (3.1), all trajectories converge to

* * *0.25, 100.2089, 149.3286.B N E ........................................................... 60

List of Figures

xx


3.3 Plot of B, N and E with respect to time t for different values of 2 ,

others values of parameter are same as figure (3.1)......................................... 61

3.4 Plot of B, N and E with respect to time t for different values of 2

others values of parameter are same as figure(3.1).......................................... 62

3.5 Plot of B with respect to time t for different values of , and others

values are

1.6, 1.2, 100, 100, 100, 0.5, 0.01,r s K L E p q

0 1 2 1 20.1, 0.001, 0.0001, 0.01, 0.0001 ,

with 0.001 and 5.c ................................................................................ 62

3.6 Plot of N with respect to time t for different values of , and others

values are same as figure 3.5........................................................................... 63

3.7 Plot of E with respect to time t for different values of , and others

values are same as figure 3.5............................................................................ 63

4.1 Time series of B, N and E of the model system (4.2)-(4.4) for the values

of parameters

0 1

2 1 2

1.6, 1.2, 100, 100, 25, 1, 1, 0.001,

0.0001, 0.01, 0.0001, 7, 0.1, 4, 1.

r s K L p q

c m n

.............. 85

4.2 Global stability of the model system (4.2)-(4.4) for the values of

parameters

0

1 2 1 2

1

1.6, 3, 100, 100, 0.5, 0.01, 0.1,

0.001, 0.0001, 0.01, 0.0001, 0.001,

0.1, 4, 1, 1.

r s K L p q

c

m n c

.......................... 86

4.3 B, N and E for different values of 1 and others values are same as

figure (4.2)......................................................................................................... 87-88


figure (4.2)......................................................................................................... 88-89


figure (4.2).......................................................................................................... 90-91

List of Figures

xxi



figure (4.2)......................................................................................................... 91-92

4.7 B, N and E for different values of and others values of parameters

are

0

1 2 1 2

1.6, 1.2, 100, 100, 25, 1, 1,

0.001, 0.0001, 0.01, 0.0001, 7,

4, 1, 0.1.

r s K L p q

c

m n

............................... 93-94

4.8 Figures for 0.4 and others values of parameters are

0 1 2 1

2

1.6, 0.00292969, 64, 0.5, 1, 1,

2.75, 1.16875, 0.03125, 0.00146484,

0.00146484, 0.1875, 0.5, 1.6667.

r s K L p q

c m n

............................. 94-95

4.9 Figures for 0.6 and others values are same as figure 4.8............................. 96-97

5.1 Typical Turing patterns of prey and predator populations is

plotted at fixed parameters

2.33, 0.25, 2.5, 1.3,f time t=10000 and

11 12 21 220.05, 0.01, 30,d d d d

at the different values of predator’s immunity

(a) 0.3 , (b) 0.32 and (c) 0.33 ........................................................ 117-118


plotted at fixed parameters

0.3, 2.33, 0.25, 1.3,f time t=10000 and

11 0.05,d 12 21 220.01, 30d d d

at the different values of half saturation constant of predator

(a) 2.4 , (b) 2.5 and (c) 2.6 ............................................................ 119


plotted at fixed parameters: 0.3, 2.33, 0.25, 2.5,

time t=10000 and 11 0.05,d 12 21 220.01, 10d d d

at the different values of fish predation rate

(a) 1.0f , (b) 1.2f and (c) 1.35f .......................................................... 120-121

List of Figures

xxii


5.4 Snapshots of prey and predator populations at fixed parameters

0.3, 2.33, 0.25, 0.0001, 2.5, f

11 12 21 220.05, 0.01, 1,d d d d

at different values of time

(a) t =200, (b) t = 400, and (c) t = 1000. ........................................................... 121-122

5.5 Snapshots of prey and predator populations at fixed parameters

11 12 21 22

0.3, 2.33, 0.25, 0.0001, 2.5,

0.05, 0.01, 1

f

d d d d

at different values of time

(a) t =200, (b) t = 400 and (c) t = 1000. ......................................................... 122-123

6.1 Interactions incorporated in the model system (6.1)-(6.3)................................. 127

6.2 Chaotic attractor of the model system (6.4)-(6.6) for the parameter

values

0 1

0 2

0.2, 0.001, 3.33, 0.25, 2.5, 1.9, 150,

4.1, 5.5, 3.9, 0.17.

s K

qE

............. 140

6.3 Time series for u, v, x species vs t for the values of the parameters

same as figure 6.2. ............................................................................................. 141

6.4 SIC in species u and v vs t for different initial values

(a) (2,2,15) and (2.01,2,15) (b) (2, 2.01,15) and (2, 0.01,15)............................. 141-142

6.5 Time series for u, v, x species showing the stable focus for the

parameter values

0 1

0 2

1.8, 0.25, 0.001, 4.93, 0.25, 2.5,

3.2, 150, 5.2, 5.5, 3.8.s K qE

......................... 142

6.6 Bifurcation diagram in the diagram in the ranges

[ 2,2], [0.1,3]u ............................................................................................. 143

6.7 Magnified bifurcation in the ranges [ 2,2], [0.34,0.36]u . .................. 143

6.8 Magnified bifurcation diagram in the ranges

[ 2,2], [0.33,0.35]u ............................................................................... 144

7.1 Schematic diagram of the SEIR model (7.1)..................................................... 158

List of Figures

xxiii


7.2 (a) Susceptible individuals, (b) Exposed individuals and (c) Infected

individuals approaching endemic equilibrium

*(252.0905,8.9009,2.5893)P of the model system (7.3) for the set

of parameters

0 1 2 37.0, 0.02, 0.2, =0.025, 0.03, 0.003,

=0.3, =0.04.

A

......................... 172

7.3 Global stability of P*=( 14.9409, 1.8118, 1.0105) of the model system

(7.3). Trajectories initiating at different initial values and enter the

infected steady state *P for the values of parameters

0 1 2 31.2, 0.05, 0.2, 0.003, 0.03, 0.03,

=0.3, =0.00027.

A

........................ 172

7.4 Effect of variation in α on the solution trajectories of susceptible and

infectious populations........................................................................................ 174

7.5 Behaviour of infectives without and with treatment for the model

system (7.3)......................................................................................................... 175

7.6 Effect of different treatment on infectives with Holling type-III and

Holling type-IV removal rate function.............................................................. 176

7.7 Effect of removal rate functions on Infected population (I) with Holling

type-III and Holling type-IV removal rates for different initial values.

The small inset windows in both figures represent trajectories for initial

period.................................................................................................................. 178

7.8 Effect of removal rate functions on Susceptible population (S) with

Holling type-III and Holling type-IV removal rates for different initial

values.................................................................................................................. 179

List of Abbreviations and Symbols

Symbol Abbreviations

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Adjoint Variable

Ri Basic Reproductive Number

q Catchability Coefficient

D12 , D21 Cross Diffusion Coefficient

DFE Disease Free Equilibria

H Hamiltonian Function

MSY Maximum Sustainable Yield

Net Economic Revenue

D11, D22 Self Diffusion Coefficient

SIC Sensitive to Initial Condition

SEIR Susceptible, Exposed, Infectious and

Recovered

Tax

, 1 The Region of Attraction

Chapter 1

General Introduction

1.1 Introduction

We live on a planet with finite resources. The developed technologies can rearrange

matter to better suit our convenience, and extract energy by reaching farther and farther into

the depths of the Earth, the limits to this process have become evident.

World population is currently growing fast and will continue to grow in the future. It is

confronted with a natural resource supply that is ultimately limited. Total use of natural

resources and energies will have to shrink in future centuries, even when several energy and

raw material deposits have not been fully exploited so far. This is a fundamental change in

economic history because up to now, the expanding world economy has relied on growing

resource input. Hence, a classical theme of economics has again become very topical: the

relationship between population growth, natural resource scarcity, and the sustainability of

development.

One of the greatest challenges facing humanity is environmental degradation, including

deforestation, desertification, pollution, and climate change - an issue of increasing concern

for international community. Environmental degradation increases the vulnerability of the

societies it affects and contributes to the scarcity of resources. A return to "ecological

thinking" is necessary if we have to survive.

Studying ecology is very important, because a simple change in the environment can

have a profound effect on all living things; the destruction of one species can mean the death

of many others. We cannot continue to harm our environment due to a poor understanding of

ecology. Humans continue to destroy wildlife habitats in order to build cities; we introduce

contaminants such as pesticides and industrial wastes into the environment; and we deplete

non-renewable natural resources. These behaviour, if unchanged, could someday render the

Earth uninhabitable. There is a world-wide movement to better understand ecology and take

necessary steps to reduce pollution and end other destructive human activities.

An ecosystem consists of the biological community that occurs in some locale, and the

physical and chemical factors that make up its non-living or a biotic environment. There are

many examples of ecosystems : a pond, a forest, an estuary, and a grassland. The boundaries

Ch.-1 Introduction

2

are not fixed in any objective way, although sometimes they seem obvious, as with the

shoreline of a small pond. Usually the boundaries of an ecosystem are chosen for practical

reasons having to do with the goals of the particular study.

Studies of individuals are concerned mostly about physiology, reproduction, develop-

ment or behaviour, and studies of populations usually focus on the habitat and resource needs

of individual species, their group behaviours, population growth, and what limits their

abundance or causes extinction. Studies of communities examine how populations of many

species interact with one another, such as predators and their prey, or competitors that share

common needs or resources. The effect of infectious disease on the above system has a very

vital role to play in the dynamics of the system.

The dynamics of growth of a population can be described if the functional behaviour of

the rate of growth is known. Of course, it is this functional behaviour which is usually

measured in the laboratory or in the field when the ecologist is interested in single-species

population growth.

It is well known that the resource carrying capacity of our planet is limited. Therefore

the growth and development in various sectors of the economy caused by a rapid pace of

industrialization, a rising population and an increasing energy requirement have stressed our

environment to such an extinct that if concrete steps are not taken soon to control this

menace, many undesirable would occur leading to disastrous consequences for mankind. It

was along these lines that Lotka (1925) and Volterra (1927) established their original works

on the expression of predator-prey and competing species relations in terms of simultaneous

non-linear differential equations, making the first breakthrough in modern mathematical

ecology. The model ecosystem of Lotka-Volterra has since been completed by a probabilistic

model; this discipline is still one of the mainstreams of mathematical ecology.

The usual assumption made is that the rate of growth is in some sense proportional to the

number of species present. The proportionality ''constant'' may be dependent or independent

of the number of the species present, and it may be dependent or independent of time. The

environmental crisis which society faces today can be controlled by:

i) restoring the productivity of the ecosystem and by conserving genetic resources,

ii) insisting and assessing the environmental impact of developmental projects to ensure

harmonization of environment with development,

Ch.-1 Introduction

3

iii) incorporating environmental safeguards and beneficial analysis for development of

industrial projects,

iv) affecting better land use, and

v) promoting environmental education awareness among people.

Keeping the above view, the present thesis deals with some mathematical models of

ecosystem, taking into account birth, death and interaction rates, habitat-size, species

migration (diffusion), species-competition and the surrounding environment, rate of

harvesting, treatment rate of infected population (Clark (1976, 1985, 1990), Janssen and

Heuberger (1995), Wolpert (1977), Ragozin and Brown (1985), Freedman and Shukla (1991),

Shukla et al. (1981, 1997, 2007), Kitabatake (1982), Chaudhuri and Saha Roy (1991, 1996),

Malchow (1994, 2000), Zhang and Suo (2000), Alsad (2001), Dubey et al. (2001, 2002a, b,

2003a, b), Kar (2004a, b), Malchow et al. (2004a, b), Kar and Misra (2006), Kar and Matsuda

(2006), Hu et al. (2008), Kar et al. (2009a, b, 2010), Elbasha et al. (2011)) .

1.2 Objectives of the Thesis

The objective of the thesis is to study the problems of survival of species subject to

ecological stability. Mainly the following four types of topics are discussed in this thesis

using mathematical models:

i) a dynamics of a resource biomass and population taking into account different

control variables to control harvesting of the resource biomass,

ii) dynamics of predator-prey model with self and cross diffusion in an aquatic system,

iii) a dynamics of fishery resource under harvesting with phytoplankton and zooplankton

in an aquatic environment, and

iv) analysis of an SEIR model with different treatment rates.

Some of the topics used in the present research work are presented below in order to

discuss the above mentioned problems in a proper prospective way.

Ch.-1 Introduction

4

1.3 Depletion of Resource Biomass and its Conservation

Since the beginning of life in the earth the natural renewable resources such as fish,

whale, deer, forest etc. constitute an increasingly important class of economic resources for

human welfare on the planet earth. The common characteristic of these resources are that they

are for direct human consumption and they can reproduce themselves with a specific spread

of renewal in a given specific environment.

Manmade projects such as the expansion of agricultural land for food production,

growing of mining and wood based industries, development of housing complexes, timber

trade, and cutting of trees for fuel and fodder, pollutants in the form of sulphur dioxide,

nitrogen oxide, carbon monoxide, hydrocarbons, fluorines, fly ash, pollulated water , etc. are

responsible for the ecological degradation.

Due to all the above facts the exploitation of resources gradually comes to the limit that

makes some species endangered. This will create an imbalance in the ecosystem. Thus

management strategies for exploited biological resources is required. Government and other

authorities manage resources because the biological, social and economic consequences of an

unregulated resources are undesirable. Their management objectives may be intended to

ensure the economic and social well being of future generation or to protect habitats and

species of conservation concern. Effective management requires clear objectives support by

the best scientific advice and appropriate management actions. In practice, the management of

resource is a decision with multiple objectives. Some of desirable objectives in the

management of resources are as follows:

i) the provision of good biomass, ii) the conservations of population, iii) the conservation of

genetic variability of the resource, iv) the provision of good economic returns, and v) the

provision of steady employment. The formation of good harvesting policies that take account

these objectives is a complex and difficult task even when the dynamics of resources are

known accurately. These objectives are fully quantified.

1.3.1 Harvesting Model

The basic harvesting model is

( ) ( )dx

F x h tdt

, (1.1)

Ch.-1 Introduction

5

where ( )x x t denotes the size of the resource population at time t, F(x) is given function

representing the natural growth rate of the population, and h(t) represents the rate of removal

or harvesting.

1.3.2 Maximum Sustainable Yield (MSY)

The MSY is a simple way to manage resources taking into consideration that over-

exploiting resources lead to a loss in productivity. When the rate of exploitation increases in a

resource we see that at first there is an almost proportionate increase in total catches. Then

this growth rate drops steadily and the curve finally shows a maximum level. Its success can

be explained by the simplicity of the model which did much to convince administrators of the

need to limit exploitation of resource.

The concept of MSY itself is based on a model of biological growth that assumes that at

any given population level less than a certain level (let K), a surplus production exists that can

be harvested in perpetuity without altering the stock level. If the surplus is not harvested, on

the other hand, this causes a corresponding increase in the stock level, which ultimately

approaches the environment carrying capacity (K), where surplus production is reduced to

zero. Since surplus production equals sustainable yield at each population level, it follows

that MSY is achieved at the population level where surplus production is greatest (i.e. at the

level where the growth rate of the population is maximized). The main problem of the MSY

is economical irrelevance. It is so since it takes into the consideration of the benefits of

resource exploitation, but completely disregards the cost operation of resource exploitation.

(Clark (1976)).

1.3.3 Output Feedback Control

Feedback management is an effective strategy for taking into account unpredictable

changes in the ecosystem, such as recruitment failure, as major cause of stock collapse.

Tanaka (1982) proposed a simple feedback management strategy in which catch quotas are

manipulated on the basis of the difference between the present and target stock levels.

Mazoudi et al. (2008) and Louartassi et al. (2012) also described this policy in a model.

Usually, a feedback policy is used to stabilize a state variable, while adaptive management

attempts to learn about the system dynamics. Both feedback and adaptive learning are

important for bio-resources whose dynamics are uncertain. Grafton and Kompas (2005)

proposed a six steps process: 1) properties and quantity management goals, 2) socio-

Ch.-1 Introduction

6

economic-ecological system appraisal, 3) choose and apply ecological socio-economic

criteria, 4) determine reserve size, location, number and duration, 5) stakeholder and peer

review, and 6) active learning, experimentation and evolution, for establishing and adaptively

managing reserves for fishery purpose.

1.3.4 Optimal Harvesting Policy

In order to prevent extinction of these resources (fish, whale, deer, and forest) regulatory

agencies have been set up to regulate and limit consumption. At the same time in order to

maintain a certain degree of economic growth, consumption must be allowed to be an

adequate level.

Before any regulations were introduced, the history of consumption of many such

resources (such as fish, whale, deer, etc.) were at a level high enough to make it impossible

for sustained growth to take place. When consumption regulations were introduced, the

problems that were faced centered around what policies should the regulatory agencies

impose in order to insure that an adequate supply of the resource is available at all times.

Similarly, from the producer's point of view, the problem remains that of determining

production policies in order to maximize production without violating the regulatory agency's

requirements.

To find the optimal harvesting policy for models with complex functional response in

prey-predator system and some other relevant results of different mathematical models, are

the targeted tasks in the area of this work which includes some basic necessities like resources

conservation, regulation of catch composition, regulation of the amount of catch, etc.

1.3.5 Taxation

The exploitation of biological resources which is a major problem now a days can be

regulated by imposition of taxes and allowing subsidies. Taxation, license fees, lease of

property rights, seasonal harvesting, reserve area etc. are usually considered as possible

governing instruments in resource regulation. Economists are particularly attracted to

taxation, partly because of economic flexibility and partly of many of the advantages of a

competitive economic system can be better maintained under taxation than other regulatory

mechanisms (Clark (1976)).

The calculation of optimal tax, which would require the management authority to know

the operating cost structure as we as the biological characteristics of the resource, is a difficult

Ch.-1 Introduction

7

task in terms of mathematical, social and economic respect. Unpredictable fluctuation of

resource also creates difficulty in calculating the optimal tax.

Clark (1985, 1990) described first a single-species fishery model using taxation as a

control instrument. Based on the work of Clark, many researchers (Ragozin and Brown

(1985), Chaudhuri (1986), Mesterton-Gibbons (1988), Gunguly and Chaudhuri (1995), Fan

and Wang (1998), Pradhan and Chudhuri (1999a, b), Zhang et al. (2000), Dubey et al.

(2002b, 2003a, b), Kar and Misra (2006), Peng (2008), Ji and Wu (2010), Kar et al. (2009a,

b, 2010), Misra and Dubey (2010), Chakraborty et al. (2010), Yunfeia et al. (2010),

Chakraborty et al. (2011a, b), Olivares and Arcos (2011), Omari and Omari (2011), Shukla et

al. (2011a), Huo et al. (2012), Xue et al. (2012), Hong and Weng (2013)) have discussed the

utilization of various renewable resource by optimal management policy.

1.4 Effects of Diffusion

Reaction–diffusion systems are mathematical models which explain how the concen-

tration of one or more substances distributed in space changes under the influence of two

processes: local chemical reactions in which the substances are transformed into each other,

and diffusion which causes the substances to spread out over a surface in space.

Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial

differential equations.

The concept of diffusion may be viewed naively as the tendency for a group of biological

population initially concentrated near a point in ecological niche space to spread out in time,

gradually occupying an ever large area around the initial point. Consider the inorganic world,

for diffusion when an inhomogeneity occurs in the concentration of matter, matter flows from

high to low concentration region.

The term self-diffusion implies the movement of individuals from a higher to lower

concentration region and it assumed to be always positive. Cross-diffusion express the

population fluxes of one species due to the presence of the others species. The value of cross-

diffusion coefficient may be positive, negative or zero. The term positive cross-diffusion

coefficient denotes the movement of the species in the direction of lower concentration of

another species and negative cross-diffusion coefficient denotes that one species tends to

diffuse in the direction of higher concentration of another species. Self-diffusion mechanisms

form is the most widely studied class of models for ecological pattern formation and in these

https://en.wikipedia.org/wiki/Chemical_reaction

https://en.wikipedia.org/wiki/Diffusion

https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation



Ch.-1 Introduction

8

cases, the system parameters are usually treated as independent of time. However, when

diffusion is combined with intra- and interspecies relations, instability of the ecological

system may arise. This can be realized from an application of the Turing (1952) effect,

familiar in the dynamic theory of morphogenesis.

Due to environmental factors and others related effects the tendency of any species living

in a given habitat is to migrate to better suitable regions for its survival and existence. In

general, the movement of the species arises due to certain factors such as overcrowding,

climate, predator-prey relationship, refuge and fugitive strategies and more importantly due to

resource limitations in the given habitat. In general, a diffusion process in an ecosystem tends

to give rise to a uniform density of population in the habitat. As a consequence, it may be

expected that diffusion, when it occurs, plays a general role of increasing stability in a system

of mixed populations and resources. However, there is an important exception, known as

diffusive instability. This exception might not be a rare event especially in a prey-predator

system.

A determination of the effects of adding diffusion to ecological models is of biological

interest in recent years. The first successful attempt to study the migration of species

mathematically is due to Skellam (1951) using the concept of random dispersal. The

diffusive instability is further studied in an ecological context by Levin (1974, 1977), Levin

and Segal (1976), Okubo (1978), Mimura (1978). Since then, several investigators studied the

effects of diffusion on local and global stability of an interacting species system by

considering Lotka-Volterra and others types of prey-predator and competition models

(Comins and Blat (1974), Murray (1975), Chow and Tam (1976), Mimura and Kawasaki

(1980), Cohen and Murray (1981), Hastings (1982), Freedman (1987), Cantrell and Cosner

(1989, 2003), Dubey and Das (2000), Dubey et al. (2001), Dubey et al. (2002a), Freedman

and Krisztin (1992), Freedman and Wu (1992)) .

1.4.1 Pattern Formation and Turing Instability

Patterns are formed through the instability of the homogeneous steady-state solution to

small spatial perturbations. If the homogeneous steady-state solution was stable, then small

perturbations from the steady-state would converge back to the steady-state. Alan Turing, in

(1952), showed how a Reaction-Diffusion system can exhibit such instabilities to form

patterns.

Turing analysis involves:

Ch.-1 Introduction

9

i) symmetric (e.g. spatially uniform) equilibria,

ii) bifurcations in various parameters, e.g. diffusivity, domain size, and

iii) activation/inhibition type reactions.

Turing concluded that the Reaction-Diffusion model may exhibit spatial patterns under

the following two conditions:

i) the equilibrium solution is linearly stable in the absence of diffusion, and

ii) the equilibrium solution is linearly unstable in the presence of diffusion.

Such an instability is called a Turing instability or diffusion-driven instability.

Turing proposed a mechanism for growth and development of patterns (morphogenesis)

in biological systems (embryonic development). According to Turing (1952):

- active genes stimulate production/activation of chemical agents (morphogenes),

- chemical reactions alone is too “symmetric” for pattern generation,

- but diffusion-driven instabilities create initial patterns and those can lead to further

development.

Turing pattern formation has received a great attention in non-linear science as well as

biology (Huang and Diekmann (2003), Yang et al. (2004)). Wolpert (1977) gives a very clear

and non-technical description of development of pattern formation in animals which

stimulates a number of illuminating experimental studies of corresponding aquatic

ecosystems. The mechanisms of spatial pattern formation were suggested as a possible cause

for the origin of planktonic patchiness in marine systems. Malchow (1994, 2000a, b),

Malchow et al. (2004a, b) observed Turing patches in plankton community due to the effect

of nutrients and planktivorous fish. Grieco et al. (2005) developed a hybrid numerical

approach to study the transport processes and applied it to the dispersion of zooplankton and

phytoplankton population dynamics. Liu et al. (2008) studied a spatial plankton system with

periodic forcing and additive noise. In recent years there has been considerable interest in

reaction-diffusion (RD) systems due to the fascinating patterns that occurs in ecological

systems and to investigate the stability behaviour of a system of interacting populations by

taking into account the effect of self as well as cross-diffusions (Raychaudhuri and Sinha

(1996), Dubey et al. (2009), Sherratt and Smith (2008)).

Ch.-1 Introduction

10

1.5 Different Treatment Rate of Infected Population in Epidemic Disease

The ecologists put forward an attractive hypothesis which viewed health as a dynamic

equilibrium between man and his environment, and disease a maladjustment of the human

organism to environment (Park (2002)). Human population has exhibited wide fluctuations

throughout time and their growth may still be responding to environmental changes in

modern times. It may be affected dramatically by advances in housing, agricultural practices,

health care, and so on. In retrospect, it is not surprising to see dramatic changes in the

carrying capacity of the earth over time, because these changes are driven by strong

environmental shifts and events that include the public health transformation experienced

over the last five decades via the widespread use of antibiotics and the implementation of

large-scale vaccination policies. Further, improvements in the economic state often led to

substantial declines in the birth rate. Hence, predicting how many individuals the earth can

support becomes a rather complex problem with no simple answers particularly when

different definitions of quality of life are considered.

Questions and challenges raised by complex demographic process may be addressed

practically and conceptually through the use of mathematical models. Mathematical models

describing the population dynamics of infectious disease have been playing an important role

in better understanding of epidemiological patterns and disease control for a long time.

Epidemiological models are now widely used as more epidemiologists realize the role that

modeling can play in basic understanding and policy development. The transmission

mechanism from an infected to susceptible is understood for nearly all existing infectious

disease, and the spread of disease through a chain of infections is known. However, the

transmission interactions in a populations are so complex that it is difficult to comprehend

the large-scale dynamics of the disease spread without the formal structure of a mathematical

model. The advantage of mathematical modeling is the economy, clarity and precision of

mathematical formulation.

The immune system may respond in either a non specific manner as in the natural or

innate immune system or in an antigen specific manner as in adaptive or acquired immune

system. The less specific component, innate immunity, provides the first line of defense

against infectious agent. Most disease conditions result from complex interaction between the

mechanism of pathogenicty of microbe and the multiple mechanism of resistance of host.

This interaction results in either removal of infection or progression of disease.

Ch.-1 Introduction

11

The cell mediated immune response is mediated by APCs (macrophages and dendritic

cells), B cells and two types of T cells, namely, Cytotoxic T cell (Tc) and the Helper T cell

(TH). B cells medicate the humoral immune response of the body by secretion of antibiotics.

The cytotoxic T cells act either by destroying the pathogens directly or killing the

intracellularly infected cells. The helper T cells which constitute the other class of T cells

regulates the immune response either by secretions of cytokines or by contact mediated

signaling with T cells, B cells or APCs.

Mathematical models have become important tools in analyzing the assumptions,

variables and parameters. Moreover, models provide conceptual results such as thresholds,

basic reproduction numbers, contact numbers, and replacement numbers. Mathematical

models and computer simulations are useful experimental tools for building and testing

theories, assessing quantitative conjectures, answering specific questions, determining

sensitivities to changes in parameter values, and estimating key parameters from data.

Understanding the transmission characteristics of infectious disease in communities, regions,

and countries can lead to better approaches to decreasing the transmission of these disease.

Mathematical models are used in comparing, planning, implementing, evaluating, and

optimizing various detection, prevention, therapy, and control programs. Epidemiology

modeling can contribute to the design and analysis of epidemiological survey, suggest crucial

data that should be collected, identify trends, make general forecasts, and estimate the

uncertainty in forecasts.

Epidemic is a derivation of two Greek words: epi (upon/among) and demos (people). It is

the 'unusual' occurrence in a community or region of a disease specific health related events

''clearly in excess'' of the ''expected occurrence''. Thus, any disease, which occurs in numbers

more than the expected occurrence, constitute an epidemic. It includes heart disease, or even

psychosomatic disorders.

Health affecting lifestyle like smoking, drug addiction and health related events like

accidents also fall into the category of epidemics. But during disasters we are more concerned

about the epidemics of communicable disease. Having learn that epidemic is the occurrence

of a particular disease in unexpected numbers. The characteristic features of epidemic disease

are that it appear quite suddenly, grow in intensity, and then disappear, leaving part of the

population untouched. Perhaps the first epidemic to be examined from a modeling point of

view was the Great Plague in London (1665-1666). In the 19th century recurrent invasions of

cholera killed millions in India.

Ch.-1 Introduction

12

In order to model such an epidemic we divide the population being studied into four

classes labeled S, E, I, and R, where S(t) denotes the number of individuals who are

susceptible to the disease, that is not infected at time t, E(t) denotes the number of individuals

who are in exposed class, I(t) denotes the number of infected individuals, assumed infectious

and able to spread the disease by contact with susceptible and R(t) denotes the number of

individuals who have been removed from the possibility of being infected again or of

spreading infection. Removal is carried out either through isolation from the rest of the

population or through immunization against infection or through recovery from the disease

with full immunity against reinfection or through death caused by disease.

1.5.1 Basic Reproductive Number

One of the most important concerns about any infectious disease is its ability to invade a

population. Many epidemiological models have a disease free equilibrium (DFE) at which the

population remains in the absence of disease. These models usually have a threshold

parameter, known as basic reproductive number:

0

infection contact time

contact time infectionR

. (1.2)

It is important to note that 0R is a non-dimensional number, not a rate. It is also defined

as the average number of secondary infections produced when one infected individual is

introduced into host population where everyone is susceptible. It is implicitly assumed that

the infected outsider is in the host population for the entire infectious period and mixes with

the host population in exactly the same way that a population native would mix. Classical

disease transmission models typically have at most one endemic equilibrium. If 0R <1, then

the DFE (Disease free equilibria) is locally asymptotically stable, and the disease cannot

invade the population, but if 0R >1, then the DFE is unstable and inversion is always

possible. Thus for many deterministic epidemiology models, an infection can get started in a

fully susceptible population if and only if 0R >1. So, the basic reproduction number 0R is

often consider as a threshold quantity that determines when an infection can invade and

persist in a new host population. In mathematical epidemiology, Kermack and MaKendric

(1927) was first to formulate a simple model. Over the years many researchers (Liu et

al.(1987), Greenhalgh (1992), Li et al. (1999), Zhang and Suo (2010), Li et al. (2001),

Korobeinikov (2004), Wang and Ruan (2004), Li et al. (2006), Li. et al. (2008), Hu et al.

Ch.-1 Introduction

13

(2008), Mukhopadhyay and Bhattacharyya (2008), Ghosh et al. (2006), Naresh et al. (2008,

2009a, b), Gerberry and Milner (2009), Acedo et al. (2010), Bai and Zhou (2012), Elbasha

et al. (2011), Liu et al. (2012)) followed that model and explore many interesting

phenomena's related to epidemiology.

1.6 Basic Definition and Tools

1.6.1 Malthusian Growth

Here the simplest case is considered, namely the case where the species grows (i.e. birth-

death) at a constant rate times the number present, with no limitations on its resources. The

equation of growth is then

x kx , k is constant. (1.3)

If 0x is the population at time 0t , then 0

ktx x e . Such a population growth may be

valid for a short time, but it clearly cannot go on forever.

1.6.2 Logistic Growth

Due to the drawbacks of Malthusian growth, it is assumed that ( ) 1x

g x rK

which

yields the model

1x

x rxK

, (1.4)

where r is the intrinsic growth rate and K is the carrying capacity that the environment can

support. This model simulates the following conditions:

for small x the population behaves as in the Malthusian growth model, but for large x the

members of the species compete with each other for the limited resource.

1.6.3 Basic Prey-Predator Model

This is concerned with the functional dependence of one species on another, where the

first species depends on the second for its food. Such a situation occurs when a predator lives

off its prey or a parasite lives off its host. The functional dependence in general depends on

many factors : the various species densities, the efficiency with which the predator can search

out and kill the prey, and the handling time.

Ch.-1 Introduction

14

Predation is a mode of life in which food is primarily obtained by killing and consuming

organisms. A predator is an organism that depends on prey for its food. A prey is an organism

that is or may be seized by a predator to be eaten.

The basic prey-predator model is

( ),

( ),

, , , 0 and , 0.

x x y

y y x

x y

(1.5)

where x , y are densities of prey and predator, respectively, is the growth rate of prey, ,

the depletion rate of prey due to predation, is the growth rate of predator due to prey and

be the death rate of predator .

1.6.4 Ecological tool: Holling Type Functional Response

A functional response in ecology is the intake rate of a consumer as a function of food

density. Non-linear functional response were on the basis of a general argument concerning

the allocation of a predator's time between two activity: 'prey searching' and 'prey handling'.

These functional responses are constructed in terms of behaviour of an individual predator,

they are routinely incorporated at the population level in models that include reproduction and

death. The term behavioural density is more appropriate than functional response because this

describes the hunting and attack behaviour of the consumer for the resource. It is associated

with the numerical response, which is the reproduction rate of a consumer as a function of

food density. Following C. S. Holling (1959), functional response are generally classified into

four types, Holling type-I, II, III and IV.

Let ( )p x is the response of a predator with respect to the particular prey and it is said to

be a Holling type functional response if the following conditions hold

) (0) 0,i p i.e. all pass through the origin, (1.6)

) ( ) 0ii p x , all are increasing . (1.7)

Type-I (Linear):

( ) ; >0.p x x (1.8)

Ch.-1 Introduction

15

This functional response in which the attack rate of the individual consumer increases

linearly with prey density but then suddenly reaches a constant value when consumer is

satisfied.

Type-II (Cyrtoid):

( ) ; , 0.1

xp x

x

(1.9)

Type-II functional response describes the attack rate increases at a decreasing rate with

prey density until it becomes constant at saturation. Cyrtoid behavioural response are typical

of predators that specialize on one or a few prey.

Type-III (Sigmoid):

2

2 2( ) ; , 0

xp x

x

. (1.10)

Sigmoid functional response in which the attack rate accelerates at first and then

decelerates towards satiation. This functional response are typical of generalists natural

enemies which readily switch from one food species to another and which concentrate their

feeding in areas where certain resource are most abundant.

Type-IV:

2( ) ; , , 0.

xp x

xx

(1.11)

If , then this type of functional response approaches to type-II. In this case, the

attack rate initially increases slowly, attains its peak and then decreases to zero. Such a

situation may arise due to limitation in availability of food for large number of species but

when suppliers of food are depleted, then in spite of high number of species the available

food becomes very low.

1.6.5 Stability for Ordinary Differential Equations

1.6.5.1 Linear Stability

Consider the following system of ordinary differential equation:

( )dx

F xdt

. (1.12)

Ch.-1 Introduction

16

Here1 2( , ,....... )T

nx x x x and 1 2( , ,....... )T

nF F F F . We assume that

1 ,n nF C D R R , which ensures the existence and uniqueness of solutions of the

initial value problem for (1.12).

The steady state solution for above system (1.12) is found by setting

0idx

dt , for 1,2,.....,i n ; which gives

1 2( , ,....., ) 0i nF x x x , 1,2,.....,i n ,

for 1 2( , ,...., )T

nx x x x in nR .

To determine the linear stability of steady state solutions, we shift the origin to x i.e.

using the transformation,

i i iX x x , 1,2,.....,i n .

With this change of variable and neglecting higher order terms in equation (1.12), we

get the following linearized system:

dX

AXdt

,

where 1 2( , ,...., )T

nX X X X and i

j x x

FA

x

is called Jacobian matrix of the

system (1.12).

The steady state x x is said to be linearly asymptotically stable if all the eigenvalues

of the Jacobian matrix are with negative real part. Thus the question of linear asymp-

totic stability of a steady state reduces to finding the sign of the roots of the charac-

teristic equation of the corresponding Jacobian matrix.

In this context the Routh Hurwitz criterion (Kot (2001)) is very helpful, which gives

necessary and sufficient condition for a polynomial to have all the roots with negative

real part.

1.6.5.2 Nonlinear stability

Consider the system of differential equations

( )x f x

, (1.13)

where : nf D R is locally Lipschitz map from domain nD R into nR .

Let x D is an equilibrium of (1.13), i.e., ( ) 0f x .

The equilibrium point x x of (1.13) is

Ch.-1 Introduction

17

(i) stable if, for each 0 , there exists 0 such that

(0) ( )x x x t x , 0t ,

(ii ) unstable, if it is not stable,

(iii ) asymptotically stable if it is stable and can be chosen such that

(0) ( )t

x x Lt x t x

.

(See Kot (2001) for details).

1.6.6 Limit Cycle

A closed trajectory of a dynamical system is called orbit of the system. The motion along

orbit is periodic. An orbit is said to be a limit cycle if every trajectory that starts at a point

close to the orbit converges towards the orbit as t . According to the general theory of

dynamical systems, any orbit that is not one of a family of concentric orbits must be either a

limit cycle or an originating cycle in the sense that all neighbouring trajectories diverge from

the orbit. An originating cycle is clearly a limit cycle for the time-reversed dynamical system.

Thus a limit cycle is a isolated periodic solution. Then neighbouring trajectories are

attracted or repelled from the limit cycle.

A limit cycle is

i) a stable limit cycle if for all x in some neighbourhood; the nearby trajectories are

attracted to the limit cycle,

ii) a unstable limit cycle if for all x in some neighbourhood; the nearby trajectories are

repelled from the limit cycle,

iii) a semistable limit cycle if it is attracting on one side and repelling on the other.

(See Clark (1976) and Stephen Lynch (2004) for details).

1.6.7 Hopf-bifurcation

If the behaviour of the dynamical system changes suddenly as a parameter is varied, then

it is said to have undergone a bifurcation and at a point of bifurcation, stability may be gained

or loss. Limit cycle are generated through bifurcation (perturbations).

Consider a two dimensional system:

Ch.-1 Introduction

18

( , , );

( , , )

dxP x y

dt

dyQ x y

dt

(1.14)

Let ( *, *) x y be a fixed point of the system. The eigenvalues , 1, 2i i of the

Jacobian matrix about this fixed point gives the local dynamical behavior of the system. If the

fixed point is stable the eigenvalues 1 2, must both lie in the left-plane Re 0 . To

destabilize the fixed point, we need one or both of the eigenvalues to cross into the right half-

plane as the parameter changes its values. The critical value c of the parameter for

which the eigen values are purely imaginary is the hopf bifurcation point. The stability

behavior of the system changes as the parameter crosses this critical value. Let the associated

Jacobian matrix is given by

* *( , )x y

P P

x yM

Q Q

x y

Let M has a pair of purely imaginary eigenvalues ( ) i at c . Consider the

following conditions to be satisfied:

(a) For some c , Re ( )

0c

d

d

, (1.15)

(b) 0x yP Q at * *( , )x y (1.16)

(c)

1( ) ( )

16 ( *, *)

16

0

P Q P Qxxx xxy xyy yyy

P P Q Q Q Q P Q P Qxy xx yy xy xx yy xx xx yy yyx y

a

(1.17)

then a periodic solution occurs for c

if ( ) 0a P Qx y or for

c if

( ) 0x ya P Q .The equilibrium point is stable for c

and unstable for

c if

Ch.-1 Introduction

19

( ) 0x yP Q .On the other hand, the equilibrium is stable forc

and unstable forc

if ( ) 0.x yP Q In both the cases the periodic solution is stable if the equilibrium point is

unstable while it is unstable if the equilibrium point is stable on side of c

for which the

periodic solution exists.

(See Lakshmanan and Rajasekar (2003) for details).

1.6.8 Pontryagin's Maximum Principle (Clark (1976))

Consider the differential equation

[ , , ]dx

f x t udt

, 0 t T , (1.18)

with initial condition

0(0)x x (1.19)

where ( , , )f x t u is a given continuously differentiable function of three real variables

, and x t u .

The maximum principle is formulated in terms of the following expression, called the

Hamiltonian:

, , ; , , , ,x t u g x t u f x t u . (1.20)

Here ( , , )g x t u is flow of accumulated ''dividends'' to the objective functional

0

( ) ( , , )

t

J u g x t u dt , (1.21)

and ( , , )f x t u is flow of ''investment'' in capital. To express the investment flow in value

terms, it must be multiplied by a variable ( )t , called adjoint variable, which is Shadow price

of the resource stock at time t (discounted back to time 0t ).

So the Hamiltonian ( , , , )H x t u , expresses the total rate of increase of total assets

(accumulated dividends + capital assets).

The Pontyagin’s Maximum Principle asserts that, an optimal control ( )u t must maximize

the rate of increase of total assets, and

Ch.-1 Introduction

20

d H

dt x

. (1.22)

1.6.9 Bendixon-Dulac Theorem (Clark (1976))

Consider the dynamical system

( , )

( , )

dxf x y

dt

dxg x y

dt

(1.23)

in which functions ( , )f x y and ( , )g x y are assumed to be smooth in a simply-connected

region D. Let ( , )x y be smooth function in D such that the expression

( ) ( )f g

x y

does not change sign in D. Then the system of equations (1.23) has no closed trajectories in

D.

1.7 Mathematical Techniques Used in the Thesis

In the deterministic analysis of evolution and stability of the system described above

many mathematical approaches have been adopted. We will adopt the following methods :

1.7.1 The Method of Characteristic Roots

The conclusion regarding asymptotic stability of the systems very much lie in the

eigenvalues of the variational matrix, a Jacobian matrix of first order derivatives of

interactions-functions. As this Jacobian is determined by Taylor expansion of the interaction-

function and neglecting nonlinear higher order terms, this method studies only the local

stability of the system in vicinity of its equilibrium state. Being a straight forward method,

based purely on the sign of real parts of eigenvalues , Routh-Hurwitz criterion is very useful

to study the local stability of wide range of systems in homogeneous environment (Rao and

Ahamed (1999)).

1.7.1.1 Routh-Hurwitz criteria

Let the constant 1 2, ,......, na a a be real numbers. The equation

Ch.-1 Introduction

21

1

1( ) 0n n

nL a a , (1.24)

has roots with negative real parts iff the values of determinant of the following matrices

1

1

1 1 2 3 3 2 1

3 2

5 4 3

1 01

( ), , ,.................,

aa

H a H H a a aa a

a a a

1

3 2 1

1 0 0

0

0

0 0 0

n

n

a

a a a

H

a

are all positive.

Here ( , )l m entry in the matrix jH is

2 , for 0 2 -

1 for 2

0 for 2 or 2

l ma l m n

l m

l m l n m

.

In particular, for quadratic and cubic polynomials these condition reduce to

i) 1 20, 0a a , and

ii) 1 3 1 2 30, 0, a a a a a

respectively.

(See Kot (2001) for details).

1.7.1.2 Descartes' Rule of sign

The characteristic polynomial of nth order can be taken in the form

1

1 0( ) n n

n nP a a a

, (1.25)

where the coefficient , 0,1,...........,ia i n are all real and 0na . Let N be the number

of sign changes in the sequence of coefficient 1 0, ,...........,n na a a , ignoring any which

are zero. Descartes's rule of signs says that there are at most N roots of (1.25), which are

real and positive, further, that there are N, N-2 or N-4,........real positive roots. By setting

and again applying the rule, information is obtained about the possible real

Ch.-1 Introduction

22

negative roots. Together these often give invaluable information on the sign of all the

roots, which from a stability point of view is usually all we require.

(See Murray ( 2002) for details).

1.7.2 Liapunov’s Direct Method

The physical validity of this method is contained in the fact that stability of the system

depend on the energy of the system which is a function of system variables. Liapunov’s direct

methods consists in finding out such energy functions termed as Liapunov functions which

need not be unique. The major role in this process is played by positive or negative definite

functions which can be obtained in general by trial of some particular functions of state

variables, and in some case with a planned procedure.

1.7.2.1 Sylvester Criterion

Let us consider an autonomous differential system of the form

( )x f x

, (1.26)

where ,n nf C R R and assuming that f is smooth enough to ensure the existence

and uniqueness of the solution of (1.26). Let (0) 0f and ( ) 0f x for 0x in some

neighbourhood of the origin so that (1.26) admits the so-called zero solution ( 0x )

and the origin is an isolated critical point of (1.26).

Let

, 1

( )nT

ij i ji jV x x Bx b x x

, (1.27)

be a quadratic form with the symmetric matrix ( )ijB b , that is, ij jib b .

The necessary and sufficient condition for ( )V x to be positive definite is that the

determinant of all the successive principal minors of the symmetric matrix ( )ijB b be

positive, that is,

11 0b , 11 12

21 22

0b b

b b ,...................,

11 12 1

21 22 2

1 2

...

...0

... ... ... ...

...

n

n

n n nn

b b b

b b b

b b b

.

Ch.-1 Introduction

23

Let S be a set :nS x R x and let 0,R and 0 0, , 0J t t .

Suppose 0 0( ) ( , , )x t x t t x is any solution of (1.26) with the initial value 0 0( )x t x such

that x for .t J Also, since (1.26) is autonomous, we can further suppose,

without any loss of generality, that 0 0t .

We shall state the following results (Ahmad and Rao (1999)) which are used in the thesis:

Theorem 1.1

If there exists a positive definite scalar function ( )V x such that ( ) 0V x

(the derivative

of (1.27) with respect to (1.26) is non-positive) on S , then the zero solution of (1.26) is

stable.

Theorem 1.2

If there exists a positive definite scalar function ( )V x such that ( )V x

is negative

positive) on S , then the zero solution of (1.26) is asymptotically stable.

Theorem 1.3

If there exists a scalar function ( )V x , (0) 0V , such that ( )V x

is positive definite on

S and if in every neighbourhood N of the origin, N S , there is a point 0x ,

where 0( ) 0V x , then the zero solution of (1.26) is unstable.

1.7.3 Numerical Simulations :

Numerical experiments is carried out with the help of Mathematica / Matlab (Pratap

(1999)).

__________________________________________________________________________________ *The content of this chapter has been published in Journal of Mathematics, ID-613706, Vol.2013, 9 pages, 2013.

Chapter 2

A Mathematical Model for Optimal Management and Utilization

of a Renewable Resource by Population*

2.1 Introduction

Renewable resources are very important source of food and materials which are essential

for the growth and survival of biological population. The continuous and unplanned use of

these resources may lead to the extinction of resources and thereby affecting the survival of

resource dependent species. There has been considerable interest in the modeling of rene-

wable resources such as fishery and forestry. Dynamic models for commercial fishing have

been studied extensively taking into account the economic and ecological factors (Clark

1976, 1990). Based on the works of Clark (1976, 1990), several investigations have been

conducted (Bhattacharya and Begam (1996), Chaudhuri and Jhonson (1990), Chaudhuri and

Saha Roy (1991, 1996), Kar and Chaudhuri (2004), Kar et al. (2006), Kar and Masuda

(2006), Dubet et al. (2002b, 2003a, b)). Leung and Wang (1976) proposed a simple economic

model and investigated the phenomena of non-explosive fishing capital investment and non-

extinctive fishery resources. Chaudhuri (1986) proposed a model for two competing fish

species each of which grows logistically. He examined the stability analysis and discussed the

bionomical equilibrium and optimal harvesting policy. It was also shown (Chaudhuri (1986))

there is no limit cycle in the positive quadrant. Ragozin and Brown (1985) proposed a model

in which prey has no commercial value and predator is selectively harvested. When the prey

and predator both are harvested, then an optimal policy for maximizing the present value and

an estimation to the true loss of resource value due to catastrophic fall in stock level has been

discussed in detail by Mesterton-Gibbons (1988). In another paper, Mesterton-Gibbons

(1996) proposed a Lotka-Volterra model of two independent populations and studied an

optimal harvesting policy. Fan and Wang (1998) generalized the classical model of Clark

(1976, 1990) by considering the time-dependent Logistic equation with periodic coefficients

and they showed that their model has a unique positive periodic solution, which is globally

asymptotically stable for positive solutions. They also investigated the optimal harvesting

policies for constant harvest and periodic harvest. The optimal harvesting policy of a stage

structure problem was studied by Zhang et al. (2000). They found conditions for the

Ch-2 Resource Management Model-I

25

coexistence and extinction of species. Song and Chen (2001) proposed two-species

competitive system and discussed the local and global stability analysis of the positive

equilibrium point of the system. They have also discussed the optimal harvesting policy for

the mature population. Dubey et al. (2002b) proposed a model where the fish population

partially depends on a resource and is harvested. They examined stability analysis and the

optimal harvesting policy with taxation as a control variable. Dubey et al. (2003a) discussed a

model of a fishery resource system in an aquatic environment which was divided into two

zones- free fishing zone and reserved zone. They discussed biological, bionomical equilibria

and optimal harvesting policy. Dubey et al. (2003b) also proposed and analyzed an inshore-

offshore fishery model where the fish population is being harvested in both areas. Then they

investigated the stability analysis and optimal harvesting policy by taking taxation as a

control instrument. Kar et al. (2009) considered two prey- one predator model where both the

preys grow logistically and harvested. Again Kar and Chattopadhay (2009) described a single

species model, which has two stages: i) a mature stage, and ii) an immature stage; and they

discussed the existence of equilibrium points and their stability analysis. They proved that the

optimal harvesting policy is much superior to the MSY policy and optimal paths always take

less time than the suboptimal path to reach the optimal steady state. Kar et al. (2010)

proposed a prey-predator model with non-monotonic functional response and both the species

are harvested. To obtain the strategies for the management of the system they used harvesting

effort as a control variable. They found the stability condition of an interior equilibrium point

in terms of harvesting effort and proved that there exists a super critical Hopf bifurcation.

From the above literature and to the best of our knowledge, it appears that harvesting of a

renewable resource which is being utilized by a population for its own growth and

development has not been considered by taking into account the effect of crowding. Hence in

this chapter, we propose a mathematical model for biological population which is being

partially dependent on a renewable resource. This resource is further harvested for the

development of the society.

The organization of the chapter is as follows. Section 2.2 describes the development of

model and Section 2.3 gives a detail outline of the stability analysis of the system. The

bionomical equilibrium and the maximum sustainable yield are presented in Section 2.4 and

2.5 respectively. The output feedback control is given in Section 2.6 and the optimal

harvesting policy in Section 2.7. A numerical simulation experiment has been presented in

Section 2.8 and the last Section 2.9 is the concluding remarks.

Ch.-2 Resource Management Model-I

26

2.2 Mathematical Model

Let us consider a renewable resource growing logistically in a habitat. Then the

dynamics of this resource biomass is governed by

2

0 1 0 1dB B

a B a B a Bdt M

, (2.1)

where ( )B t is the density of resource biomass, 0a is its intrinsic growth rate, 1a is

interspecific interference coefficient and 0

1

aM

a is the carrying capacity.

Let ( )N t be the density of the population at any time 0t which utilizes the resource

biomass ( )B t for its own growth and development. Thus the intrinsic growth rate and carrying

capacity of the resource biomass will depend on the density of population. Hence we assume

that 0a and M in equation (2.1) are functions of N . Thus, equation (2.1) reduces to

2

00 ( )

( )

f BdBa N B

dt M N , (2.2)

where 0f is a positive constant.

We consider the following assumptions:

i) The intrinsic growth rate 0 ( )a N is a decreasing function of N and it satisfies

0(0) 0a r , 0( ) 0a N for 0.N

We take a particular from of 0 ( )a N as

0 1( ) .a N r N

ii) The carrying capacity ( )M N is also a decreasing function of N and it satisfies

0(0) 0,M m ( ) 0M N for 0.N

We take a particular form of ( )M N as

0

1

( ) .1

mM N

m N


27

Let us denote 0

0

rmK

f and 1 0

2

0

.m f

m

Then equation (2.2) can be rewritten as

2

1 21dB B

rB NB NBdt K

.

Now we consider a population of density ( )N t which is also growing logistically. We

assume that growth rate and carrying capacity of the population depend on the resource

biomass density. If the resource biomass density increases, the growth rate and carrying

capacity of population also increase. Thus, the dynamics of population is governed by the

following differential equation:

2

1 21dN N

sN NB NBdt L

,

where s is the intrinsic growth rate of population, L is its carrying capacity in the absence of

resource biomass, 1 and 2

are the growth rates of population in the presence of resource

biomass.

We assume that the resource biomass is harvested with harvesting rate ( )h t qEB ,

where q is a positive constant and in fishery resource it is known as catchability coefficient,

E is the harvesting effort, which is a control variable.

Keeping the above aspect in view, the dynamics of the system can be governed by the

following system of differential equations:

2

1 21dB B

rB NB NB qEBdt K

, (2.3)

2

1 21dN N

sN NB NBdt L

, (2.4)

(0) 0, (0) 0B N .

In the next section, we present the stability analysis of model (2.3)-(2.4).


28

2.3 Stability Analysis

First of all, we state the following lemma which is a region of attraction for the model

systems (2.3)-(2.4).

Lemma 2.1

The set

2 0( , ) : 0 , 0B N R B K N L

is a region of attraction for all solutions initiating in the interior of the positive quadrant,

where

2

0 1 2

LL s K K

s .

The above lemma shows all the solutions of model (2.3)-(2.4) are non-negative and

bounded, and hence the model is biologically well-behaved.

The proof of this lemma is similar to Freedman and So (1985), Shukla and Dubey

(1997), hence omitted.

Now we discuss the equilibrium analysis of the present model. It can easily be checked

that model (2.3)-(2.4) has four non-negative equilibria, namely,

* * *

0 1 2(0,0) , ( ,0) , (0, ) and ( , ).P P B P L P B N

The equilibrium points 0 2 and P P always exist. For the existence of equilibrium point 1P ,

we note that B is given by

( )

KB r qE

r . (2.5)

This shows that 1P exists iff

r qE . (2.6)


29

For the fourth equilibrium point *P , we note that * * and B N are the positive solutions of

the following algebraic equations:

1 21 0

Br N NB qE

K

, (2.7)

2

1 21 0N

s B BL

. (2.8)

Substituting the value of N , from equation (2.8) into equation (2.7), we get a cubic

equation in B as follows:

3 2

1 2 3 4 0a B a B a B a , (2.9)

where

1 2 2 2 2 1 1 2

3 2 1 1 4 1

0, = ( + ) > 0,

> 0, ( ) .

a KL a KL

a sr KL s KL a KLs qE r Ks

The above equation (2.9) has a unique positive solution *B B , if the following

inequality holds:

1( )L qE r . (2.10)

After knowing the value of *B , the value of *N can then be calculated from the relation

* * *

1 2

LN s B B

s .

From equation (2.10), we note that for the coexistence of resource biomass and

population together in a habitat, the intrinsic growth rate of the resource biomass must be

larger than a threshold value. This threshold value depends upon the carrying capacity of the

population and the harvesting effort.

Now we discuss the local and global stability behavior of these equilibrium points. For

local stability analysis, first we find variational matrices with respect to each equilibrium

point. Then by using eigenvalue method and the Routh-Hurwitz criteria, we can state the

following Theorems 2.1 to 2.4.


30

Theorem 2.1

(i) If the equilibrium point 1( ,0)P B exists, then 0P is always unstable in the B - N

plane.

(ii) If the equilibrium point 1( ,0)P B does not exist, then 0P is a saddle point with stable

manifold locally in the B - direction and with unstable manifold locally in the N -

direction.

Theorem 2.2

The equilibrium point 1( ,0)P B , whenever it exists, is a saddle point with stable manifold

locally in the B - direction and with unstable manifold locally in the N - direction.

Theorem 2.3

(i) If 1( )r L qE , then 2 (0, )P L is a saddle point with unstable manifold locally in the

B -direction and stable manifold locally in the N - direction.

(ii) If 1( ),r L qE then 2 (0, )P L is always locally stable in the B-N plane.

Theorem 2.4

The interior equilibrium *,P whenever it exists, is always locally asymptotically stable in

the B - N plane.

In the next theorem, we are able to find a sufficient condition for *P to be globally

asymptotically stable.

Theorem 2.5

Let the following inequality holds in :

2* *

1 1 2 2 2 2

4( )

s rK B N

L K

, (2.11)


31

Then the interior equilibrium *P is globally asymptotically stable with respect to all

solutions initiating in the interior of the region defined in lemma 1.

Proof :

We consider the following positive definite function about *P :

* * * *

* *ln ln

B NV B B B N N N

B N

. (2.12)

Differentiating V with respect to time t along the solutions of model (2.3)-(2.4), a little

algebraic manipulation yields

* 2 * * * 2

11 12 22

1 1( ) ( )( ) ( )

2 2

dVa B B a B B N N a N N

dt ,

where

* *

11 2 22 12 1 1 2 2

22 , , ( )

r sa N a a B B B

K L

.

Sufficient condition for dV

dt to be negative definite is that the following inequality holds:

2

12 11 22a a a . (2.13)

We note that (2.11) implies (2.13), thus V is a Liapunov’s function for all solutions

initiating in the interior of the positive quadrant whose domain contains the region of

attraction , proving the theorem.

Our next result shows that the model under consideration cannot have any closed

trajectories in the interior of the first quadrant.

Theorem 2.6

Systems (2.3)-(2.4) cannot have any limit cycle in the interior of the positive quadrant.

Proof: Let1

( , ) 0

H B NB N

, is a continuously differential function in the interior of the

positive quadrant of the B N plane.


32

Let

2

1 1 2( , ) 1B

F B N rB NB NB qEBK

,

and

2

2 1 2( , ) 1N

F B N sN NB NBL

.

Then,

1 2

2

( , ) ( ) ( )

1 + < 0.

B N HF HFB N

r sN

N K BL

This shows that ( , )B N does not change sign and is not identically zero in the positive

quadrant of the B N plane. By Bendixon-Dulac criteria, it follows that the system (2.3)-

(2.4) has no closed trajectory, and hence no periodic solution in the interior of the positive

quadrant of the B N plane.

2.4 Bionomical Equilibrium

In this section, we study the bionomical equilibrium of the model system (2.3)-(2.4).

Bionomical equilibrium is the level at which the total revenue (TR) obtained by selling the

harvested biomass in an economic equilibrium case, is equal to the total cost (TC) to the

harvested biomass, i.e. the economic rent is completely dissipated.

The net economic revenue at time t is given by

( , , ) ( )B E t pqB c E ,

where p is price per unit biomass and c is utilized cost per unit resource biomass. The

bionomical equilibrium is ( , , ),P B N E where , and B N E are the positive solutions of

0B N

. (2.14)

Solving (2.14), we get


33

cB

pq , (2.15)

2

1 2

2 2

c cLN s

s pq p q

, (2.16)

2

11c Nr c

Eq Kpq pq q

, (2.17)

It is clear that 0E , if

211

c Nr c

q Kpq pq q

. (2.18)

Thus the bionomical equilibrium ( , , ) P B N E exists under condition (2.18).

Remark 1. From equations (2.15) and (2.18), it may be noted that .c

B Kpq

2.5 The Maximum Sustainable Yield

The maximum sustainable yield (MSY) of any biological resource biomass is the

maximum rate at which it can be harvested and any larger harvest rate will lead to the

depletion of resource eventually to zero. In absence of any population, the value of MSY is

given by (Clark (1976))

0

4MSY

rKh .

If the resource biomass is subjected to the harvesting by a population, the sustainable

yield is given by

*

* * * * * *2

1 21B

h qEB rB N B N BK

.

We note that

*

0h

B

gives

** 1

*

2

( )

2( )

K r NB

r KN

and

2

*20

h

B

.


34

Thus, *

* * *

2MSY

rBh B N B

K

when

** 1

*

2

( ).

2( )

K r NB

r KN

From the above equations, it is interesting to note that, when * 0,N then *

2

KB

and

0

4MSY MSY

rKh h .

This result matches to Clark (1976).

If MSYh h , then it denotes the overexploitation of the resource and if MSYh h , then the

resource biomass is under exploitation.

2.6 Output Feedback Control

The habitat under our consideration consists of a resource biomass which is utilized by a

population. The resource biomass is being harvested and harvesting effort u E is consi-

dered as an input. The harvesting effort is applied to the stock and produces an yield Y qB

per unit effort. We assume that the total yield Y per unit effort is subject to the constraint

min maxY Y Y . Then our objective is to construct an output feedback control

0( ) ( ( ))E Y E u Y t in such a way that the steady state* * *( , )P B N is globally asymptotically

stable for the closed-loop system. Then model equations (2.3)-(2.4) can be written as in the

vector matrix differential equation form:

( ) ( )X f X ug X

, (2.19)

where

( )u E t ,

2

1 2

2

1 2

1

( )

1

BrB NB NB

Kf X

NsN NB NB

L

and ( )0

qBg X

.

Now under an analysis similar to Mazoudi et al. [2008], and Louartassi et al. [2012], one

can prove the following theorem:


35

Theorem 2.7

For any constant fishing effort 0E , there exists an 0w such that *P is globally

asymptotically stable through the output feedback control law *( )u w Y Y , where

* *.Y qB

2.7 Optimal Harvesting Policy

In this section we discuss the optimal management of a renewable resource in the

presence of population which is to be adopted by the regulatory agencies to protect the

resource and to ensure the survival of population with a sustainable development. The present

value J of a continuous time-stream of revenues is given by

0

e ( ( ) ) ( ) tJ pqB t c E t dt

, (2.20)

where is the instantaneous rate of annual discount. Thus our objective is to

max ,J

subject to the state equations (2.3)-(2.4) and to the control constraints

max0 E E . (2.21)

For this purpose, we use Pontryagins’s Maximum Principle. The associated Hamiltonian

function is given by

2

1 1 2

2

2 1 2

e ( ) ( ) 1

+ ( ) 1 ,

t BH pqB c E t rB NB NB qEB

K

Nt sN NB NB

L

(2.22)

where 1 2 and are adjoint variables and 1( ) e ( )tt pqB c qB is called switching

function.

The optimal control ( )E t which maximizes H must satisfy the following conditions:


36

max 1, when ( ) 0 i.e. t c

E E t e pqB

, (2.23)

10 , when ( ) 0 i.e. e t cE t p

qB

. (2.24)

The usual shadow price is 1 e t and the net economic revenue on a unit harvest is

cp

qB . This shows that if the shadow price is less than the net economic revenue on a unit

harvest, then maxE E and if the shadow price is greater than the net economic revenue on a

unit harvest, then 0.E When the shadow price equals the net economic revenue on a unit

harvest, i.e. ( ) 0,t then the Hamiltonian H becomes independent of the control variable

( )E t , i.e. 0H

E

.

This is a necessary condition for the singular control *( )E t to be optimal over control set

*

max0 E E .

Thus, the optimal harvesting policy is

max

*

, ( ) 0

( ) 0 , ( ) 0

, ( ) 0

E t

E t t

E t

(2.25)

When ( ) 0,t we have

1 e .t c

pqB

(2.26)

Now in order to find the path of singular control, we utilize the Pontrygin’s Maximum

Principle. According to this principle, the ajoint variables 1 2 and must satisfy

1 2 and d dH H

dt B dt N

. (2.27)

The above equations can be rewritten as


37

11 1 2 2 1 2

2e 1 2 2td B

pqE r N NB qE N NBdt K

,

2 22

1 1 2 2 1 2

2( ) 1

d NB B s B B

dt L

.

Using equations (2.7)-(2.8), the above equations can be re-written as

11 2 2 1 22td B

e pqE r NB N NBdt K

, (2.28)

22

1 2 2

td c Ne p B B s

dt qB L

. (2.29)

Equation (2.29) can again be written as

21 2 1

tdA B e

dt

, (2.30)

where

1

2

1 1 2

,

.

sNA

L

cB p B B

qB

Solving equation (2.30), we get

112 0

1

A ttBe K e

A

. (2.31)

We note that when t , then the shadow price 2

te is bounded iff 0 0.K

Thus, we have

12

1

tBe

A

. (2.32)

Now from equation (2.28), we get


38

12 1 2

tdA B e

dt

, (2.33)

where

2 2

12 1 2

1

,

( 2 ).

rBA NB

K

BB pqE N NB

A

Eq. (2.33) yields

221 1

2

A ttBe K e

A

.

Again for 1 1the shadow price e to be bounded, we must have 0.t K

Thus,

21

2

tBe

A

. (2.34)

Equations (2.26) and (2.33) yield

2

2

B cp

A qB

. (2.35)

Substituting the values of 1 1 2 2, , and A B A B into equation (2.34), we get

2

1 2 1 22

( )( 2 )1 B B N NBc rBE p NB

sNpq qB K

L

. (2.36)

Hence solving the equations (2.7)-(2.8) with the help of equation (2.36), we get an

optimal solution ( , )B N and the optimal harvesting effort E E .


39

2.8 Numerical Simulations

In order to investigate the dynamics of the model system (2.3)-(2.4) with the help of

computer simulations, we choose the following set of values of parameters (other set of

parameters may also exist):

1 2 1 2

1.6, 1.2, 100, 100, 100, 0.1, 0.01

0.001, 0.01, 0.01, 0.1,

r s K L E p q

(2.37)

with initial conditions: (0) 5 and (0) 8.B N

For this set of parameters, condition (2.10) for the existence of the interior equilibrium is

satisfied. This shows that * * *( , )P B N exists and it is given by

* *0.4790 and 102.3108B N .

Thus the interior equilibrium point * * *( , )P B N is locally asymptotically stable. It is

noted that condition (2.11) in Theorem 2.5 is not satisfied for the set of parameters chosen in

(2.37). Since condition (2.11) is just a sufficient condition for *P to be globally

asymptotically stable, no conclusion can be drawn at this stage. The behavior of B and N

with respect to time t is plotted in Fig. (2.1) for the set of values of parameter chosen in

(2.37). From this figure we see that the density of population increases whereas the density of

resource biomass decreases with respect to time, and both settle down at its equilibrium level.

Now we choose another set of values of parameters as follows :

1 2 1 2

1.6, 1.2, 100, 100, 100, 0.1, 0.01,

0.001, 0.0001, 0.01, 0.0001,

r s K L E p q

(2.38)

with different initial values.

For the set of values of parameters given in (2.38), it may be noted that the positive

equilibrium * * *( , )P B N exists and it is given by

* *17.4311 and 117.0579B N .


40

Fig.2.1: Plot of B and N verses t for the values of parmeters given in eq. (2.37)

Fig.2.2: Global stability of * * *( , ).P B N

It may be cheeked that in this case (for the values of parameters given in (2.38)),

condition (2.11) in Theorem 2.5 is satisfied. This shows that * * *( , )P B N is locally as well as

globally asymptotically stable in the interior of the first quadrant. In figure (2.2), we have

plotted the behavior of B and N with different initial values. Figure (2.2) shows that all the

trajectories starting from different initial points converge to the point *(17.4311, 117.0579)P .

This shows that *(17.4311, 117.0579)P is globally asymptotically stable.


41

It may be noted here that 2 and 2 are important parameters governig the dynamics of

the system. Therefore we have plotted the behavior of B and N with respect to time for

different values of 2 in Fig. (2.3) and for different values of 2 in Fig. (2.4).

Fig.2.3 : Plot of B and N with respect to time t for different values of 2 , and other

values of parameters are same as given in eq. (2.38).

From Fig. (2.3a), we note that B decreases as 2 increases. If 2 is very small, then B

initially increases slightly and then decreases and converges to its equilibrium level. If 2

increases beyond a threshold value, then B always decreases and settles down at its

equilibrium level. Fig. (2.3b) shows that N also decreases as 2 decreases. This is due to the

fact that with increase in 2 , the equilibrium level of B decreases and since the population N

is dependent on the resource biomass B, thus N also decreases. However, it is interesting to

note here that N always increases with respect to time t and finally attains its equilibrium

level.


42

From Fig. (2.4a), it is noted that B decreases as 2 increases. Fig. (2.4b) shows that N

increases as 2 increases. It may also be noted that for all positive values of 2 , N increases

continuously with time and finally settles down at its steady state. However, the resource

biomass B initially increases for some time, then decreases continuously and finally gets

stabilized at its lower equilibrium level. This shows that if the population utilizes the resource

without any control, then the resource biomass decreases continuously and it may doomed to

extinction.

Fig.2.4: Plot of B and N with respect to time for different values of 2 with 2 0.0001

and other values of parameters are same as in Fig.(2.2).

In order to see the qualitative behavior of optimal harvesting resource, we solve eqs.

(2.7), (2.8) and (2.36) for the same set of values of parameters as given in (2.38) with the

additional values as 0.001 and 5.c Then we get the optimal equilibrium levels as given

below:

1.40895, 101.191 and 146.201.B N E


43

Fig.2.5: Plot of B vs t for different values of E .

Fig.2.6: Plot of N vs t for different values of E

We observe that E is a very important parameter which governs the dynamics of the

system. The behavior of B with respect to time is shown in Fig. (2.5) and behavior of N with

respect to time is shown in Fig. (2.6) for different values of E. Fig. (2.5) shows that if E is

less than its optimal level ( 146.201E ), then B increases and settles down at its equilibrium

level. If E E , then B decreases and settles down at a lower equilibrium level. If E is large


44

enough, then B tends to zero. Fig. 2.6 shows that if E E , and then N increases with time

and finally obtains its equilibrium level. If E E , then N again increases with time, but it

settles down at a lower equilibrium level. This is due to the fact that when E E , then B

decreases and hence N also decreases. This suggests that the harvesting effort E should be

always kept less than E (optimal harvesting level) so that both the resource and the

population can be maintained at an optimal level.

2.9 Conclusions

In this chapter, a mathematical model has been proposed and analyzed to study the effect

of harvesting of a renewable resource. It has been assumed that the resource is being utilized

by a population for its own growth and survival. The resource and the population both are

growing logistically. The existences of equilibria and stability analysis have been discussed

with the help of stability theory of ordinary differential equations. It has been shown that the

positive equilibrium *P (whenever it exists) is always locally asymptotically stable. But *P is

not always globally asymptotically stable. However, we have found a sufficient condition

under which *P is globally asymptotically stable. This condition gives a threshold value of

the specific growth rate (r) of resource biomass. If r is larger than this threshold value, then

*P is globally asymptotically stable. Using Bendixon-Dulac criteria, it has been observed

that the model system has no limit cycle in the interior of the positive quadrant.

We have also discussed the bionomical equilibrium of the model and it has been

observed that the bionomical equilibrium of the resource biomass does not depend upon the

growth rate and carrying capacity of population. An analysis for sustainable yield (h) and

maximum sustainable yield ( MSYh ) has been carried out. It has been shown that if MSYh h ,

then the resource biomass will tend to zero and if MSYh h , then the resource biomass and

population may be maintained at desired level. The global asymptotic stability behavior of

the positive equilibrium has also been studies through output feedback control method.

We have constructed a Hamiltonian function and then using Pontryagin’s Maximum

Principle, the optimal harvesting policy has been discussed. An optimal equilibrium solution

has been obtained. The results are validated based on the numerical simulation. Here we

observed that 2 2, and E are important parameters governing the dynamics of the system. It


45

has been found that if 2 increases, then the resource biomass and the population both

decrease. However, if 2 increases, then the population density increases and the resource

biomass density decreases. A threshold value for the optimal harvesting effort E has been

found theoretically as well as numerically. It has been shown that the harvesting effort E

should be always kept less than E to maintain the resource and the population at an optimal

equilibrium level.

_________________________________________________________________________________ *The content of this chapter has been published in Non-linear Analysis: Modeling and Control, 18(1), 37-52, 2013.

Chapter 3

Optimal Management of a Renewable Resource Utilized by a Population

with Taxation as a Control Variable*

3.1 Introduction

With the rapid growth of industrialization and population, the exploitation of several

resources has increases significantly. Although the exploitation of resource is necessary for

the growth and development of any country, however, unplanned exploitation will eventually

lead to the extinction of these resources and consequently affecting the growth and survival

of species dependent on these resources. In the last few decades, many researchers have done

work on optimal management of renewable resources. The issues associated with these

resources have been discussed in detail by Clark (1976, 1990). Clark and De Pee (1979) have

discussed the implication of restricted malleability of capital for the optimal exploitation of a

renewable resource stock. Chaudhuri (1986, 1988) proposed combined harvesting of two

competiting fish species and analyzed the bio-economic and dynamic optimization.

Kitabatake (1982) discussed a model for fishing resource and proved that if the trawling

efficiency in the catch of prey species is improved, then the use of diesel-powered trawling

may lead to the extinction of predator as well as prey species. Dai and Tang (1998) proposed

a prey-predator system with constant rate of harvesting. They studied how to approximate the

region of asymptotic stability in biological terms in the initial states that ultimately lead to co-

existence of two-species. An optimal policy for combined harvest of two ecologically

independent species which grow logistically and are harvested at a rate proportional to both

stock and effort was discussed by Mesterton-Gibbons (1987). Ragozin and Brown (1985)

proposed a prey-predator model in which predator is selectively harvested and prey has no

commercial value. Pradhan and Chaudhuri (1998) analyzed the dynamics of a single species

fishery in which fish species follows the Gompertz law of growth. A resource based model in

three species fishery consisting of two predator and one prey with competition amongst

predators was discussed by Chottopadhyay et al. (1996). They also found the conditions for

persistence and global stability of the system. Dubey et. al. (2003a) proposed a model for

fishery resource system in an aquatic environment that consists of two zones: a free fishing

zone and a reserve zone. This model is further modified by Kar and Misra (2006). Kar et al.

(2009a) proposed a model to study the dynamics of two competing species which are

harvested in the presence of a predator. Kar et al. (2010) further studied the dynamics of a

Ch.-3 Resource Management Model-II

47

prey-predator model with non-monotonic functional response where both the species are

harvested with constant effort.

Regulation of renewable resources is a very important problem where an immediate

attention is required to be paid. Taxation, license fees, lease of property rights, seasonal

harvesting, fishing period control, creating reserve zones, etc. can be used as possible control

instruments. In fishery resource management, some investigations have been carried out with

taxation as a control instrument. Pradhan and Chaudhuri (1999a) proposed a mathematical

model to study the growth and exploitation of a schooling fish species by imposing a tax on

the catch to control the overexploitation of fish species. Dubey et. al. (2002b) discussed a

dynamical model for a single-species fishery, which depends partially on a logistically

growing resource with functional response and taxation as a control instrument to protect fish

population from over-exploitation. Dubey et. al. (2003a) further analyzed a non-linear

mathematical model to study the dynamics of an inshore-offshore fishery under variable

harvesting by considering taxation as a control instrument. They also proved that the fishery

resources can be protected from overexploitation by increasing the tax and discounted rate.

Pradhan and Chaudhuri (1999b) also proposed and analyzed a dynamical reaction model of

two species fishery with taxation as a control variable and then discussed its optimal

harvesting policy. Ganguly and Chaudhuri (1995) also discussed the bionomic exploitation of

single-species fishery using taxation as a control variable. Recently, Huo et al. (2012)

extended the result of Dubey et al. (2003a) by considering the taxation as a control

parameter.

In this chapter, we generalize the model developed in the previous Chapter-2. Thus we

propose a model of resource biomass and population, where both of them grow logistically

and population utilize resource for its own growth and development. The harvesting effort is

considered to be a dynamical variable and tax as a control variable. Then we find existence of

non-negative equilibria, condition of local as well as global stability analysis. Finally,

choosing an appropriate Hamiltonian function the optimal harvesting policy is discussed. The

main objective of this paper is to find an optimal taxation policy to give maximum profit to

the harvesting community and to sustain the resource biomass at a desied level.

3.2 Mathematical model

Consider a habitat when a renewable resource is growing logistically. Then the dynamics

of this resource biomass is governed by


48

2

0 1 0 1dB B

a B a B a Bdt M

, (3.1)

where ( )B t is the resource biomass density, 0a is its intrinsic growth rate and 0

1

aM

a is the

corresponding carrying capacity which the environment can support.

Now we assume that the resource biomass is being utilized by population of density

( )N t at any time 0t which may affect the intrinsic growth rate and carrying capacity of the

resource biomass, and 0a and M in equation (3.1) may be regarded as a function of N .

Thus, equation (3.1) can be written as

2

00 ( )

( )

f BdBa N B

dt M N , (3.2)

where is a positive constant.

Under the assumptions considered in Chapter-2 and as in Dubey and Patra (2013a), we

take

00 1

1

( ) , ( )=1

ma N r N M N

m N

.

By denoting 0

0

rmK

f and 1 0

2

0

m f

m , equation (3.2) can be rewritten as (Dubey and

Patra (2013a))

2

1 21 .dB B

rB NB NBdt K

Now we assume that the population of density ( )N t is growing logistically and its

growth rate as well as carrying capacity increases as the resource biomass density increases.

Then in a similar way, the dynamics of population may be governed by the following

differential equation

2

1 21dN N

sN NB NBdt L

.


49

In this equation, s is the intrinsic growth rate of population, L is its carrying capacity in

the absence of resource biomass, 1 and 2

are the growth rates of population in the

presence of resource biomass.

Next, we assume that the resource biomass is subjected to a harvesting rate ( )h t qEB ,

where q is a positive constant and in fishery resource it is known as catchability coefficient,

E is the harvesting effort. Let p be the fixed selling price per unit biomass and c the fixed

cost of harvesting per unit of effort. Then the economic revenue is

0( ) .R t pqEB cE

In order to conserve the resource biomass, the regulating agency imposes a tax 0 per

unit harvested resource biomass. Then the net economic revenue to the harvesting agency is

( ) ( ) .R t p qEB cE

Thus, the dynamics of the harvesting effort can be governed by the following differential

equation:

0 ( )dE

E p qB cdt

,

where 0 is the stiffness parameter measuring the strength of reaction of effort to the

perceived rent.

Keeping the above aspect in view, the dynamics of the system can be governed by the

following system of differential equations:

2

1 21dB B

rB NB NB qEBdt K

, (3.3)

2

1 21dN N

sN NB NBdt L

, (3.4)

0 ( ) )dE

E p qB cdt

, (3.5)

(0) 0, (0) 0, (0) 0.B N E

In the next section, we shall discuss the stability analysis of models (3.3)-(3.5).


50


First of all, we state the following lemma which represents a region of attraction of the

model system (3.3)-(3.5).

Lemma 3.1

The set

3 0

0

2( , , ) : 0 ,0 ,0 ( ) ( )

rKB N E R B K N L B t E t

is a region of attraction for all solutions initiating in the interior of positive orthant,

where 2

0 1 2 0 0 0( ) and min , ( )L

L s K K r p qK cs

, c

pqK

.

The above lemma shows that all solutions of model (3.3)-(3.4) are non-negative and

bounded, thus the model is biologically well-behaved.

This proof of this lemma is similar to Freedman and So (1985), Shukla and Dubey

(1997), so omitted.

To study the behavior of equilibrium points, we note that the model system (3.3)-(3.5)

has the following six equilibrium points:

* * * *

0 1 2 3 4(0,0,0), ( ,0,0), (0, ,0), ( , ,0), ( ,0, ) and ( , , ).P P K P L P B N P B E P B N E

The equilibria 0 1 2, and P P P

always exist. We show the existence of other equilibria as

follows.

Existence of 3( , ,0)P B N :

Here and are the positive solutions of the following algebraic equations:

1 21 0

Br N NB

K

, (3.6)

2

1 21 0N

s B BL

. (3.7)


51

From equation (3.7), we get

2

1 2( ).L

N s B Bs

Putting the value of N in equation (3.6), we get a cubic equation of B, i.e.

3 2

1 2 3 4 0,a B a B a B a

(3.8)

where

1 2 2 2 2 1 1 2

3 1 1 2 4 1

, ( ) ,

, ( ) .

a LK a LK

a rs LK s LK a L r sK

Equation (3.8) has a positive solution in B if

1r L . (3.9)

Knowing the value of B B , the value of N can then be calculated from eq. (3.7).

Existence of 4( ,0, )P B E :

Here and B E are the positive solutions of the algebraic equations

1 0B

r qEK

, (3.10)

0 ( ) 0p qB c . (3.11)

Solving the above equations, we get

( )

cB

p q

, and 1

( )

r cE

q Kq p

.

This shows that E exists if

0c

pKq

and p (3.12)

hold.


52

Existence of * * * *( , , )P B N E :

Here * * *, and B N E are the positive solution of following algebraic equations :

1 21 0

Br N NB qE

K

, (3.13)

2

1 21 0N

s B BL

, (3.14)

( ) 0p qB c . (3.15)

Solving the above equations, we get

*

( )

cB

q p

, (3.16)

* 2

1 2

LN s B B

s , (3.17)

and

** * * *

1 2

1 rBE r N N B

q K

. (3.18)

Thus, *E exists if the following hold

* *

1 2

*

*

) ,

( )) ,

( )

) .

i p

B N Kii r

K B

iii B K

(3.19)

From these conditions, we can conclude that, the non-zero equilibrium point exists if the

intrinsic growth rate of the resource biomass must be larger than a threshold value.


local stability analysis, first we find variational matrices at each equilibrium point. Then by

using Eigenvalue method & Routh-Hurwitz criteria, we get the following results.

i) The point 0P is a saddle point with unstable manifold in the BN -plane and stable

manifold in the E -direction.


53

ii) If 0c

pKq

, then the point 1P is always a saddle point with stable manifold in

the B -direction and unstable manifold in the NE -plane .

iii) a) If the point 3P exists, then the point

2P is a saddle point with unstable manifold in

the B -direction and stable manifold in the NE -plane.

b) If the point 3P does not exists, then

2P is always locally asymptotically stable.

iv) a) If 0c

pqB

, then 3P is a saddle point with stable manifold in the BN -plane

and unstable manifold in the E -direction.

b) If c

pqB

, then 3P is locally asymptotically stable.

v) The point 4P , whenever it exists, is a saddle point with stable manifold in the BE -

plane and unstable manifold in the N -direction.

To study the local stability behavior of the interior equilibrium *P , we note that the

characteristic equation of the variational matrix computed at *P is given by

3 2

1 2 3 0,a a a

(3.20)

where

2

* ** *

1 2

* ** * * * * * * 2 * *

2 2 1 2 1 2 0

2 * * *

03

,

2 ( ),

( ).

rB sNa N B

K L

rB sNa N B B B N N B q B E p

K L

sq B N Ea p

L

By Routh-Hurwitz criteria, it follows that all roots of equation (3.20) have negative real

parts iff

1 3 1 2 30, 0 and a a a a a . (3.21)


54

Clearly 1a is always positive,

3a >0 iff p . It is easy to cheek that 1 2 3a a a

holds true.

Thus we can now state the following theorems:

Theorem 3.1

The interior equilibrium *,P whenever exists, is locally asymptotically stable .

In the next theorem, we show that *P is globally stable.

Theorem 3.2

The interior equilibrium *,P whenever exists, is globally asymptotically stable.

Proof: Consider the positive definite function about *P :

* * * * * *

1 2* * *ln ln ln

B N EW B B B c N N N c E E E

B N E

. (3.22)

Differentiating W with respect to time t along the solutions of model (3.13)-(3.15) and by

choosing 2

0

1

( )c

p

, a little algebraic manipulation yields

* * 2 * 212

* * *

1 2 1 1 2 1

( ) ( )

( ) ( )( ).

c sdW rN B B N N

dt K L

c B B c B B B N N

Sufficient condition for dW

dtto be negative definite is that the following condition holds:

2

12 11 224a a a . (3.23)

If we choose 12 *

1 2 ( )c

K B

, then condition (3.23) is satisfied. Thus, W is a

Liapunov function with respect to all solutions initiating in the interior of the positive orthan,

proving the theorem.

In the next section, we discuss the bionomical equilibrium of the model system (3.3)-(3.5).


55


The bionomical equilibrium is said to be achieved when the total revenue (TR) obtained

by selling the harvested biomass is equal to the total cost (TC) for effort, i.e. the economic

rent is completely dissipated.

Then net economic revenue at time t is given by

( , , ) ( ) ,B E t pqB c E

The bionomical equilibrium is ( , , ) ,P B N E where , and B N E

are the positive

solutions of

0.B N E

(3.24)

Solving equations (3.24), we get

,c

Bpq

(3.25)

2

1 2

2 2

c cLN s

s pq p q

, (3.26)

211 .

c Nr cE

q Kpq pq q

(3.27)

It is clear that 0E , if

211 .

c Nr c

q Kpq pq q

(3.28)

Thus the bionomical equilibrium ( , , ) P B N E exists under condition (3.28).

If E E , then the total costs exceed the total revenues. In such a case, some users

will lose money and eventually some will drop out, thus reducing the level of harvesting

effort. If E E then the total revenues exceed the total costs. In such a case, it attracts

additional user and thus increasing the level of harvesting effort.

Remark 1. From (3.25) and (3.28), it may be noted that .c

B Kpq


56


The maximum rate of harvesting of any biological resource biomass is called maximum

sustainable yield (MSY) and any larger harvest rate will lead to the depletion of resource

eventually to zero. In absence of any population, the value of MSY is given by (Clark (1976))

0 .4

MSY

rKh


yield is given by

2

** * * * * *

1 21 .B

h qEB rB N B N BK

We note that

*

0h

B

yields

** 1

*

2

( ) and

2( )

K r NB

r KN

2

2

*0.

h

B

Thus,

* 2

1

*

2

( ),

4( )MSY

K r Nh

r N K

when

** 1

*

2

( ).

2( )

K r NB

r KN


2

KB and

0 .4

MSY MSY

rKh h

This result matches with the result of Clark (1976).

If MSYh h , then it denotes the overexploitation of the resource and if

MSYh h , then the

resource biomass is under exploitation.


A regulatory agency adopt the optimal harvesting policy to maximize the total

discounted net revenue using taxation as a control instrument on the resource biomass.


57

The present value J of a continuous time-stream of revenues is given by

0

( ( ) ) ( )tJ e pqB t c E t dt

, (3.29)

where is the instantaneous rate of annual discount.

Thus our objective is to

max J

subject to the state equations (3.13)-(3.15) and to the control constraint

min max . (3.30)

To find the optimal level of equilibrium, we use Pontryagins’s Maximum Principle. The

associated Hamiltonian function

2

1 1 2

2

2 1 2 3 0

( ) 1

1 ( ) ,

t BH e pqB c E rB NB NB qEB

K

NsN NB NB E p qB c

L

(3.31)

where , 1,2,3 i i are adjoint variables.

From equation (3.31), we note that H is linear in the control variable , hence the

optimal control will be a combination of bang-bang control and singular control.

For H to be maximum on the control set min max , we must have

30 . . 0.

Hi e

(3.32)

This gives a necessary condition for a singular control to be optimal.

Note from the maximum principle,

31 2, , .dd dH H H

dt B dt N dt E

(3.33)

The above equations can be written as


58

1

1 1 2 2 1 2

3 0

21 2 2

( ) ,

td Be pqE r N NB qE N NB

dt K

E p q

2 22

1 1 2 2 1 2

21 ,

d NB B s B B

dt L

31( ) ( ) 0.td

e pqB c qBdt

Using equations (3.13)-(3.15) &(3.32), these three previous equations can be re-written as

1

1 2 2 1 22td rBe pqE NB N NB

dt K

, (3.34)

221 1 2 2

d sNB B

dt L

, (3.35)

1

t ce p

qB

. (3.36)

Thus 1( ) ( ) t c

t t e pqB

is the usual shadow price along the singular path.

Putting the values of 1 in equation (3.35), we get

21 2 2

tdA A e

dt

, (3.37)

where

2

1 2 1 2, ( ).sN c

A A p B BL qB

The solution of equation (3.37) is 122 0

1

A ttAe K e

A

.

Now when t , then the shadow price 2

te is bounded if 0 0K .

Thus the solution is


59

22

1

tAe

A

. (3.38)

Substituting the value of 1 2 and in equation (3.34), a little algebraic manipulation yields

2

1 2 1 22

( )( 2 )

cp

qB B B N NBrBE NB

sNpq K

L

, (3.39)

c

pqB

. (3.40)

Hence solving equations (3.13)-(3.15) with the help of equations (3.39) and (3.40), we get an

optimal solution ( , , )B N E and the optimal tax .

3.7 Numerical Simulation

For the numerical simulation part of the model system (3.3)-(3.5), we choose the

following set of values of parameters (others sets of values of parameters may exist):

0

1 2 1 2

1.6, 1.2, 100, 100, 0.5, 0.01, 0.1,

0.001, 0.0001, 0.01, 0.0001, 0.001, 0.1.

r s K L p q

c

(3.41)

For the above set of values of parameters, we note that the positive equilibrium

* * * *( , , )P B N E exists and is given by

* * *0.25, 100.2089, 149.3286.B N E

It may also be noted that *P is locally as well as globally asymptotically stable.

Now we plot the dynamics of the system for the set of values of parameters given in

(3.41) with the help of MATLAB 6.1. The behavior of B, N and E with respect to time t is

plotted in Fig.(3.1). From this figure, we note that B and N increase for a very short time and

then they decrease and finally settle down at its steady state. However, E increases with time

and attains its equilibrium level.


60

Fig 3.1: Plot of B, N and E verses t for the values of parameter given in equation (3.41).

Fig. 3.2: Global stability of *P

Figure (3.2) shows the behavior of , and B N E with different initial values. From fig.

(3.2), we see that all trajectories starting with different initial points converge to

*(0.25,100.2089,149.3286)P . Thus *P is globally asymptotically stable.


61

Again we observe that 2 2 and

are important parameters in the model. We plot

, and B N E with respect to time t for different values of 2 2 and . Here we observe the

change of the behavior of , and B N E for different values of 2 2 and as shown in

Fig.(3.3) and Fig.(3.4), respectively. From these figures we note that if 2 increases, then B

and N initially decrease. But after a threshold value of2 , the behavior is changed. If

2

increases beyond this threshold value, then B and N decrease as 2 decreases. Again B

tends to zero level and N tends to its equilibrium level.

Fig (3.3): Plot of B, N and E with respect to time t for different values of 2 , others values of

parameter are same as given in equation (3.41).

For the set of parameters given in (3.41) with 5 , solving equations (3.39) and (3.40)

with the help of equations (3.13)-(3.15), we get optimal level of solution

0.28491, 100.2381, 149.2348 and 0.149.B N E

We observe that is also an important parameter which governs the dynamics of the

system. The behavior of , and B N E with respect to time t for different values of are


62

Fig.3.4: Plot of B, N and E with respect to time t for different values of 2 , others values of

parameter are same as given in equation (3.41).

Fig.3.5: Plot of B with respect to time t for different values of , others values of parameters

are same as given in Eq. (3.41).


63

Fig.3.6: Plot of N with respect to time t for different values of , others values of parameters


Fig.3.7: Plot of E with respect to time t for different values of , others values of parameters


shown in fig (3.5-3.7) respectively. From these figures, we note that the densities of the

resource biomass and population increases as increases, but the density of effort decreases


64

as increases. For an optimal level of the tax imposed on per unit of harvested biomass, the

resource biomass, the population and the effort settles down at their respective optimal level.

3.8 Conclusions

In this chapter, a mathematical model has been discussed where the resource biomass,

which has commercial importance, is harvested according to catch-per-unit-effort hypothesis.

The harvesting effort has been considered as a dynamical variable. The population utilizes the

resource for its own growth and development. The population and the resource both are

growing logistically. The existence of equilibrium points has been discussed and stability

analysis has been carried out by eigenvalue method, Routh-Hurwitz criteria and Liapunov

direct method. A threshold level of the intrinsic growth rate of resource biomass has been

found and it has been shown that if the intrinsic growth rate of the resource biomass is larger

than a threshold value and price of the per unit harvested biomass is larger than the tax

imposed on it, then the non-zero equilibrium point *P exists. And whenever *P exists, it is

always locally and globally asymptotically stable. When the population does not have any

direct effect on the resource biomass, and the resource biomass is being continuously

harvested, then equation (3.12) gives a range of tax. This range of tax may be very useful by

the regulatory agency at the time of formulating tax structure on per unit harvested resource

biomass. But when the resource biomass is utilized by a population and it is also harvested,

then the range of tax should be slightly modified as given in equation (3.19) keeping in view

some other thresholds on the intrinsic growth rate of resource biomass. It has been also found

that if the price of the harvested resource increases faster than cost of harvesting, then the

resource biomass density shifts to a lower equilibrium level. This shows that price of the per

unit harvested resource should not increase beyond a critical level, otherwise the survival of

resource biomass will be threatened. It has also been observed that q (harvesting coefficient

or catchability coefficient) increases with the advancement of technology due to which the

resource biomass may further shift to a lower equilibrium level. Equation (3.16) shows that

the equilibrium level of resource biomass may be increased by increasing tax to a certain

level on the harvested resource biomass. The bionomical equilibrium of the model has been

found and it has been shown that the bionomical equilibrium of the resource biomass does

not depend upon the growth rate and carrying capacity of population utilizing the resource

biomass. The maximum sustainable yield (MSY) of the model has been obtained and it has


65

been observed that if MSYh h , then the resource biomass will tend to zero and if

MSYh h ,

then the resource biomass and population can be maintained at appropriate level. In section

3.5, we have derived a formula for MSYh , which shows that maximum sustainable yield

depends upon the carrying capacity of the resource biomass and the equilibrium level of

population. The optimal harvesting policy has been discussed using Pontryagin’s Maximum

Principle. Constructing an appropriate Hamiltonian function the optimal tax policy has been

found. A computer simulation has been performed to illustrate all theoretical results. We

have used tax on the per unit harvested resource biomass as a regulatory instrument to derive

the optimal tax trajectory. It has been shown that if resource biomass, population and

harvesting effort all kept along this path, then resource biomass and population both can be

maintained at an appropriate level.

__________________________________________________________________________________*The content of this chapter has been submitted to The Scientific World Journal.

Chapter 4

Modeling the Dynamics of a Renewable Resource under Harvesting

with Taxation as a Control Variable*

4.1 Introduction

Conservation of renewable resources is a very important task for ecologists to maintain

the ecological balance. The first work on renewable resource using mathematical modelling

was done by Clark (1976, 1990). The capital for the optimal exploitation of a renewable

resource stock of restricted malleability has been discussed by Clark and De Pee (1976).

Many researchers have done work on the renewable resources. Chaudhuri (1986) described a

model of competiting logistically growing fish species which is being harvested according to

catch-per-unit-effort hypothesis. He described optimal harvesting policy using Pontryagin’s

Maximum Principle. Dubey et al. (2003a) investigated a model in an aquatic environment

that consists of two zones: i) free fishing zone, and ii) reserved zone by taking fishing effort a

control variable. They also proved that if fishery is exploited continuously in the unreserved

zone, fish population can be maintained at an appropriate equilibrium level in the habitat.

The model proposed by Dubey et al. (2003a) was extended by Kar and Misra (2006). A

ratio-dependent model with selective harvesting of prey species has been discussed by Kar

(2004a). Again Kar (2004b) proposed a model of prey-predator system with delay and

harvesting. He showed that both delay and harvesting effort play an important role on the

stability of the system. Dhar et al. (2008) proposed and discussed a phytoplankton-fishery

model, where fish depends upon plankton which grows logistically and the revenue is

generated from fishing. Then they converted this model with delay in digestion of plankton

by fish. They found a threshold of conversional parameter for Hopf-bifurcation. The

dynamics of two competing prey and one predator species was proposed and discussed by

Kar et al. (2009a). Here both the prey are harvested according to catch-per-unit effort

hypothesis. In the above model, tax on per-unit harvested biomass has been used to control

the over-exploitation of the resource biomass. Misra and Dubey (2010) analysed a prey-

predator model with discrete delay and the predator is harvested. Then they discussed

stability analysis of equilibrium points and Hopf bifurcation taking delay as a bifurcation

parameter. Ji and Wu (2010) proposed a prey-predator model with Holling type II functional

response incorporating a constant prey refuge and a constant rate prey harvesting. They also

Ch.-4 Resource Dependent Model-III

67

discussed instability, global stability, and existence and uniqueness of limit cycles of the

model. A ratio dependent eco-epidemiological system where prey population is harvested,

has been discussed by Chakraborty et al. (2010). They also studied a suitable condition for

non-existence of a periodic solution around the interior equilibrium. Yunfei et al. (2010)

described a phytoplankton-zooplankton model, in which both species are harvested for food.

They found stability conditions of equilibria and conditions for the existence of Hopf-

bifurcation. They also discussed the existence of bionomic equilibria and the optimal

harvesting policy. Sadhukhan et al. (2010) described a three competing species model: i)

prey, ii) predator and iii) super predator. These three species are harvested in this model.

They described the global stability, bionomic equilibrium and optimal harvesting policy. A

prey-predator harvested model with non-monotonic functional response has also been studied

by Kar et al. (2010). In this model, they introduced scaled harvesting efforts for both the

species. Chakraborty et al. (2011a) discussed a prey-predator fishery model with stage

structure for prey, where the adult prey and predator populations are harvested. They also

observed singularity induced bifurcation phenomena when variation of the economic interest

of harvesting was taken into account. Olivares and Arcos (2011) investigated a model of a

renewable resource in an aquatic environment composed of two different patches. They also

discussed the optimal harvesting policy using Pontryagin’s maximum principle. Shukla et al.

(2011a) proposed and analysed the depletion of a renewable resource by population and

industrialization with resource dependent migration. The resource biomass, which grows

logistically, is depleted by population and industrialization but it is conserved by

technological effort. The growth rate of technological effort depends on difference between

carrying capacity and current density of resource biomass. They proved that the resource

never become extinct by population and industrialization, if technological effort is applied

appropriately for its conservation. Dubey and Patra (2013a) discussed a resource and a

population model where both are growing logistically and resource is harvested according to

catch-per-unit effort hypothesis. Using optimal harvesting policy they proved that the

harvesting effort should be always kept less than optimal harvesting effort to maintain the

resource and the population at an optimal equilibrium level. In all of the previous cases

harvesting effort is taken to be a control variable.

In addition to the harvesting effort as a control variable, there are some other tools

which have been used as control variables. Taxation, lease of property rights, seasonal

harvesting, license fees, creating reserve zones, fishing period control are taken as a control


68

instrument. Among all of these taxation is assumed to be best because of its flexibility, and

many of the advantages of a competitive economic system can be better maintained by taking

taxation as a control instrument than other. A dynamical model of single-species fishery is

described by Dubey et al. (2002b) using taxation as a control instrument to protect the fish

population from over exploitation. The dynamics of inshore-offshore fishery under variable

harvesting has been discussed by Dubey et al. (2003b). They proved that by increasing tax

and discount rates, the overexploitation of fishery resources can be protected. The work of

Dubey et al. (2003a) was further extended by Huo et al. (2012) taking into account

harvesting effort as a dynamical variable and taxation is a control variable. They also

examined the optimal harvesting policy using Ponyryagin’s Maximum Principle. Dubey and

Patra (2013b) described model of resource biomass and population by taking into account the

crowding effect. The harvesting effort is dynamical variable and taxation is a control

variable.

Gunguly and Chaudhuri (1995) proposed and analysed a single species fishery model

with realistic catch rate function instead of usual catch-per-unit-effort hypothesis. The fishing

effort is dynamical variable in their model. They also discussed the stability analysis and

optimal harvesting policy. Pradhan and Chudhuri (1999a) investigated a mathematical model

for growth and exploitation of a schooling fish species by taking into account a realistic catch

rate function and taxation as a control instrument. Peng (2008) discussed a mathematical

model concerning continuous harvesting of a single species fishery. He assumed a reasonable

catch-rate function and a suitable tax per unit biomass of landed fish imposed by an external

energy. A prey-predator fishery model incorporating prey refuge, where prey is harvested, is

proposed and analysed by Chakraborty et al. (2011b). They also discussed Hopf-bifurcation

through considering density dependent mortality for the predator as bifurcation parameter.

Then they found optimal tax with the help of Pontrygin's Maximum Principle.

In most of the harvesting models, the harvesting rate function is proportional to catch-

per-unit effort hypothesis i.e. ( ) ( ) ( )h t qE t B t , where q is catchability coefficient, E is the

harvesting effort and B is the resource biomass. But in fishery models, this catch-rate-

function has some unrealistic features such as

i) random search for the resource,

ii) equal likelihood of being captured for every resource,

iii) unbounded linear increase of h with E for fixed B , and


69

iv) unbounded linear increase of h with for B fixed E .

In order to avoid the above circumstances, Ganguly and Chaudhuri (1995), Pradhan and

Chaudhuri (1999a), Peng (2008), and Chakraborty et al. (2011b) have taken the harvesting

rate

,qEB

hmE nB

where m and n are positive constant. This catch rate function is always saturated with

respect to effort level and stock abundance. The parameter m is proportional to the ratio of

the stock-level to the catch rate at higher level of effort and n is proportional to the ratio of

the effort level to the catch rate at higher stock levels.

In this catch rate function we observe the following:

a) qB

hm

as E for fixed value of B ,

b) qE

hn

as B for fixed value of E , and

c) h has singularity at 0B and 0E .

In order to remove the singularity of h, we modify the harvesting rate h in the following

form

( ) ( )

( ) .1 ( ) ( )

qE t B th t

mE t nB t

(4.1)

So in this chapter, we will formulate a dynamical model of resource biomass and

population, both growing logistically. The resource biomass, which is of commercial

importance, is harvested according to the harvesting rate function h(t) defined in equation

(4.1). The harvesting effort is taken as a dynamical variable and taxation as a control variable.

Then we analyse the existence of non-negative equilibria and their local and global stability.

We also discuss the maximum sustainable yield (MSY). Choosing an appropriate

Hamiltonian function and using Pontryagin’s maximum principle, we discuss optimal

harvesting policy.


70


Let us consider a resource biomass of density B(t) and a population of density N(t), both

growing logistically in absence of each other. Following Chapter-2, the dynamics of resource

biomass and population may be governed by the following system of ordinary differential

equations:

2

1 211

dB B qEBrB NB NB

dt K mE nB

,

2

1 21dN N

sN NB NBdt L

.

In the above model, r and s are the intrinsic growth rates of resource and population

respectively, K and L are their respective carrying capacities, 1 and 2 are the depletion rates

of resource biomass due to population, 1 2 and are growth rates of population due to the

presence of resource biomass. Now we assume that the resource biomass is being harvested

according to the modified harvesting rate function h(t) given in equation (4.1) and a

regulatory agency imposes a tax ( 0) per unit resource biomass to protect the over-

exploitation of the resource.

Thus the net economic revenue is

( )

( )1

p qBR t E c

mE nB

,

where p is the fixed selling price per unit biomass and c is the fixed cost of harvesting per

unit of effort.

Thus the harvesting effort E is now dynamical variable and can be governed by the

following differential equation

0

( )

1

dE p qBE c

dt mE nB

,

where 0 is the stiffness parameter measuring the intensity of reaction between the effort and

the prevent rent.


71

Therefore we consider the following dynamical system of equations:

2

1 211

dB B qEBrB NB NB

dt K mE nB

, (4.2)

2

1 21dN N

sN NB NBdt L

, (4.3)

0

( )

1

dE qB pE c

dt mE nB

, (4.4)

(0) 0, (0) 0, (0) 0.B N E

In the next section, we shall discuss the stability analysis of the model system (4.2)-(4.4).


For a region of attraction of the model system (4.2)-(4.4), we state the following lemma.

Lemma 4.1:

The

0{( , , ) : 0 ( ) , 0 N(t) , 0 ( ) ( ) }B N E B t K L B t E t

is a region of attraction of all solution initiating in the interior of the positive orthant,

where 2 00 1 2

2 ( )( ), , .

r p KLL s K K p

s

The above lemma shows that all solutions of this model (4.2)-(4.4) are non-negative and

bounded, so our model is biologically well-behaved.

The proof of this lemma is same as Freedman and So (1985), Shukla and Dubey (1997),

hence omitted.

To discuss the biological equilibrium of the system (4.2)-(4.4), we note that the model

system has six non-negative equilibrium points, viz,

* * * *

0 1 2 3 4ˆ ˆ(0,0,0), ( ,0,0), (0, ,0), ( , ,0), ( ,0, ), ( , , ).P P K P L P B N P B E P B N E

The equilibrium points 0 1 2, and P P P always exists. The existence of other equilibrium

points are given below:


72

Existence of 3( , ,0)P B N :

B and N are the positive solutions of the following two equations:

1 21 0,

Br N NB

K

(4.5)

2

1 21 0.N

s B BL

(4.6)

From equation (4.6), we have

2

1 2

LN s B B

s . (4.7)

Putting the value of N , in equation (4.5), we get a cubic equation in B , i.e.

3 2

1 2 3 4 0a B a B a B a , (4.8)

where

1 2 2 2 2 1 1 2

3 1 1 2 4 1

, ( ) ,

, ( ) .

a LK a LK

a rs LK s LK a L r sK

Equation (4.8) has a positive real root if 1r L .

Putting the value of B , we can calculate N from equation (4.7), and thus we can state

the following result.

Theorem 4.1

The equilibrium point 3( , ,0)P B N exists if 1r L .

The above theorem shows that for the co-existence of resource biomass and population,

the intrinsic growth rate of resource biomass should be larger than a threshold value which

depends on the carrying capacity L of population and the bilinear depletion rate coefficient

1 of the resource biomass.

Existence of the point 4ˆ ˆ( ,0, )P B E :

In this case, ˆ ˆ and B E are positive solutions of the following equations:


73

1 01

B qEr

K mE nB

, (4.9)

( )

01

qB pc

mE nB

. (4.10)

From eq. (4.10), we have

1

[ ]E q p nc B cmc

. (4.11)

From eq. (4.11), we note that

0E if ( 1)

0c nB

pqB

. (4.12)

Then putting the value of E from equation (4.11) in equation (4.9), we get an quadratic

equation in B, i.e.

2

1 2 3 0b B b B b , (4.13)

where

1

( )q pb

cK

, 2

( )( )

q p nc qb r p

m m c

, 3

qb

m .

Thus equation (4.13) has always a positive real root.

Thus we can state the following result.

Theorem 4.2

The equilibrium 4ˆ ˆ( ,0, )P B E exists if

( 1)0

c nBp

qB

.

Existence of * * * *( , , )P B N E :

Here* * *, and EB N are the positive solutions of the following equations:

1 21 0,

1

B qEr N NB

K mE nB

(4.14)


74

2

1 21 0N

s B BL

, (4.15)

( )

01

qB pc

mE nB

. (4.16)

From equations (4.15) and (4.16), we get

2

1 2

LN s B B

s , and

( ) 1q p B nE B

mc m m

. (4.17)

Putting these values in equation (4.14), we get a biquadrate equation in B, namely,

4 3 2

1 2 3 4 5 0c B c B c B c B c , (4.18)

where

2 21 2 2 1 1 2 3 1 1 2

4 1 5

, , ,

( ), .

( ) ( )

L L L rc c c L

s s s K

q p nc cc L r c

m p m p

Using Descartes' rule of sign, we note that equation (4.18) has always a positive real root

*B B .

Putting the value of B , we can get the values of N and E from equation (4.17) .

Now we can state the following result.

Theorem 4.3

The equilibrium * * * *( , , )P B N E exists if

( 1)0

c nBp

qB

. (4.19)

Now we analyse the local as well as global stability behaviour of these non-negative

equilibrium points. For local stability behaviour, at each equilibrium point, first we find the

variational matrices and then using Eigenvalue method and Routh-Hurwith criteria, we can

conclude the following results.


75

i) The point 0P is a saddle point with unstable manifold in the B-N-plane and stable

manifold in the E-direction.

ii) a) The point 1P is always a saddle point with stable manifold in the B-direction and

unstable manifold in the N-E-plane if (1 )c

p nKqK

,

b) If (1 )c

p nKqK

, then 1P is again a saddle point with stable manifold in the

B-E-plane and unstable manifold in the N-direction.

iii) a) The point 2P is locally asymptotically stable if 1r L ,

b) If 1r L , then 2P is a saddle point with unstable manifold in the B-direction

and stable manifold in the N-E-plane.

iv) a) (1 )c nB

pqB

, then the point 3P is locally asymptotically stable,

b) If (1 )c nB

pqB

, then 3P is a saddle point with stable manifold in the B-N-

plane and unstable manifold in the E-direction.

v) The point 4P is always a saddle point with unstable manifold in the N-direction and

stable manifold in the B-E-plane if

2

ˆ

ˆ ˆ(1 )

r qnE

K mE mB

.

Let be an eigenvalue of the variational matrix *M evaluated at the interior equilibrium

point * * * *( , , )P B N E . Then the characteristic equation is given by

3 2

1 1 1 0A B C , (4.20)

where,

* ** * * *

* * 01 2 * * 2 * * 2

( ),

(1 ) (1 )

qm p B ErB qnE B sNA N B

K mE nB L mE nB


76

* ** * * ** * 0

1 2 * * 2 * * 2

* * * * *

0

* * 2 * * 2

* *2 * * *

1 2 1 2

( )

(1 ) (1 )

( ) (1 )(1 )

(1 ) (1 )

( )( 2 ),

qm p B ErB qnE B sNB N B

K mE nB L mE nB

q p B E smN q nB mE

mE nB L mE nB

B B N N B

* ** * * ** * 0

1 2 * * 2 * * 2

* ** * *

0

* * 2 * * 2

* ** *2 * * * 0

1 2 1 2 *

( )

(1 ) (1 )

( ) (1 )(1 )

(1 ) (1 )

( ) ( )( 2 )

(1

qm p B ErB qnE B sNC N B

K mE nB L mE nB

q p E mEqB nB sN

mE nB L mE nB

qm p B EB B N N B

mE nB

* 2.

)

Using the Routh-Hurwitz criteria, we note that all roots of equation (4.20) have negetive

real parts iff

1 1 1 1 10, 0 and 0.A C A B C (4.21)

Thus, we are now able to state the following results.

Theorem 4.4

The interior equilibrium point *P is locally asymptotically stable iff conditions in eq.

(4.21) hold.

In the following theorem, we state sufficient conditions under which *P is globally


Theorem 4.5

The interior equilibrium point *P is globally asymptotically stable in the region if

the following conditions hold:

* * * *

2 (1 )r

N mE nB nqEK

, (4.22)


77

*

2* * 1

1 2 1 1 2 2 * *( )

(1 )

k sr nqEK k K B N

K mE nB L

. (4.23)

Proof : Consider a positive definite function about *P :

* * * * * *

1 2* * *ln ln ln

B N EW B B B k N N N k E E E

B N E

, (4.24)

where 1k and 2k are positive constants to be chosen suitably later on.

Differentiating W with respect to time t along the solutions of model (4.2)-(4.4), a little


* 2 * * * 2

11 12 22

* 2 * * * 2

11 13 33

* 2 * * * 2

22 23 33

1 1( ) ( )( ) ( )

2 2

1 1 ( ) ( )( ) ( )

2 2

1 1 ( ) ( )( ) ( ) ,

2 2

dWa B B a B B N N a N N

dt

a B B a B B E E a E E

a N N a N N E E a E E

(4.25)

where

**

11 2 * *(1 )(1 )

r nqEa N

K mE nB mE nB

,

122 0,

k sa

L

*

2 033 * *

( )0

(1 )(1 )

k mq p Ba

mE nB mE nB

,

*

12 1 2 1 1 1 2 ( )a B k k B B ,

23 0a ,

**

2 013 * * * *

( )(1 )(1 )

(1 )(1 ) (1 )(1 )

k q p mEq nBa

mE nB mE nB mE nB mE nB

.

Sufficient conditions for 1dW

dt to be negative definite are that the following inequalities hold:

11 0,a (4.26)

2

12 11 22 ,a a a (4.27)

2

13 11 33,a a a (4.28)


78

2

23 22 33.a a a

(4.29)

Clearly, equation (4.29) holds as 23 0.a

If we choose *

2 *

0

(1 )

( )(1 )

nBk

p mE

, then condition (4.28) is satisfied.

Again equation (4.22) (4.26) and equation (4.23) (4.27). Thus, 1W is a liapunov

function for all solutions initiating in the interior of the positive orthant whose domain

contains the region of attraction , proving the theorem.


The maximum rate of harvesting of any biological resource biomass is called maximum

sustainable yield (MSY) and any larger harvest rate will lead to the depletion of resource

eventually to zero. In absence of any population, the value of MSY is given by (Clark (1976))

0 .4

MSY

rKh


yield is given by

* * *

* * * * *2

1 2* *1 .

1

qE B Bh rB N B N B

mE nB K

We note that

*

0h

B

yields

** 1

*

2

( ) and

2( )

K r NB

r KN

2

*20.

h

B

Thus,

* 2

1

*

2

( ),

4( )MSY

K r Nh

r N K

when

** 1

*

2

( ).

2( )

K r NB

r KN


2

KB and


79

0 .4

MSY MSY

rKh h

This result matches with the result of Clark (1976).

If MSYh h , then it denotes the overexploitation of the resource and consequently the

resource biomass decreases. IfMSYh h , then the resource biomass is under exploitation and

the resource biomass may be maintained at an appropriate level.


The bionomic equilibrium is said to achieved when the total revenue obtained by selling

the harvested biomass is equal the total cost to harvest the biomass, i.e. the economic rent is

completely dissipated.

The economic revenue at time t is given by

.1

pqBc E

mE nB

(4.30)

The bionomic equilibrium is ( , , )P B N E , where , and B N E are the positive

solutions of

0.B N E

From 0 , we get 1

1pq

E E n Bm c

, (4.31)

0N

gives us 2

1 2

LN s B B

s . (4.32)

Putting the values of E E and N N , then 0B

gives a biquadratic equation in B i.e.

4 3 2

1 2 3 4 5 0d B d B d B d B d , (4.33)

where

2 21 2 2 1 1 2, ,

L Ld d

s s


80

1 13 2 4 1 5 , , .

Lr pq cn cd L d L r d

K s mp mp

We note that equation (4.33) has always a positive root B B .

Putting the value of B in equations (4.32) and (4.31), we can find the value of

and N E . It may be noted that E exists if 1pq

n Bc

.


The net revenue to the society, ( , , , , )B N E t

= the net economic revenue to the harvesting agency + the net economic revenue to

the regulatory agency

=1

pqBc E

mE nB

.

So, our aim is to solve the maximization problem

0

( )1

t pqBJ e c E t dt

mE nB

, (4.34)

subject to the state equations (4.14)-(4.16) and to the control constraints

min max . (4.35)

To solve the maximization problem, we adopt Pontryagin’s Maximum Principle. The

Hamiltonian function H is given by

2

1 1 2

2

2 1 2

3 0

( ) 11 1

+ ( ) 1

( ) ( ) ,

1

t pqB B qEBH e c E t rB NB NB

mE nB K mE nB

Nt sN NB NB

L

qB pt E c

mE nB

(4.36)

where , 1,2,3i i are adjoint variables.


81

The optimal control will be a combination of bang-bang control and singular control as

H is linear in the control variable in equation (4.36).

H will be maximized under the control set (4.35), if

30 =0.

H

(4.37)

This is a necessary condition for singular control to be optimal. Using the Maximum

Principle, we get

31 2, , .dd dH H H

dt B dt N dt E

(4.38)

From the previous equation we get the following results

1

2

1 1 2 2 1 22

(1 )

(1 )

2 (1 ) 1 2 2 ,

(1 )

td mE Epqe

dt mE nB

B qE mEr N NB N NB

K mE nB

(4.39)

2 221 1 2 2 1 2

21 ,

d NB B s B B

dt L

(4.40)

312 2

(1 ) (1 )0

(1 ) (1 )

td pqB nB qB nBe c

dt mE nB mE nB

. (4.41)

Using equations (4.14)-(4.16), we can re-write these previous equations as follows:

1

2

1 2 2 1 22

(1 )

(1 )

+ 2 ,(1 )

td mE Epqe

dt mE nB

rB qnEBNB N NB

K mE nB

(4.42)

22

1 1 2 2 ,d sN

B Bdt L

(4.43)

2

1

(1 )

(1 )

t c mE nBe p

qB nB

. (4.44)


82

The shadow price along the singular path is 2

1

(1 )( ) ( ) .

(1 )

t c mE nBt t e p

qB nB

Putting the value of 1 in equation (4.43), we get an equation

22 2 2

tdA B e

dt

, (4.45)

where

2

sNA

L ,

22

2 1 2

(1 )( )

(1 )

c mE nBB p B B

qB nB

.

The solution of this equation is

2

22

2 0 1 2

(1 )( )

(1 )

tA t e c mE nB

K e p B BsN qB nB

L

.

Now when t , then 2


Thus

22

2 1 2

(1 )( )

(1 )

te c mE nBp B B

sN qB nB

L

. (4.46)

Putting the value of 2 in equation (4.42), we get the equation

13 1 3

tdA B e

dt

, (4.47)

where

3 2 2(1 )

rB qnEBA NB

K mE nB

22

1 2 1 2

3 2

(1 )( )( 2 )

(1 )(1 )

(1 )

c mE nBp B B N NB

qB nBpqE mEB

sNmE nB

L

.


83

The solution of the equation (4.47) can be written as

3 31 1

3

tA t B e

K eA

.

Now when t , then 1


Thus

31

3

tB e

A

. (4.48)

Equating both the values of 1 , from equations (4.44) and (4.48), we get

2

3

3

(1 )0.

(1 )

Bc mE nBp

qB nB A

(4.49)

This equation gives us the optimal level of resource biomass and population i.e.

, and .B B N N E E Then tax are given by

(1 )c

p mE nBqB

. (4.50)

4.7 Hopf-Bifurcation

Mathematical analysis of the present model suggests that the system has so many stable

equilibrium points. Since the model is highly non-linear, it is interesting to explore the non-

linear behaviour characteristic of the system in the form of existence of limit cycle and Hopf

bifurcation. The characteristic polynomial for the system at *P is given in equation (4.20).

i) The Routh-Hurwitz Criteria and Hopf Bifurcation:

If is the bifurcation parameter, then for some cr , necessary and sufficient

conditions for Hopf Bifurcation to occur are

1 1

1 1 1

1. ( ) 0, ( ) 0,

2. ( ) ( ) ( ) ( ) 0, and

3. Re 0, 1, 2, 3.

cr

cr

cr cr cr cr

c

j

rA C

f A B C

dj

d

(4.51)


84

The condition 1 1 1 0f A B C gives an equation in having one of the root as cr .

Since we have 1 0B at cr , there exists an interval containing cr , say ,cr cr

for some 0 for which 0cr such that 1B remains positive for ,cr cr .

Thus, for ,cr cr the characteristic polynomial of P cannot have real positive

roots. For cr we get

21 1 0B A . (4.52)

This has three roots 1 1 2 1 3 1, ,i B i B A

For ,cr cr the roots of characteristic polynomial are of the general forms

1 1 1 2 2 2 3 1, , .i i A

The third condition can be verified as follows:

Substituting 1 1j i into (4.52) and taking its derivative, we have

1 1

1 1

0,

0,

R S T

S R U

where

2 2

1 1 1 2 1

1 1 1 1

2 2

1 1 2 1 3 1 1

1 1 1 2 1

3 2 3 ,

6 2 ,

,

2 .

R d d

S d

T d d d d

U d d

Hence,

2 2

Re 0,

crcr

jd SU RT

d R S

since, 0cr cr cr cr

S U R T and also 3 1 0.cr crA

Hence, regarding the bifurcation at equilibrium point P we have the following theorem:

Theorem 4.6

Under the assumptions (4.51), there is a simple Hopf bifurcation at equilibrium point

P at some critical value of the parameter given by the equation 0crf .


85


In this section, we present numerical simulation results. For the model system (4.2)-(4.4),

we choose set of values of parameters

0 1

2 1 2

1.6, 1.2, 100, 100, 25, 1, 1, 0.001,

0.0001, 0.01, 0.0001, 7, 0.1, 4, 1,

r s K L p q

c m n

(4.53)

with initial conditions (0) 5, (0) 25, (0) 10.B N E

For the above set of values of parameters, condition (4.19) for the existence of the

interior equilibrium is satisfied. Thus positive equilibrium point * * * *( , , )P B N E is given by

* * *41.2666, 148.5799, 26.1311.B N E

We also note that all conditions of Theorem 4.4 are satisfied for the set of parameters

chosen in equation (4.53). Thus the equilibrium point* * * *( , , )P B N E is locally asymptotically

stable.

Fig.4.1: Time series of B, N and E.

The time series of B, N and E are presented in Fig.(4.1). This figure shows that B, N and

E increase as time increases and finally they settle down at their steady states. It is also

observed here that the increase in the population density is much more in comparison to the

increase in the density of B and E.


86

It may be pointed out that values of parameters chosen in (4.53) satisfy local stability

conditions but they do not satisfy global stability conditions.

Now we choose the following set of values of parameters :

0 1

2 1 2 1

1.6, 3, 100, 100, 0.5, 0.01, 0.1, 0.001,

0.0001, 0.01, 0.0001, 0.001, 0.1, 4, 1, 1,

r s K L p q

c m n c

(4.54)

with different initial conditions.

Fig.4.2: global stability

These values of parameters satisfy the global stability conditions of Theorem 4.5. The

trajectories of B, N and E with different initial values are plotted in Fig. (4.2). From this

figure, we note that all the trajectories starting from different initial conditions converge to

the equilibrium point *(54.6889, 111.7918, 40.7652)P . This shows that *P is globally


In this model we observe that, 1 2 1 2, , and are important parameters governing the

dynamics of the system.

In figures (4.3), we have plotted the trajectories of B, N and E respectively for different

values of 1 . Fig. (4.3a) shows that B increases with time and after little decrease it settles

down at its equilibrium level. It may be noted that B decreases as 1 increases due to which

N and E also decrease as 1 increases (see Fig. (4.3b) and (4.3c)). We also observe that N


87

increases with time and finally stabilized at its steady state level. E first decreases with time,

then increases and settle down at its equilibrium level.

The effect of 2 on B, N and E are shown in Figs. (4.4a), (4.4b) and (4.4c) respectively.

We notice that B, N and E all decreases as 2 increases. By comparing Fig. (4.3a) with Fig.

(4.4a), Fig. (4.3b) with Fig. (4.4b) and Fig. (4.3c) with Fig. (4.4c), we note that 2 is a very

sensitive parameter in comparison to 1 .

a)

b)


88

c)

Figs. (4.3a), (4.3b) and (4.3c): B, N and E vs t for different values of 1

and others values are same as (4.54)

The behaviour of B, N and E with respect to time t for different values of 1 are shown

in Figs. (4.5a), (4.5b) and (4.5c) respectively.

If 1 increases, then the population N increases and after that it settles down at its

equilibrium level. We know that population utilizes the resource for its own growth and

development.

a)


89

b)

c)

Figs. (4.4a), (4.4b) and (4.4c): B, N and E for different values of 2

and others values are same as (4.54).

So, if the population increases, then obviously resource biomass decreases. Thus if 1

increases, then B and E decrease and then settle down at its lower equilibrium levels.

The effect of 2 on B, N and E are shown in Figs. (4.6a), (4.6b) and (4.6c) respectively.

From the figure (4.6b), we note that if 2 increases then N also increases. N increases very


90

quickly with respect to time t and goes to the peak after that it decreases very quickly and

settle down at its equilibrium level. Again, B and E decrease as 2 increases and then obtain

its respective equilibrium levels. It is also observed here that dynamics of the system is

highly sensitive with respect to the parameter 2 .

For optimal harvesting part, we choose these values of parameters:

0 1

2 1 2

1.6, 1.2, 100, 100, 25, 1, 1, 0.001,

0.0001, 0.01, 0.0001, 7, 4, 1, 0.1.

r s K L p q

c m n

(4.55)

a)

b)


91

c)



Solving (4.49) and (4.50) with the help of equations (4.14)-(4.16), we get the optimal

values

42.908, 151.099, 9.20654 and 11.829.B N E

a)


92

b)

c)



The behaviour of B, N and E with respect to time t for different values of are given in

Figs. (4.7a), (4.7b) and (4.7c) respectively. We see that B and N increase with time t if

increases. Both B, N initially increase and go to the peak and slightly decrease and settle

down at its lower equilibrium level. But when 20 , then B, N increase with time and

then settle down at higher equilibrium levels. We also note that E decreases as increases


93

(see Fig. (4.7c)). If is greater than , then E increases slowly as t increases and then settles

down at its equilibrium level. If is large enough then E goes to the zero level.

For existence of periodic solution of the underlying equation (4.2)-(4.4), we consider the

parametric values of the system as follows:

0 1 2 1

2

1.6, 0.00292969, 64, 0.5, 1, 1,

2.75, 1.16875, 0.03125, 0.00146484,

0.00146484, 0.1875, 0.5, 1.6667.

r s K L p q

c m n

(4.56)

a)

b)


94

c)

Figs. (4.7a), (4.7b) and (4.7c): B. N and E for different values of


For the above set of values of parameters given in (4.56) with 0.6 , the equilibrium

point* * * *( , , )P B N E is given by

* * *1.2264, 1.1826, 0.3711.B N E

For parameter, 0.4 , the system has periodic solution (Limit cycle) and as increase

to a value 0.6, the system converges to stable equilibrium point * * * *( , , )P B N E . This is

illustrated in Fig. 4.8 and Fig. 4.9 respectively.

a)


95

b)

c)

d)

Figs. (4.8a, 4.8b, 4.8c, 4.8d) : for 0.4 and others values are same as (4.56)


96

a)

b)

c)


97

d)

Figs. (4.9a, 4.9b, 4.9c, 4.9d): for 0.6 and others values are same as (4.56)

4.9 Conclusions

This chapter analysed a model of resource biomass and population in which both are

growing logistically. The resource biomass, which has commercial importance, is harvested

according to a realistic non-linear catch-rate function. The population utilizes the resource for

its own growth and development. The harvesting effort is a dynamical variable and taxation

as measure as a control variable. Here we discussed the existence of equilibria, local stability

by Eigenvalue method and Routh-Hurwitz criteria and global stability using Liapunov direct

method. The non-negative equilibrium point *P exists if tax on per unit harvested biomass is

less than a threshold value. This threshold value depends on the selling price per unit

biomass, the fixed cost of harvesting per unit of effort and the equilibrium value of resource

biomass. This point *P is locally and globally asymptotically stable under certain conditions.

When the population does not utilize the resource directly, but the resource biomass is

harvested according to same catch-rate function, then equation (4.12) gives us the range of

the tax for the regulatory agency to find the range of tax on per unit harvested biomass. But,

when the population utilizes the resource biomass for its growth and development and the

resource is harvested, then equation (4.19) gives the range of the tax. This is same as (4.12),

but in some modified form. The maximum sustainable yield (MSY), MSYh has been computed

for our model system. It has been found that 0

4MSY

rKh obtained by Clark (1976) is a special

case of MSYh which has been proposed in this paper. Then bionomic equilibrium has been


98

obtained and we observed that for this model bionomic equilibrium point exists under certain

condition. Choosing an appropriate Hamiltonian function and using Pontryagin's Maximum

Principle, we analysed optimal harvesting policy. Finally, a numerical simulation

experiments have been carried out with the help of MATLAB 7.1. It has been observed that

2 2 and are very sensitive parameters in comparison to 1 1 and . An optimal level of tax

to be imposed by the regulatory agency has been suggested. It has been shown that if

, then E decreases and goes to the zero level but B and N increases with respect to time

t. Thus the regulatory agency should keep , so that one can maintain resource and

population at an optimal level.

The present system can also become unstable under certain parametric values once the

conditions of Routh-Hurwitz criteria violated as shown in Fig 4.8. The Hopf-bifurcation

analysis with respect to parameter , suggest that the under certain conditions, imposing a

tax by regulatory agency per unit resource biomass to protect the over-exploitation of the

resource could decide the fate of the system in terms of stability and unstability.

__________________________________________________________________________________*The content of this chapter has been submitted to Journal of Biological System.

Chapter 5

A Predator-Prey Interaction Model with Self and Cross-Diffusion in an

Aquatic System*

5.1 Introduction

The fundamental goal of ecological research is to understand the interactions of

individual organisms with each other and with the environment that determines the

distribution of populations and the structure of communities. The most challenging and

crucial role of modeling is to examine and validate whether the designed model system can

exhibit the proper behavior of the system under consideration. The empirical evidence

suggests that the spatial scale and structure of environment can influence population

interactions (Cantrell and Conser (2003)). In recent years there has been considerable interest

in reaction-diffusion (RD) systems due to the fascinating patterns that occurs in ecological

systems and to investigate the stability behavior of a system of interacting populations by

taking into account the effect of self as well as cross-diffusion (Roychaudhuri and Sinha

(1996), Dubey et al. (2001), Sherratt and Smith (2008)). The phenomena pursuit-evasion in

predator-prey systems, there is a tendency that predators pursuing prey and prey escaping

predators (Biktashev et al. (2004)). The escaping velocity of preys may be considered as

dispersive velocity of the predators and the chase velocity of predators may be considered as

dispersive velocity of prey (Shukla and Verma (1981)). The escaping- chasing phenomena

arises the idea of cross-diffusion first proposed by Kerner (1959), and first applied in

competitive systems by Shigesada et al. (1979).

From the pioneering work of Turing (1952), the reaction-diffusion equations have been

intensively used to describe mechanisms for pattern formation (Huang and Diekmann

(2003)). Turing pattern formation has attracted great attention in nonlinear science as well as

in biology (Yang and Zhabotinsky (2004)). On the other hand, in the past several decades, the

issue of chaos in spatiotemporal systems has become one of the focuses in nonlinear science

(Egolf et al. (2000)). It seems very unlikely that chaotic systems could generate spatially

ordered Turing patterns, because temporally chaotic motions (trajectories are sensitive to

extremely small perturbations) are not supposed compatible with ordered space structures

(which should be stable against rather strong noise). The spatiotemporal pattern formation

like patchiness and blooming have been studied in (Segel and Jackson (1972), Steel and

Henderson (1981), Scheffer (1991a, b), Malchow (1993), Pascual (1993), Malchow et al.

Ch.-5 Model With Self & Cross Diffusion

100

(2001) and Dubey et al. (2009)). Wolpert (1977) that gives a very clear and non-technical

description of development of pattern formation in animals which stimulates a number of

illuminating experimental studies of corresponding aquatic ecosystems. The mechanisms of

spatial pattern formation were suggested as a possible cause for the origin of planktonic

patchiness in marine systems. Malchow (1994, 2000a, b) and Malchow et al. (2004a, b)

observed Turing patches in plankton community due to the effect of nutrients and

planktivorous fish. Grieco et al. (2005) developed a hybrid numerical approach to study the

transport processes and applied it to the dispersion of zooplankton and phytoplankton

population dynamics. Liu et al. (2008) studied a spatial plankton system with periodic forcing

and additive noise.

Most reaction-diffusion models assume that the diffusion matrix is a diagonal matrix,

i.e., the diffusive flux of a given species is only driven by the gradient of that species.

Recently the models with off-diagonal terms in the diffusion matrix, representing cross-

diffusion, have attracted attention. The effect of cross-diffusive terms on pattern formation

has been in the context of population dynamics (Biktashev and Tsyganov (2005)), activator-

inhibitor system (Chattopadhyay et al. (1994), Mendez et al. (2007)), predator-prey systems

((Dubey et al. (2001), Chen and Wang (2008)). The term self-diffusion i.e., dij = 0 for i j of

the diffusion matrix d implies the movement of individuals from a higher to lower

concentration region and it is assumed to be always positive. Cross diffusion expresses the

population fluxes of one species due to the presence of the other species. If the cross-

diffusion term is positive, dij > 0, then the flux of ith

species in the direction of lower

concentration of jth

species and for dij < 0 the flux of the ith

species towards the higher

concentration of jth

species.

Alstad (2001) explains that the functional response encapsulates attributes of both the

prey and predator biology. Hence, the handling time, search efficiency, encounter rate, prey

escape ability, etc. should alter, in general, the functional responses. Therefore, predator’s

functional response may be different when a particular predator preys different prey having

different escape ability and if a particular prey is predated by different predators having

different hunting ability (KO and Ryu (2008), Zhang et al. (2009)). Zeng and Liu (2010)

studied the non-constant positive steady states for a ratio-dependent predator-prey system

with cross diffusion.


101

In this chapter, we have considered the reaction–diffusion system with Holling type IV

functional response studied by Upadhyay et al. (2008, 2010). Here we have considered the

effect of self and cross-diffusion on the predator-prey interaction and observed that the

positive cross diffusions i.e., the movement of the species in the direction of lower

concentration of another species leads to uniform steady state unstable. It seems that cross-

diffusion is able to generate many different kinds of spatiotemporal patterns. We observed

that by increasing the coefficients of self- diffusion the irregular pattern leads to stationary

pattern. Therefore, we can predict that the interaction of self and cross-diffusion can be

considered as an important mechanism for the appearance of complex spatiotemporal patterns

in aquatic ecological model. We have also investigated the effect of critical wave length on

the stability of two species of the system.


We consider a four-component nutrient-phytoplankton-zooplankton-fish model where at

any location (X,Y) and time T, the phytoplankton ( , , )P X Y T and zooplankton ( , , )Z X Y T

populations satisfy the reaction-diffusion equation

2 2 2 22

11 122 2 2 2 2

N

P NP cPZ P P Z ZP D D

PT H N X Y X YP a

i

, (5.1)

2 2 2 2 2

22 212 2 2 2 2 2 2

z

Z bcPZ FZ Z Z P PmZ D D

PT H Z X Y X YP a

i

, (5.2)

where 11 22 and D D are the self-diffusion coefficient and 12 21 an dD D are the cross diffusion

coefficient. N is the nutrient level of the system which is assumed to be constant. The

phytoplankton equation is based on the logistic growth formulation, with a Monod type of

nutrient limitation. Growth limitations by different nutrients are not treated separately.

Instead, it is assumed that there is an overall carrying capacity which is a function of the

nutrient level of the system and the phytoplankton do not deplete the nutrient level. ξ is the

maximum per capita growth rate of prey population, NH is the phytoplankton density at

which specific growth rate becomes half its saturated value. is the intraspecific

interference coefficient of phytoplankton population, c is the rate at which phytoplankton is


102

eaten by zooplankton and it follows Holling Type-IV functional response, i is the direct

measure of the predator’s immunity from or tolerance of the prey, a is the half saturation

constant in the absence of any inhibitory effect, b is the conversion coefficient from

individuals of phytoplankton into individuals of zooplankton, m is the morality rate of

zooplankton. F be the predation rate of zooplankton population, which follows Holling Type-

III functional response and zH be the zooplankton density at which specific growth rate

becomes half its saturation value. Although the basic functional response of fish in the

presence of a single prey-species is generally type-II, there are good reasons to assume that

under natural condition the type-III response to zooplankton density is a more realistic

assumption. This type of functional response has been broadly adopted to describe

zooplankton predation in many theoretical studies. The units of the parameters are as follows:

time T and length X ,Y 0, L are measured in days d and meters m respectively. N, P,

Z, , ,N P ZH H H are usually measured in mg of dry weight per liter . /mg dw l ; b is a

dimensionless parameter; the dimension of ξ and c is 1d , and m are measured in

1 1. /mg dw l d

and 1d respectively. The diffusion coefficient ijD are measured in

2 1m d . F is measured in 1. /mg dw l d . The dimension and values of the other

parameters are chosen from literature.

We introduce the following substitution and notations to bring the system of equation

into non-dimensional form

x

N

NR

H N

, x

xR

Rar

R

,

i

a ,

bc

R ,

m

R ,

2

Fcf

aR ,

2 22

22

Z

R

c H

a ,

P

ua

, cZ

vaR

, X

xL

, Y

yL

, t RT , 1111 2

Dd

L R , 12

12 2

Dd

cL ,

2222 2

Dd

L R , 21

21 22

cDd

L R .

Then the model system (5.1)-(5.2) in dimensionless form reduced to


103

2 2 2 2

11 122 2 2 2 21

1

x

u uv u u v vr u u d d

ut x y x yu

, (5.3)

2 2 2 2 2

22 212 2 2 2 2 2 2

1

v uv fv v v u uv d d

ut v x y x yu

, (5.4)

with initial conditions

( , ,0) >0 , ( , ,0) 0 for ( , )u x y v x y x y (5.5)

and

0, ( , ) , 0

u vx y t

n n

(5.6)

where n is the outward normal to .

The zero flux boundary condition (5.6) imply that no external input is imposed from

outside. ( , , ) and ( , , )u x y t v x y t are phytoplankton and zooplankton densities respectively. is

the parameter measuring the ratio of the predator’s immunity from or tolerance of the prey to

the half-saturation constant in the absence of any inhibitory effect, is the parameter

measuring the ratio of product of conversion coefficient with consume rate to the product of

intensity of competition among individuals of phytoplankton with carrying capacity, be

the per capita predator death rate, f is the non-dimensional form of fish predation rate on

zooplankton. 11 22 and d d are self-diffusion and 12 21 and d d are cross -diffusion coefficients

for non-dimensionalzed model.

It may be noted that the proposed model system (5.3-5.4) and (5.5-5.6) in the

generalization of the previous model (Upadhyay et al. (2010)). Our main purpose is to see the

effect of cross-diffusion in the phytoplankton-zooplankton interaction.

5.3 Stability Analysis of a Non-Spatial Model

In this section, we consider the model system (5.3)-(5.4) in the absence of diffusion.

In this case, the model system reduces to the form (Upadhyay et al. (2010))

21

1x

du uvr u u

udtu

, (5.7)


104

2

2 2 2

1

dv uv fvv

udt vu

, (5.8)

(0) 0 , (0) > 0u v .

First of all, we state the following lemma which establishes a region of attraction for the

system (5.7)-(5.8). The proof of this lemma is similar to Freedman and So (1985), Freedman

and Shukla (1991) and Upadhyay et al. (2010) and hence omitted.

Lemma 5.1

The set

1

2, : 0 1, 0 xru v u u v

is a region of attraction for all solutions initiating in the interior of the positive quadrant

1 , where min , xr .

The above lemma shows that all solutions of model system (5.7-5.8) are bounded which

implies that the model is biologically well-behaved.

Now we shall find the non-negative equilibrium points of the above model and discuss

their stability behavior. It can be seen that model system has three non-negative feasible

equilibria, namely * * *

0 10,0 , 1,0 , and ( ),E E E u v .

The first two equilibria 0 1 and E E

obviously exist. The existence of the positive

equilibrium point *E can be shown as follows:

It may be noted that * u , *v are the positive solution of the following algebraic equations:

2

1 1x

uv r u u

, (5.9)

22 2

1

fv u

uvu

. (5.10)

From equation (5.9) we note the following:

when 2

0, then 1, as 1 0u

v u u

.

when 0, the .n xu v r


105

3 2(1 )

3

xurdvu

du

=

2 0 if 1

3

2 10 if

3

2(1 )0 if

3

u

u

u

(5.11)

From equation (5.10) we note the following:

when 0 , then 0 , 2u v if f n ,

when 0 , then v u u ,

where u is the solution of 2 0u u ,

and

22

2 is given by ( )

4

2u u

.

It may be noted that u is always real and positive if 2

1 .

We further note that equation (5.10) can be re-written as

2 22 2 1 ( ) 1

u ufv u v u u

.

From the above relation, we obtain

2 2 2 2

2

( ) (2 )

2

)(

2

v u v fvdv

du u u f v uv

. (5.12)

It is clear from the above equation that dv

du is positive if

(i) 2 2 2 22v u v fv

, (5.13)

(ii) 2( 2 )( ) 2f v u u uv . (5.14)

We note that if the following inequalities hold:

2 2 2 2 2 2(2 ) (4 ) 2x xr f r , (5.15)

24 xf r . (5.16)

then (5.15) implies (5.13) and (5.16) implies (5.14).


106

Remark:

It may be noted that dv

du will remain positive even if both the inequalities (5.13), (5.14)

are reversed.

From the above analysis we note that the two isoclines (5.9) and (5.10) intersect at a

unique point * *( , ) u v in the positive quadrant of the uv -plane if in addition to condition

(5.15) and (5.16) the following inequalities hold

) 0 1,

) 2 ,

2) 1 .

i u

ii f

iii

(5.17)

5.3.1 Stability Analysis

Now we study the stability behavior (Ahmad and Rao (1999)) of each equilibrium points

of the model system. To study the local stability behavior of the equilibria of model system,

we compute the variational matrices corresponding to each equilibrium point. From these

matrices, we note the following obvious results:

(i) 0E is a saddle point with an unstable manifold along the u-direction and stable

manifold along the v-direction.

(ii) If (1 2 ) , then 1E is locally asymptotically stable in the u-v plane.

(iii) If 1 2 , then 1 E is a saddle point with stable manifold locally along the

u - direction and unstable manifold locally along the v -direction.

We take the following notation:

* * ** * 2

1 2 2*2 * 2

*

2

2

2

1

x

v u fvB u r v

u vu

, (5.18)

* ** * *

*2

2

2

* 2 *

2 2 2 3*2 *2* 2

* *

1 1

12

x

uu v

v u fvB u r v

u uvu u

. (5.19)


107

Now under an analysis similar to Upadhyay et al. (2010), we state the following two

theorems:

Theorem 5.1

The unique non-trivial positive equilibrium point * E is locally asymptotically stable iff

the following inequalities hold

1 20, and 0B B . (5.20)

Remark:

It may be noted that if the following inequalities

2

** * *

*

2

2* 2

2

1 2 ,

,

,

x

ur u v u

u

v

(5.21)

hold, then 1 20, and 0.B B

We state sufficient conditions for * E to be globally asymptotically stable in the following

theoram.

Theorem 5.2

Let the following inequalities hold in the region 1 :

* 22 ,xr v

(5.22)

2

2

* * 2 **

* * 2 2 2 2 * 2*

2*

2

1 δ ( δ 2 )( )1 4

4 δ1 1

xx

x

v u f r vur

u u r vu u

. (5.23)

Then *E is globally asymptotically stable with respect to all solutions initiating in the

interior of the positive quadrant 1 .

Remark:

Define

ln ln

u vW u u u v v v

u v

. (5.24)

Then 0dW

dt under conditions (5.22) and (5.23).


108

5.4. Stability Analysis with Diffusion

In this section, we investigate the effect of diffusion on the dynamics of the model

system (5.7)-(5.8). It may be noted here that when conditions obtained in Theorem 5.1 is

violated, instability will occur. Again due to nonlinearity character in the proposed model,

Turing instability may also arise. We shall study this phenomenon in one and two

dimensional cases.

5.4.1 One Dimensional Case

In this section we consider the full model equations (5.3)-(5.4) together with initial and

boundary conditions (5.5) and (5.6). As a consequence of initial–boundary conditions (5.5)

and (5.6), ** *, E u v is a uniform steady state for the following system:

2 2

11 122 2 21

1x

u uv u vr u u d d

ut x xu

, (5.25)

2 2 2

22 212 2 2 2 2

1

v uv fv v uv d d

ut v x xu

, (5.26)

with initial and zero flux boundary conditions

0, , 0, ,

,0 0, ,0 0, 0, ,

| | | | 0.t R t t R t

u x v x for x R

u u v v

x x x x

(5.27)

In order to study the effect of self and cross-diffusion, we consider the following linear

version of the model system (5.25)-(5.26) about the positive equilibrium * * *, :E u v

2 2

11 12 11 122 2

U U Vm U m V d d

t x x

, (5.28)

2 2

21 22 22 212 2

V V Um U m V d d

t x x

, (5.29)

where

* * **

11 122 **

* *2 * * 2

21 222 * 2 2

2*2

*

2

*

22*

2,

1

( ) ( ),

( )

1

1

x

v u um u r m

uu

v u fv vm m

vuu

uu

, (5.30)


109

and U, V are small perturbations about * *,u v , and they are given by

* *, .u u U v v V

With the boundary conditions under consideration, we look solutions of the linear model

(5.28)-(5.29) of the form

exp cosU a n x

tV b R

, (5.31)

where /R n is the critical wave length, R is the length of the system, 2

n

is the period

and is the frequency.

Then the characteristic equation of the system is given by

2

1 2 1 2 1 2 0r s , (5.32)

where,

2 2

1 1 11 22 2,

nB d d

R

(5.33)

2

2 2 2 2

2 2 11 22 22 11 11 222 2

n nB m d m d d d

R R

, (5.34)

21r m , (5.35)

12 0s m , (5.36)

2 2

1 122

nd

R

, (5.37)

2 2

2 212

nd

R

. (5.38)

Now we assume that inequalities in equation (5.21) are satisfied. In such case, we note

that 1 2 11 22 0, 0, >0, <0, 0 and B B r m m E is locally asymptotically stable in the abs-

ence of diffusion. This shows that 1 20 and >0 .

From Routh-Hurwitz criteria, the stability of the equilibrium * E in the presence of

diffusion depends on the sign of


110

2 1 2 1 2r s . (5.39)

We shall discuss the following cases under the assumption that 1 2 0, and 0 .B B

Case 1: Let

11 12 21 22 0.d d d d

Then the characteristic equation (5.32) becomes

2

1 2 0B B , (5.40)

which is same as, in the case of no diffusion.

Thus the equilibrium point * * *( , )E u v is locally asymptotically stable iff 1 2 0, and 0B B .

Case 2: Let 12 210 and 0 , d d then 1 0 and equation (5.32) reduces to

2

1 2 2 0s

In this case * * *( , )E u v is locally stable if

2 2

2 21 20

nsd

R

,

i.e.2

21

2 2

2

sdR

n , 0.n (5.41)

We note that if 21 0 ,d then equation (5.41) is automatically satisfied. This shows that

if the predator species tends to diffuse in the direction of higher concentration of the prey

species and the prey species moves along its own concentration gradient, then the equilibrium

* E remains locally asymptotically stable. This situation is a usual phenomena in nature.

If 21 0d , then we note that one can always find values of n such that

2

21

2 2

2

sdR

n . (5.42)

This implies that by increasing the value of n, the critical wave lengthR

ln

can be

made small and in such a case * E will be unstable. Such situation occurs in nature when the

predator species moves in the direction of lower concentration of the prey species. Such

examples are observed in nature where the predator prefers to avoid group defense by a large

number of prey and chooses to catch its prey from a smaller group which is weak and unable

to resist.


111

Remark 1

Let 1 20 or 0 .B B Then *E is unstable. We assume that 12 210, and 0.d d Then

unstable equilibrium *E will be stable if the following inequalities hold :

1 0 , (5.43)

2 2 0s . (5.44)

It may be noted that if 21 0d , then by increasing 11 22 d and d to a sufficient large

value, 1 and 2 2 0s can be made positive. This shows that an unstable

equilibrium *E can be made stable by increasing self-diffusion coefficients

appropriately.

Case 3: Let 12 210 and 0d d . Then equation (5.32) reduces to

2

1 2 1 0r .

We note that *E remains locally stable in the presence of diffusion if

2 1 0 ,r

i.e. 2

12

2 2

2

> 0rdR

n

0 n . (5.45)

Further if 12 0,d then the inequality (5.45) is satisfied. This implies that if the prey

species moves in the direction of lower concentration of the predator species and the predator

species moves along its own concentration gradient, then the equilibrium *E remains locally

asymptotically stable. This situation can be compared in nature where the prey moves

towards the lower concentration of the predator in search of new food. It may be noted that r

is positive under condition (5.21). If 12 d 0, then we can always find n such that

2

12

2 2

2

rdR

n . (5.46)

In such a case * * *( ),E u v will be unstable.

This shows that when the critical wave length is too small that the prey species moves

towards the higher concentration of the predator species, and the predator species moves

along its own concentration gradient, then the equilibrium state which is stable without self

and cross-diffusion becomes unstable.


112

Remark 2

Let 1 2 < 0 or B 0 ,B then the interior equilibrium *E is unstable in the absence of self

and cross-diffusion. But the unstable equilibrium *E becomes stable in the presence of

self and cross-diffusion if the following inequalities hold:

1 0 , (5.47)

2 1 0.r (5.48)

It may be noted that if 12 > 0 ,d

then by increasing 11 22 and d d to large values

inequalities (5.47) and (5.48) can be made satisfied. This shows that if the prey species

tend to more in the direction of lower concentration of the predator species, then the

unstable equilibrium can be made stable by increasing the self diffusion coefficient of

prey and predator species appropriately.

Case 4: If 12 210 and 0 .d d In this case * * *( ),E u v is locally stable if

2 1 2 1 2 0r s , (5.49)

and unstable if the inequality is reversed.

Again, we note that if 1 20 and 0

i.e., if 12 210 and 0 ,d d

then the condition (5.49) is automatically satisfied. This shows that if the prey species tends

to move in the direction of lower concentration of the predator species, and the predator

species tends to move in the direction of higher concentration of the prey species, then the

stable equilibrium state without self and cross-diffusion remains stable.

Remark 3

Let 1 2 0 or B 0B , then the interior equilibrium *E is unstable in the absence of self

and cross diffusion. If 12 210 and 0d d , it may be seen that the unstable equilibrium

point *E becomes stable if

1 0, (5.50)

2 1 2 1 2 0 .r s

(5.51)

In particular it may be noted that the above two inequalities are satisfied if

1 12 2 210 . ., 0 , 0 ( . ., 0)i e d i e d , and 11 22 and d d are increased to a large value.


113

To study the global stability behavior of the positive equilibrum *E , we consider the

following region:

2{( , ) : 0 < 1 , 0 < },x

m m

ru v u u v u v

which is the interior of the first quadrant, where is the same as defined in Lemma 5.1.

Now we state the following theorem :

Theorem 5.3

Let 12 210 and 0.d d If the following condition holds

2 2 2 2 2 2 4

12 21 11 22(4 + ) 4x m mr u d v d d d u v u v , (5.52)

then the uniform steady state *E of the initial-boundary-value problems eqns. (5.25)-

(5.26) is globally asymptotically stable with respect to all solutions initiating in the

region .

Proof: For the sake of notation, let ( , )u x t u and ( , )v x t v . Now, using the positivity of

, u v for 0, and 0,u R t , we define a functional

1

0

,

R

W t W u v dx , (5.53)

where W is defined in equation (5.24).

Taking derivative of 1W with respect to time t along the model equations (5.25)-(5.26), we

obtain,

2 2

1 11 122 2

0 0 0

2 2

22 212 2

0 0

,

.

R R R

R R

W u W vW W u v dx d dx d dx

u x u x

W v W ud dx d dx

v x v x

Using the boundary conditions we get

2 **

1 11 122 2

0 0 0

2* *

22 212 2

0 0

.

R R R

R R

dW u u u u vW dx d dx d dx

dt u x u x x

v v v u vd dx d dx

v x v x x

(5.54)

From Eq. (5.54), we note that 1W is negative definite under condition (5.52), proving the

theorem.


114

Remark 4

If 12 21 0,d d then equation (5.52) is automatically satisfied. This shows that if the

equilibrium point *E is globally asymptotically stable, then the uniform steady state *E

of initial-boundary value problems without cross-diffusion remains globally


Remark 5

If 12 210 a nd 0d d , and if W

is positive definite, then from eq. (5.54), we note that

1W

can be made negative definite by increasing 11 22 and d d to sufficiently large values.

This implies that the unstable equilibrium *E of model system (5.3)-(5.4) can be made

stable by increasing self-diffusion coefficients to sufficiently large values.

5.5 Two Dimensional Case:

In this section, we consider the complete model (5.3)-(5.4) and state the main results in

the form of following theorem:

Theorem 5.4

(i) In addition to condition (5.52), let the following inequality hold

0dW

dt , (5.55)

where W is given by equation (5.24). Then the uniform steady state of the model

system (5.3)-(5.4) is globally asymptotically stable.

(ii) If the equilibrium *E of model system (5.7)-(5.8) is unstable, even then the

corresponding uniform steady state of model system (5.3)-(5.4) in two dimensional

case can be made globally asymptotically stable by increasing the self-diffusion

coefficient 11 22, and d d to a sufficiently large value.

Proof: Let

2

* * * *

* *

( ) ( , ) ,

where ln ln .

W t W u v dA

u vW u u u v v v

u v

Differentiating with respect to time t along the solution model (5.3)-(5.4), we get


115

21 2

dWI I

dt ,

where

1

dWI dA

dt

,

and

2 2

2 11 21 12 22 W W W W

I d d u dA d d v dAu v u v

.

Using Green's first identity in the plane

2 ( . ) ,G

F GdA F ds F G dAn

We note that

2 2* * * *

2 11 22 21 122 2 2 2

2 2* * * *

11 22 21 122 2 2 2

+

.

u u v v v u u vI d d d d dA

u x v x v u x x

u u v v v u u vd d d d dA

u y v y v u y y

We note that (5.55) 1 0I and (5.52) 2 0.I This shows that under conditions

(5.52) and (5.55)

2 0.dW

dt

This implies the if in the absence of diffusion *E is globally asymptotically stable, then in

the presence of diffusion *E will remain globally asymptotically stable. This proves the first

part of the theorem.

From Theorem-5.4, we also note if > 0,dW

dt then 1 0.I In such case *E will be

unstable in the absence of diffusion. However, even if 1 0,I 2dW

dt can be made negative by

increasing 11 22 and d d to sufficiently large values.

Remark 1

If 12 21 0d d , then the condition (5.52) is automatically satisfied. This shows that if

equilibrium*E of model equation (5.3)-(5.4) with self diffusion is globally asymptotically


116

stable, then the uniform steady state *E of initial boundary value problems equation

(5.25) and (5.26) without cross diffusion is also globally asymptotically stable.

Remark 2

If 12 210 and 0d d , and if W

is positive definite, then we note that 2W

can be made

negative definite by increasing 11 22 and d d to sufficiently large values. This implies that

the unstable equilibrium *E of model equation (5.3)-(5.4) can be made stable by

increasing self-diffusion coefficient to sufficiently large values.

Remark 3

We note that the equilibrium *E is stable in one-dimension case under condition (5.52)

and it is stable in two-dimensional case under condition (5.52) and (5.60). This shows

that if *E is stable in one-dimensional case, it need not be stable in two-dimensional

case.

Remark 4

In two-dimensional case, the number of negative term is more than one-dimensional

case. This implies that the rate of convergence towards its equilibrium in two-dimension

is faster in comparison to one-dimension case.


In this section, we perform extensive simulations of the model systems (5.3)-(5.4) with

constant diffusivity. Numerical simulations employ the zero flux boundary conditions with a

size of ,R R with R = 200 for figs. (5.1)-(5.3) and 900 600x yR R for figs. (5.4) and

(5.5) with step lengths 1x y and 0.1t respectively. We use the standard five-point

approximation for the two dimensional Laplacian such that the concentrations 1 1

, ,,n n

i j i ju v at

the moment 1n at the mesh position ,i jx y are given by

1

, , 11 , 12 , , ,

1

, , 21 , 22 , , ,

, ,

, .

n n n n n n

i j i j h i j h i j i j i j

n n n n n n

i j i j h i j h i j i j i j

u u d u d v f u v

v v d u d v g u v

with the Laplacian defined by


117

1, 1, , 1 , 1 ,

, 2

4.

n n n n n

i j i j i j i j i jn

h i j

u u u u uu

h

To obtain a nontrivial spatiotemporal dynamics, we have perturbed the homogenous

initial distribution. We consider the following initial conditions (Medvinsky et al. (2002))

0 0*

1 1

0 0*

1 1

2 2, , 0 sin sin ,

2 2, , 0 sin sin ,

x x y yu x y u

S S

x x y yv x y v

S S

(5.56)

where, 4

1 0 05 10 , 0.1, 0.2.x y S and * *( , ) (0.1266,1.0306)u v .

(a) 0.3

(b) 0.32


118

(c) 0.33

Fig.5.1. Typical Turing patterns of prey [first column figures] and predator populations

[second column figures] is plotted at fixed parameters 2.33, 0.25, 2.5, 1.3,f time t=10000 and 11 12 21 220.05, 0.01, 30d d d d at the different values of predator’s

immunity (a) 0.3 , (b) 0.32 and (c) 0.33 .

We observed the Turing pattern for the model system (5.3)-(5.4) obtained by performing

numerical simulations with initial condition given in (5.56) for the different set of parameters

value with domain size 200 200 at time level t =104.

From Fig. (5.1) and (5.2), we observed the typical Turing pattern of phytoplankton and

zooplankton population with small random perturbation of stationary point *u and *v of the

spatially homogenous system.

From these figures, one can observe that on increasing the parameter measuring the

ratio of Zooplankton’s immunity from or tolerance of the prey to the half-saturation constant

in the absence of any inhibitory effect and half saturation constant of Zooplankton ,

stationary spotted, spot-stripe mixture patterns emerge in the distribution of phytoplankton

and zooplankton population density. From Fig.(5.3) one can see that on increasing the fish

predation rate f, stationary stripe, stripe-spot mixture and finally, the regular spot patterns

prevail over the whole domain.

The spatiotemporal dynamics of the system depends to a large extent on the choice of

initial conditions. In a real aquatic ecosystem, the details of the initial spatial distribution of

the species can be caused by spatially homogenous initial conditions. However, in this case,

the distribution of species would stay homogenous for any time, and no spatial pattern can

emerge.


119

(a) 2.4 .

(b) 2.5 .

(c) 2.6 .

Fig. 5.2. Typical Turing patterns of prey [first column figures] and predator

populations[second column figures]is plotted at fixed parameters 0.3, 2.33, 0.25,

1.3f time t=10000 and 11 12 21 220.05, 0.01, 30d d d d at the different values of half

saturation constant of predator (a) 2.4 , (b) 2.5 and (c) 2.6 .


120

To get a nontrivial spatiotemporal dynamics, we have perturbed the homogenous initial

distribution. Here, we present the spiral spatial pattern and the irregular patchy pattern by

performing numerical simulations for two sets of initial conditions given in equations (5.57)

and (5.58) respectively.

For this purpose, in the first case we consider the following initial conditions (Medvinsky

et al. (2002))

2

3 4

( , , 0)=0.1266- ( -0.1 - 225)( -0.1 -675),

( , , 0) 1.0306 - ( - 450) - ( -150),

u x y x y x y

v x y x y

(5.57)

where, 7 5 4

2 3 42 10 , 3 10 , 1.2 10 with fixed parameter set

0.3, 2.33, 11 12 21 220.25, 0.0001, 2.5, 0.05, 0.01, 1.f d d d d

(a) f =1

(b) f =1.2


121

(c) f =1.35

Fig. 5.3. Typical Turing patterns of prey [first column figures] and predator populations

[second column figures] is plotted at fixed parameter 0.3, 2.33, 0.25, 2.5, time

t=10000 and 11 12 21 220.05, 0.01, 10d d d d at the different values of fish predation rate

(a) 1.0f , (b) 1.2f and (c) 1.35f .

a) t = 200

b) t =400


122

c) t =1000

Fig. 5.4. Snapshots of prey [first column figures] and predator populations [second column

figures] at fixed parameters 110.3, 2.33, 0.25, 0.0001, 2.5, 0.05,f d

12 21 220.01, 1,d d d at different values of time (a) t =200, (b) t = 400 and (c) t = 1000.

From Fig.(5.4), we observe that at time t = 200 regular spiral spatial pattern evolves,

on increasing the time t = 400 the destruction of the spirals begins in their centers and finally

at time t =1000 patchy irregular patterns prevails over the whole domain. In the second case,

the initial conditions describe a phytoplankton patch placed into a domain with a constant-

gradient zooplankton distribution:

*

5 6

*

7 8

, , 0 180 720 90 210 ,

, , 0 450 135 .

u x y u x x y y

v x y v x y

(5.58)

where, 7 7 5 5

5 6 7 82 10 , 6 10 , 3 10 , 6 10 with fixed parameter set 0.3

11 12 21 222.33, 0.25, 0.0001, 2.5, 0.05, 0.01, 1f d d d d .

a) t =200


123

b) t=400

c) t=1000

Fig. 5.5. Snapshots of prey [first column figures] and predator populations [second column

figures] at fixed parameters

11 12 21 220.3, 2.33, 0.25, 0.0001, 2.5, 0.05, 0.01, 1f d d d d

at different values of time (a) t =200, (b) t = 400 and (c) t = 1000.

From Fig.(5.5), we observed that in the second case also the system behaves more or less

similar behavior as previous one. Again at time t =200 regular spiral spatial pattern evolves

but the spiral structure is not perfect as in the Fig. (5.4), at t =400 destruction of the spirals

begins in their centers and finally t =1000 irregular patchy structure prevails over the whole

domain.

5.7 Conclusions

In this chapter, we have considered a 2D spatial Rosenzweig-MacArthur type model with

Holling type-IV functional response for plankton-fish interaction with self and cross-

diffusion. The existence of positive equilibrium has been established. Criteria for local


124

stability, instability and global stability of the positive equilibrium have been obtained. It has

been found that critical wave length of the system plays an important role in stabilizing the

system. It has been observed that if the critical wave length is very small so that the prey

species moves towards the higher concentration of the predator species and the predator

species moves along its own concentration gradient, then in such a case the positive

equilibrium which is stable without self and cross-diffusion becomes unstable in the presence

of self and cross-diffusion. It has also been noted that unstable equilibrium can be made

stable by increasing self diffusion coefficients to an appropriate large value. We have

investigated the effect of , and f , the ratio of the predator’s immunity from or tolerance

of the prey to the half-saturation constant in the absence of any inhibitory effect, non-

dimensional half- saturation constant and non-dimensional form of fish predation rate on

zooplankton respectively and observed the following:

(i) For the increasing values of the ratio of the predator’s immunity from or tolerance of

the prey to the half-saturation constant in the absence of any inhibitory effect ( )

and non-dimensional half- saturation constant ( ) we observed the spot and spot-

strip mixture in the whole domain (cf. Fig 5.1 and 5.2).

(ii) For the increasing values of fish predation, f we observed the strip, spot-strip

mixture and spot patterns in the whole domain for both phytoplankton and

zooplankton population (cf. Fig. 5.3).

(iii) As we increase the time steps from t =200 to t =1000, we observed the destruction

of regular spiral pattern to patchy irregular patterns which spread in the whole

domain. This phenomena also persists for the longer period. We have changed the

initial condition into a constant –gradient zooplankton distribution and observed the

similar dynamics.

In conclusion, we have revealed different types of Turing and spatial patterns in the

designed model system with specific coupling structure with self and cross-diffusion. Further

exploration of Turing and spatial patterns in experimental set up and in natural pattern

formation process will stimulate a new area of pattern formation.

__________________________________________________________________________________*The content of this chapter has been accepted in Application and Applied Mathematics.

Chapter 6

Dynamics of phytoplankton, zooplankton and Fishery Resource Model*

6.1 Introduction

The study of prey-predator models have been of great interest for ecologists in the past

few decades. The prey-predator models have been also used in phytoplankton-zooplankton-

fish interactions to study the spatiotemporal pattern (Steele and Henderson (1981), Scheffer

(1991b), Pascual (1993)) and to study local and temporal chaos (Sherratt et al.(1995),

Petrovskii and Malchow (1999, 2001), Malchow et al. (2002), Upadhyay et al. (2008)).

Modeling of phytoplankton-zooplankton interaction takes into account zooplankton grazing

with saturating functional response to phytoplankton abundance called Michaelis-Menten

models of enzyme kinetics (Michaelis and Menten (1913)). The oscillatory behavior of

phytoplankton and zooplankton has been extensively studied by several researchers (Steele

and Handerson (1981), Scheffer (1991a, 1998), Steele and Henderson (1992a, b), Truscott

and Brindley (1994a, b)). Dubois (1975) proposed a nonlinear partial differential equation

model with Lotka-Volterra type ecological interaction taking into account advection and eddy

diffusivity. Vilar et al. (2003) showed that biotic fluctuations and turbulent diffusion in

standard prey-predator models are able to explain plankton field observations which include

not only the spatial pattern but also its temporal evolution. Morozov and Arashkevich (2008)

proposed a simple model explaining the observed alternations of functional response. They

observed that the overall response of zooplankton exhibits different behavior compared to the

patterns of the local response.

To study effects of space and time on the interacting species, Jansen (1995) has

extended the scope of the simple Lotka-Volterra system and Rosenzweig-McAurthur model

to a patchy environment. Temporal and spatiotemporal chaos in population dynamics have

been observed by many authors (Hanski et al. (1993), Becks et al. (2005), Xiao et al. (2006),

Liu et al. (2008), Malchow et al. (2008)). Upadhyay et al. (2008) proposed a phytoplankton-

zooplankton -fish interaction model with Holling type-IV functional response and studied the

wave of chaos and pattern formation. Comparing with the empirical evidence from a different

predator-prey model, Skalski and Gilliam (2001) pointed out that the predator-dependent

functional responses could provide better descriptions of predator feeding over a range of

Ch.-6 Food Chain Fishery Model

126

predator-prey abundance, and in some cases, the Beddington–DeAngelis type functional

response performed even better (Liu and Edoardo (2006)). Upadhyay et al. (2009, 2010)

investigated the wave phenomena and nonlinear non-equilibrium pattern formation in a

spatial plankton-fish system with Holling type-II and IV functional responses.

Holling type-III functional response has also been used to demonstrate cyclic collapses

for representing the behavior of predator hunting (Real (1977), Ludwig et al. (1978)). This

response function is sigmoid, rising slowly when prey are rare, accelerating when they

become more abundant, and finally reaching a saturated upper limit. Keeping the above

mentioned properties in mind, we have considered the zooplankton grazing rate on

phytoplankton and the zooplankton predation by fish follows a sigmoidal functional response

of Holling type III. Misra (2011) also studied the depletion of dissolved oxygen due to algal

boom in a lake with Holling type III interaction. Shukla et al. (2011a) proposed and analyzed

a mathematical model to study the depletion of a renewable resource by population and

industrialization. They showed that density of the resource biomass decreases due to increase

in densities of population and industrialization. It decreases further as the resource dependent

industrial migration increases. However, the resource biomass can be maintained at an

appropriate level if suitable technological efforts are applied for its conservation. Recently,

Shukla et al. (2012) studied the effect of acid rain formed by precipitation on the plant

species. They showed that the plant species may become extinct if the rate of formation of

acid rain is very high. In these studies (Misra (2008, 2011), Shukla et al. (2007, 2011a,

2012)) interaction considered are either linear or bilinear. Kar and Chudhuri (2004) studied

the bio-economic equilibrium and optimal harvesting policy of Lotka-Volterra model

consisting of two prey species in the presence of a predator. In this article, they have

considered the predator functional response in which the feeding rate of predator increases

linearly with the prey density. Kar et al. (2009a) further extended the above idea for two prey

and one predator system by consdering the predator functional response as Holling type I for

one prey while for other prey it is of Holling type II. Kar et al. (2010) further proposed a

prey-predator model with non-monotonic functional response and showed the existence of

super critical Hopf bifurcation.

In this chapter, we extend the two-dimensional model studied by Upadhyay et al. (2010)

into a three-dimensional model by considering the fish population as a dynamical variable.

We assume that the grazing rate of zooplankton is dependent on the phytoplankton


127

concentration according to type IV functional response (Upadhyay et al. (2008)) while the

predation rate of fish on zooplankton is of type III and on phytoplankton is of type II

(Gentleman et al. (2003)). We further assume that fish population is growing logistically and

is harvested according to catch-per-unit-effort (CPUE) hypothesis (Clark (1976)). It may be

pointed out here that our model proposed in this chapter is much more general than those

studied in Kar and Choudhuri (2004), Kar et al. (2009a, 2010) and Upadhyay et al. (2010).


Let us consider a habitat consisting of the phytoplankton of density P , zooplankton of

density Z and fish population of density F , at any time 0T . Keeping the assumptions

discussed in the introduction in our mind, and following our previous work (Upadhyay et al.

(2010)), the dynamics of the system may be governed by the following system of differential

equations:

2 0

2

0

1N

PFdP NP cPZP

PdT H N hPP a

i

, (6.1)

2

2 2 2

z

dZ bcPZ FZmZ

PdT H ZP a

i

, (6.2)

2

0 1 00 12 2

0 0

1 ,1z

FZ PFdF Fs F q EF

dT K H Z hP

(6.3)

Interactions involved in the model system (6.1)-(6.3) are pictorially represented in Fig.

(6.1).

Figure 6.1: Interactions incorporated in the model system

The interaction part of the model system investigated is depicted by solid arrows. The

interaction involving dotted arrows indicates that either of the positive or negative effects

involved is considered in the model.


128

In this model system (6.1)-(6.3), N is the nutrient level of the system which is assumed

to be constant. The nutrient level is increased due to eutrophication. The phytoplankton

require both inorganic (phosphorus, nitrogen, iron, silicon, etc.) and organic (vitamins)

nutrients for growth. However, excessive nutrients in coastal water can cause excessive

growth of phytoplankton, microalgae and macroalgae. An excessive increase in phyto-

plankton and algae can lead to severe secondary impacts such as - i) reduction of light which

decreases the subaquatic vegetation, ii) inhibition of the growth of coral reef as nutrient levels

favor algae growth over coral larvae, and iii) reduction in the level of dissolved oxygen

forming an oxygen-depleted water zone which may cause the ecosystem to collapse. In the

present work, we assume that the phytoplankton grows with a Monod type of nutrient

limitation. Growth limitations by different nutrients are same. Instead, it is assumed that there

is an overall carrying capacity which is a function of the nutrient level of the system and the

phytoplankton do not deplete the nutrient level. is the maximum per capita growth rate of

prey population, NH is the phytoplankton density at which specific growth rate becomes half

its saturated value. is the intraspecific interference coefficient of phytoplankton popula-

tion, c is the rate at which phytoplankton is consumed by zooplankton and it follows Holling

Type-IV functional response, i is the direct measure of the predator’s immunity from or

tolerance of the prey, a is the half saturation constant in the absence of any inhibitory effect,

b is the conversion coefficient from individuals of phytoplankton into individuals of

zooplankton, m is the morality rate of zooplankton. It is assumed that the zooplankton

population is predated by fish which follows Holling Type-III functional response and zH is

the zooplankton density at which specific growth rate becomes half its saturation value. It is

further assumed that phytoplankton is predated by fish according to type-II functional

response. The fish population grows logistically with intrinsic growth rate 0s and carrying

capacity 0K . 0 1 and are conversion coefficients ( 0 10 , 1 ) of phytoplankton and

zooplankton respectively. The fish population is harvested according to catch-per unit-effort

(CPUE) hypothesis and 1q is the catchability coefficient. Here harvesting effort is a control

variable.

We introduce the following substitution and notations to bring the system of equations

into non-dimensional form


129

N

Nr

H N

,

ra

,

i

a ,

bc

r ,

m

r ,

2

Fcx

ar , ZcH

ar ,

Pu

a ,

cZv

ar , t rT , 0

0

ar

c

,

1 0a h , 1 02

a

r

, 0s

sr

,

0

2

cKK

ar , 0

0r

, 1q

qr

.

Using these parameter the model system (6.1)-(6.3) in dimensionless form reduced to

0

2

1

1 , 1

1

uxdu uvu u

udt uu

(6.4)

2

2 2 2

1

dv uv xvv

udt vu

, (6.5)

2

0 2

2 2

1

1 ,1

xv xudx xsx qEx

dt K v u

(6.6)

with initial condition (0) 0, (0) 0, (0) 0.u v x

is the parameter measuring the ratio of the predator’s immunity from or tolerance of

the prey to the half-saturation constant in the absence of any inhibitory effect, is the

parameter measuring the ratio of product of conversion coefficient with consume rate to the

product of intensity of competition among individuals of phytoplankton with carrying

capacity, is the per capita predator death rate.

In the next section, we present the analysis of model (6.4)-(6.6).


In the following lemma, we state a region of attraction for the model system (6.4)-(6.6).

Lemma 6.1

The set

1

2( , , ) : 0 ( ) ( ) ,0 ( )u v x u t v t x t L

(6.7)

is a region of attraction of all solutions initiating in the interior of positive orthant ,

where


130

1

2

02

2 2 2

1 1

min (1, ),

4.

1 4

KL s

s

The above lemma shows that all solutions of the model (6.4)-(6.6) are non-negative and

bounded, which shows that the model is biologically well-behaved.

The proof of this lemma is similar to Freedman and So (1985), Shukla and Dubey (1997),

hence omitted.

Now we discuss the equilibrium analysis of the model. The model (6.4)-(6.6) has six

non-negative equilibrium points, viz,

* * * *

0 1 2 3 4ˆ ˆ(0,0,0), (1,0,0), (0,0, ), ( , ,0), ( ,0, ) and ( , , ).P P P x P u v P u x P u v x

It may be noted here that the equilibrium points 0 1, and P P always exist.

For the point 2P ,

x is given by ( )K

x s qEs

and it exists iff .s qE

In the equilibrium point3P , and u v are the positive solutions of the following equations:

2

(1 ) 0

1

vu

uu

, (6.8)

2

0

1

u

uu

. (6.9)

From equations (6.8) and (6.9) we get

2

(1 ) 1u

v u u

, (6.10)

and

2 ( ) 0u u . (6.11)

From equation (6.11), we note the following:

i) when 2

, then equation (6.11) has a unique root, namely

1

( )

2u u

.


131

In such case, equilibrium 1

3 1 1( , ,0)P u v exists,

where

1

( ),

2u

2

11 1 1 1(1 ) 1 ,

uv v u u

under the condition 0 1 .

ii) when 2 2

. . 1i e

,

then equation (6.11) has two distinct positive real roots, namely

2

2

2

2( ) ( )

2u

,

2

2

3

2( ) ( )

2u

.

Thus, in this case we have two equilibrium points 2 3

3 2 2 3 3 3( , ,0) and ( , ,0)P u v P u v , where

2 3 and v v are calculated from equation (6.10).

Now to show the existence of the equilibrium point 4ˆ ˆ( ,0, )P u x , we note that ˆ ˆ and u x are

the positive solutions of the two equations:

0

1

(1 ) 01

xu

u

, (6.12)

2

1

1 01

uxs qE

K u

. (6.13)

From equation (6.12),


132

1

0

1(1 )(1 )x u u

. (6.14)

Clearly

0 if 1x u . (6.15)

Substituting the value of x from equation (6.14) into equation (6.13), we get

3 2

1 2 3 4 0a u a u a u a , (6.16)

where

2

1 1 2 1 1

3 0 1 1 0 2 0 1 4 0 0

, (2 ),

( 2 ), .

a s a s

a s K s s K q KE a s K s q KE

Using the Decarte’s rule of sign, it may be noted that equation (6.16) has a positive real root

if

0 ( ) 1K

s qEs

. (6.17)

This shows that4

ˆ ˆ( ,0, )P u x exists under conditions (6.15) and (6.17).

Existence of interior equilibrium point * * * *( , , )P u v x :

In this case * * *, and u v x are the positive solutions of following three equations:

0

2

1

(1 ) 0,1

1

xvu

u uu

(6.18)

2 2 20,

1

v xv

u vu

(6.19)

2

0 2

2 2

1

1 0.1

v uxs qE

K v u

(6.20)

From equation (6.18), we get

2

0

1

(1 ) 11

x uv u u

u

. (6.21)

Clearly

0v if 1 0(1 )(1 )u u x . (6.22)


133

Putting the value of v from (6.21) into equations (6.19 and 6.20) we get two functions in

and u x .

These two functions are given below:

2

0

1

1222 22 0

1

1 (1 )1

0 ( , ),

1 1 (1 )1

xux u u

uuF u x

u xuu u uu

(say) (6.23)

222

00

1

222

2 0

1

22

1

1 (1 )1

1

1 (1 )1

0 ( , ).1

xuu u

uxs

K xuu u

u

uqE F u x

u

(say)

(6.24)


i) when 0,u 1(0, ) 0F x has a real root ax which is given by

2

0 0 0

0 0

(2 1) 1 4 (1 )

2 ( 1)ax

.

We note that

0ax if 0 1. (6.25)

ii) when 0x , then 1( ,0) 0F u has a real root au which is given by

2

2 2( ) ( ) .

2au

We note that 0au if

21

. (6.26)

Let 1 1 and x uF F be the partial derivatives of 1F with respect to and ,x u respectively.

Let 11

1

, where 0.xu

u

FduF

dx F

It may be noted that 0du

dx if


134

1 1

1 1

either ) 0 and 0,

or, ) 0 and 0.

x u

x u

i F F

ii F F

(6.27)


i) when 0u ,2(0, ) 0F x has a positive real root

bx if

2

0( )( 1) ,qE s (6.28)

ii) 2

2

x

u

Fdu

dx F .

Clearly 0du

dx if

2 2

2 2

either ) 0 and 0,

or, ) 0 and 0.

x u

x u

i F F

ii F F

(6.29)

hold.

From the above analysis, we note that the isoclines (6.23) and (6.24) intersect at a unique

point * *( , )u x , if in addition to conditions (6.25)-(6.29), the following condition

b ax x . (6.30)

This completes the existence of interior equilibrium * * * *( , , )P u v x .


local stability analysis, first we find the variational matrices with respect to each equilibrium

point. Then by using eigenvalue method and the Routh-Hurwitz criteria, we get the following

results.

i) The point 0 (0,0,0)P is always a saddle point. In fact, 0P has a unstable manifold in the

u direction and stable manifold in the v direction. 0P is stable or unstable in the x-

direction according as ( )s qE is negative or positive.

ii) The point 1(1,0,0)P is locally asymptotically stable if 2 1

and 2

11qE s

,

otherwise it is a saddle point. It may be noted that 1P has always stable manifold in

the u direction.


135

iii) The point 2P , whenever it exists, is locally asymptotically stable if

0 ( )

sK

s qE

.

If 0 ( )

sK

s qE

, then

2P is a saddle point with stable manifold in the v x -plane

and unstable manifold in the u -direction.

iv) If the following inequalities hold :

a) 1

12

,

b) 2

0 1 2 1

2 2

1 1 1

,1

v uqE s

v u

1 1where , (1 )(2 ),u v

then the point 1

3 1 1( , ,0)P u v is locally asymptotically stable.

Otherwise, 1

3 1 1( , ,0)P u v is a saddle point with stable manifold in the v direction and

unstable manifold in the u x -plane.

v) If the following inequalities hold:

22

2

0 222

12 2

(1 ) 1

a) 0,1

(1 ) 1

ii i

i

iii i

uu u

us qE

uuu u

b) 2 1,iu

c) 2

(1 )3

iu ,

where 2,3,i

then 3 ( , ,0)i

i iP u v is locally asymptotically stable.

vi) The point 4ˆ ˆ( ,0, )P u x is locally asymptotically stable if

2ˆˆ ˆ 1 .

uu u

If 2ˆ

ˆ ˆ 1u

u u

, then 4P

is a saddle point whose stable manifold is locally in

the u x -plane and unstable manifold is locally in the v -direction.


136

The stability behavior of the interior equilibrium point *P is not obvious from the

variational matrix. However, with the help of Liapunov’s direct method, we are able to find

sufficient conditions for *P to be locally asymptotically stable. We state these results in the

following theorem.

We use the following notations:

** *

0 11 2 * 2*2

1*

21

(1 )1

xv uL

uuu

, (6.31)

** *

0 12 **2

* 1

( 1)

(1 )1

xv uL

uuu

, (6.32)

2 * * * *

0 0 1 0 11 2* 2 *2 *

2 2

2 (1 ) (1 ), ,

( )

u u u uc c

v v x

(6.33)

22

* 2

22 * *0 00 0 1 0 1

1 2* 2 *2 *22 2

0

2 4

2 (1 ) (1 ), .

( ) 2

vK u u

d ds x v v

(6.34)

Theorem 6.1

Let the following inequalities hold :

i) 1 1L , (6.35)

ii) *v , (6.36)

iii)

2*2

*

* * * 2 *2*

1 1 12 2 *2 2*2 *2* *

1( )

(1 ) .( )

1 1

uv

u v x vc c u L

vu uu u

(6.37)

Then *P is locally asymptotically stable.

Proof: First of all, we linearize the model system (6.4)-(6.6) by using the following

transformation:


137

* * *, , u u U v v V x x X ,

where ( , , )U V X are small perturbations about the interior equilibrium * * *( , , )u v x .

Then the model system (6.4)-(6.6) can be written as

11 12 13

21 22 23

31 32 33

dUM U M V M X

dt

dVM U M V M X

dt

dXM U M V M X

dt

, (6.38)

where

**

* *** 0 1 0

11 12 132 * 2 *2 **2*1 1*

*2*

* * 2 *2 *2

21 22 232 *2 2 2 *2 2*2*

*

231

21

1 , , ,(1 ) 1

11

1( )

, , ,( )

1

(1

uv

x uuM u M M

u u uu uu

uv

x v v vM M M

v vuu

xM

* * 2 *

032 33* 2 *2 2 2

1

2, , .

) ( )

x v sxM M

u v K

(6.39)

Let

2 2 2

1 2

1 1 1

2 2 2W U c V c X

is a positive definite function, where 1 2&c c are positive constants as chosen in equation

(6.33).

Differentiating W with respect to time t along the solution of model (6.38), a little


2 2

11 12 1 21 1 22

2 2

11 13 2 31 2 33

2 2

1 22 1 23 2 32 2 33

1 1( )

2 2

1 1 ( )

2 2

1 1 ( ) .

2 2

dWM U M c M UV c M V

dt

M U M c M UX c M X

c M V c M c M VX c M X

(6.40)


138

Sufficient conditions for dW

dtto be negative definite are that the following inequalities

hold:

0, 1,2.iiM i (6.41)

2

12 1 21 1 11 22 ,M c M c M M

(6.42)

2

13 2 31 2 11 33,M c M c M M (6.43)

2

1 23 2 32 1 2 22 33.c M c M c c M M

(6.44)

For the values of 1 2&c c , as given in equation (6.33), we note that condition (6.43) and

(6.44) are satisfied, and (6.35) implies 11 0M and (6.36) implies

22 0M . Thus (6.41) holds

true. Again (6.37) implies (6.42). This shows that dW

dtis negative definite under conditions

stated in Theorem 6.1. This completes the proof of theorem.

Theorem 6.2

Let the following inequalities hold in :

i) 2 1L , (6.45)

ii) * 2

02 v , (6.46)

iii)

22 *

* 01* *

011 2*2 2

* * 2 *2

2

0

2

( 1)1 1 .

41 ( )

vd x

d u ud L

uu v

(6.47)

Then *P is globally asymptotically stable with respect to all solutions initiating in the

interior of the positive orthant .

Proof : Let us choose a positive definite function around the equilibrium point *P as

* * * * * *

1 1 2* * *ln ln ln ,

u v xW u u u d v v v d x x x

u v x

(6.48)

where 1 2 and d d are some positive constant as given in equation (6.34).

Differentiating 1W with respect to time t along the solution of model (6.4)-(6.6), a little



139

* 2 * * * 2111 12 22

* 2 * * * 2

11 13 33

* 2 * * * 2

22 23 33

1 1( ) ( )( ) ( )

2 2

1 1 ( ) ( )( ) ( )

2 2

1 1 ( ) ( )( ) ( ) ,

2 2

dWa u u a u u v v a v v

dt

a u u a u u x x a x x

a v v a v v x x a x x

(6.49)

where

** *

0 111 *2 *2

* 1 1

( )1

(1 )(1 )1 1

xv u ua

u uu uu u

,

* 2 *

122 2 2 2 *2

( )

( )( )

d x vva

v v

, 2

33 ,d s

aK

* *1 1

12 2 2 *2*

1 ( )

1 1 1

d d u u ua

u u uu u u

,

*

13 2 2 0 0 1*

1 1

1

(1 )(1 )a d u

u u

,

2 *

2 0123 2 2 2 *22 2

( )

( )( )

d v vd va

v vv

.

Sufficient conditions for 1dW

dtto be negative definite is that the following inequalities

hold:

0, 1,2.iia i (6.50)

2

12 11 22 ,a a a

(6.51)

2

13 11 33,a a a

(6.52)

2

23 22 33.a a a

(6.53)

For the chosen values of 1 2&d d (see eq. (6.34)), conditions (6.52) and (6.53) are

satisfied. We further note that (6.45) implies 11 0a , (6.46) implies 22 0a . After subs-

tituting the values of ija , in Eq. (6.51), we maximize the LHS and minimize the RHS using

Lemma 6.1. Then we note that (6.47) implies (6.51). Thus 1W is a liapunov function for all


140

solutions initiating in the interior of the positive orthant whose domain contains the region of

attraction , proving the theorem.


For the numerical integration of the model system, we have used the Runge-Kutta fourth

order procedure on the MATLAB 7.0 platform. The dynamics of the model system (6.4)-

(6.6) is studied with the help of numerical simulation. We choose the following set of values

of parameters:

0 1 0

2

0.2, 0.001, 3.33, 0.25, 2.5, 1.9, 150, 4.1,

5.5, 3.9, 0.17.

s K

qE

(6.54)

These parameter values are selected on the basis of values given by Letellier and Aziz-

Alaoui (2002). It is observed that model system (6.4-6.6) has a chaotic solution for the above

set of parameter values (see Fig. 6.2). The time series for these populations are presented in

Fig. (6.3). The chaotic nature of the model system is confirmed by SIC test and is presented

in Fig. (6.4a) and (6.4b).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 6.2: Chaotic attractor for the parameter values given in (6.54).

The stable focus is obtained (see Fig. (6.5)) for the following set of parameter values

0 1

0 2

1.8, 0.25, 0.001, 4.93, 0.25, 2.5, 3.2, 150,

5.2, 5.5, 3.8.

s K

qE

(6.55)


141

A bifurcation diagram of model system (6.4-6.6) is plotted in Fig. (6.6) and the blow-up

bifurcation diagram is presented in Fig. (6.7) and (6.8) in the different ranges for the control

parameter, , measuring the ratio of the predator’s immunity from or tolerance of the prey to

the half-saturation constant in the absence of any inhibitory effect. This figure exhibits the

transition from chaos to order through a sequence of period halving bifurcations. The blow up

bifurcation diagrams show that the model system possesses a rich variety of dynamical beha-

0 200 400 600 800 1000 1200 1400 1600 1800 2000-1

0

1

2

3

4

5

6

7

8

9

10

time

u,v

,x

u

v

x

Fig. 6.3:Time series for u, v, x species.

a)

500 1000 1500 20000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time(t)

u

(u,v,x)= (2,2,15)

(u,v,x)=(2.01,2,15)


142

b)

500 1000 1500 20000

0.5

1

1.5

2

2.5

3

3.5

4

time(t)

v

(u,v,x)=(2,2.01,15)

(u,v,x)=(2,0.01,15)

Fig. 6.4a and 6.4b: SIC in species u and v.

0 500 1000 1500 2000 2500 3000-1

0

1

2

3

4

5

6

time

popula

tion d

ensity

u

x

v

Fig. 6.5:Time series for u, v, x species showing the stable focus for the parameter

values given in (6.55).

vior including KAM tori for bifurcation parameter in the ranges [0.34, 0.36] and [0.33,

0.35]. Closed curve in this diagram correspond to invariant KAM tori in the phase space.

Later on, theses curves break and give rise to chaotic dynamics. The chaotic behavior of the

system is not continuing further, as the unstable period-3 orbits which originate at the time of

saddle-node bifurcation do not allow it to move further. The bifurcation diagram are plotted


143

with the help of software-''Dynamics: Numerical Exploration'' developed by Nusse and Yorke

(1994).

Fig. 6.6: Bifurcation diagram in the diagram in the ranges [ 2,2], [0.1,3]u .

Fig. 6.7: Magnified bifurcation in the ranges [ 2,2], [0.34,0.36]u .


144

Fig. 6.8: Magnified bifurcation diagram in the ranges

[ 2,2], [0.33,0.35]u .


The net economic revenue at time t is given by

( , , ) ( ) ,x E t pqx c E (6.56)

where p is the price per unit harvested fish and c is the utilized cost per unit effort. The

bionomical equilibrium is ( , , , )P u v x E , where , , and u v x E are positive solutions

of

0u v x

. (6.57)

Solving equation (6.57) we get ,c

xpq

substituting this value in equations (6.18) &

(6.19), these equations reduce to

0

2

1

1 0(1 )

1

cvu

u pq uu

, (6.58)

2 2 20

( )1

u cv

u pq vu

. (6.59)

We show that the above two isoclines (6.58) and (6.59) intersect at a unique point in the


145

interior of the positive quadrant. For this purpose we note the following from equations

(6.58):

i) when 0,u then 011

cv v

pq

(say),

1 0v if

0 .c pq

ii) when 0v , then 1u u is given by

2 2 2

1 1 1 0

1

1

(1 ) (1 ) 4 ( )0,

2

pq p q pq c pqu

pq

iii) assume that 0.dv

du

Now from equation (6.59), we note the following:

iv) when 0u , then 2v v is given by

2 2 2 2 2

2

4.

2

c c p qv

pq

We note that 2v is always real and negative if 2 .c pq

v) when 0,v then

2u u is given by

2

2

2

2( ) ( )

2u

.

If 2

1

, then 2u is always positive.

vi) assume that 0.dv

du

The above analysis shows that the two isoclines (6.58) and (6.59) intersect at a unique

point ( , )u v if in addition to assumptions (i)-(vi), the following holds:

2 1 .u u

After knowing the values of u and v , we may calculate E from (6.20), that is,

2

0 2

2 2

1

1 .1

v us cE

q pqK v u

It is clear that 0E if pqK c .


146

This shows that bionomical equilibrium ( , , , )P u v x E exists.


It is well known (Clark (1976)) that the value of Maximum sustainable yield (MSY) in

absence of any alternative resource is

0

4MSY

sKh .

In the present case when the fish population depends on phytoplankton and zooplankton

both, then we have

*2 * * *** * 0 2

*2 *2 *

1

1 .1

v x u xxh qEx sx

K v u

We note that *

0h

x

yields

*2 ** 0 2

2 *2 *

1

11

2 1

v uKx

s v u

and

2

*20

h

x

.

Thus,

2*2 *

0 2

2 *2 *

1

11 .

4 1MSY

v usKh

s v u

From the above equations, if * * 0u v , then 0

4MSY MSY

sKh h .This result matches with the

result of Clark (1976).

Thus MSYh h , then it denotes the overexploitation of fish population and if MSYh h ,

then the fish population is under exploitation.


In this section, we explain the optimal harvesting policy to be adopted by a regulatory

agency. The net economic revenue to the fisherman

= revenues obtained by selling the fishescost of harvesting

( ) ( ) ( ) ( ) ( )pqx t E t cE t pqx t c E t .

Now, our objective is to solve the following optimization problem:

max J

subject to the state equations (6.18)-(6.20) and to the control constraints max0 E E ,

where


147

0

( ( ) ) ( ) tJ e pqx t c E t dt

,

is the continuous time-stream of revenues and is instantaneous rate of annual discount.

Now to find the optimal level of equilibrium we use Pontryagins’s Maximum Principle.

The associated Hamiltonian function is given by

01 2

1

2

2 2 2 2

2

3 0 22 2

1

( ( ) ) ( ) (1 )1

1

( )

1

( ) 1 ,1

t uxuvH e pqx t c E t u u

u uu

uv xvt v

u vu

x xv uxt sx qEx

K v u

(6.60)

where1 2 3, and

are adjoint variables, 3( ) ( )tt e pqx c qx is the switching function.

The optimal control ( )E t which maximizes H must satisfy the condition

max 3

3

( ) 0 i.e.

( ) .

0 ( ) 0 i.e.

t

t

cE t e p

qxE t

ct e p

qx

Now the usual shadow price is 3

te and the net economic revenue on a unit harvest is

cp

qx

. Thus, if the shadow price is less than the net economic revenue on a unit harvest,

then maxE E , the shadow price is greater than the net economic revenue on a unit harvest,

then 0E and when shadow price equals the net economic revenue on a unit harvest, i.e.

( ) 0t , then the Hamiltonian becomes independent of the control variable ( )E t i.e. 0H

E

.

This is necessary condition for singular control *( )E t to be optimal over control set

*

max0 E E .

Hence the optimal harvesting policy is


148

max

*

, ( ) 0

( ) 0, ( ) 0

( ), ( ) 0

E t

E t t

E t t

. (6.61)

For the singular control to be optimal, we must have ( ) 0H

tE

. This gives

3

t ce p

qx

. (6.62)

According to this principle, the adjoint equations are

31 2, , dd dH H H

dt u dt v dt x

. (6.63)

From the first adjoint equation, we have

2 2

01 21 2 32 22 22 2

1 1

(1 2 ) .(1 ) (1 )

1 1

u v u vv v

xd xu

dt u uu uu u

Using equation (6.18), this equation becomes

0 111 2 22

1

2

22 32 22

1

21

(1 )1

1

.(1 )

1

uuv

uxdu

dt uuu

uv

x

uuu

(6.64)

The second adjoint equation can be written as

22

021 2 32 2 2 2 2 22 2

22.

( ) ( )1 1

vxd u u vx

dt v vu uu u


22 2

021 2 32 2 2 2 2 22

2( ).

( ) ( )1

vxd u vx v

dt v vuu

(6.65)


149

The third adjoint equation can be written as

2

3 01 2 2 2

1

2

0 23 2 2

1

(1 )

2 1 .

(1 )

td u ve pqE

dt u v

v uxs qE

K uv


2

3 01 2 32 2

1

.(1 )

td u v sxe pqE

dt u v K

(6.66)

From equation (6.62), differentiating with respect to t , we have

33.

d

dt

(6.67)

With the help of equation (6.67), equation (6.66) reduces to

2

01 2 2 2

1

.1

tu v sx ce pqE p

u v K qx

(6.68)

Putting the values of 1 in equation (6.65), we get

21 2 2

tdA A e

dt

, (6.69)

where

2 2 21

1 2 2 2 2 22

0

1 ( )

( ) ( )1

u v xv vA

v vuu

,

21 0

2 2 2 22

0

1 2.

( )1

u xvsx c cA pqE p p

K qx qx vuu

Solving equation (6.69), we get 122 0

1

.A ttA

e k eA

We note that when t , then shadow price 2

te is bounded iff 0 0k .

Thus

22

1

.tA e

A

(6.70)

Putting the values of 2 3 and in equation (6.64), we get


150

11 1 2 ,td

B B edt

(6.71)

where

0 11 2 22

1

21

(1 )1

uuv

uxB u

uuu

,

2

2 22 2 22

1 1

1

.(1 )

1

uv

A xcB p

A qx uuu

Solving equation (6.71), we get

121 1

1

.t

B tB ek e

B

We note that when t , then shadow price 1

te is bounded iff 1 0k .

Thus we have

21

1

.tB e

B

(6.72)

After knowing the values of 1 2, and 3 , the value of E can be calculated from

equation (6.66), and it is given by

2

02 2

2 2

1 1 1

1,

( ) (1 ) ( ) ( )

uB Ac sx vE E p

pq qx K B u A v

(6.73)

where E = optimal level of effort.

Hence solving the equation (6.18)-(6.20) with the help of equation (6.73), we get an

optimal solution ( , , )u v x and the optimal harvesting effort E E .

6.8 Conclusions

In this chapter, a mathematical model to study the dynamics of phytoplankton,

zooplankton, fish population has been proposed and analyzed in which functional response


151

are considered to be of Holling type-II, III and IV. The fish population is harvested according

to CPUE hypothesis and harvesting effort is a control variable. The existence of equilibrium

points and their stability analysis have been discussed with the help of stability theory of

ordinary differential equations. The positive equilibrium point is locally and globally

asymptotically stable under a fixed region of attraction when certain conditions are satisfied.

We have observed that model system (6.4)-(6.6) has a chaotic solution for the chosen set of

parameter values. For the different values , measuring the ratio of the predator’s immunity

from or tolerance of the prey to the half saturation constant in the absence of any inhibitory

effect, the system exhibits bifurcation phenomena. The bifurcation and blow-up bifurcation

diagrams exhibits the transition from chaos to order through a sequence of period having

bifurcation. The blow-up bifurcation diagram shows that the model system possesses a rich

variety of dynamical behavior. The chaotic behavior of the system is not continuing further,

as the unstable period-3 orbits which originate at the time of saddle-node bifurcation do not

allow it to move further. Birge (1897) observed that marine habitat to be highly

heterogeneous in factors such as light, nutrients, temperature and oxygen. In lakes, both the

medium (e.g. water and sediments) and the organisms are highly dynamic; currents, wave

action and turbulence render spatial patterns highly ephemeral (Downing (1991). It becomes

clear early in the study of temperate aquatic habitats that they varied in composition spatially

and temporally. The most important temporal variations were perceived to be seasonal

developments of pelagic communities mediated by annual cycles of temperature

(Downing(1991)).

We have discussed the bionomical equilibrium of the model and found the sustainable

yield (h) and maximum sustainable yield ( hMSY ). It has been shown that if h > hMSY, then the

over-exploitation of fish population takes place and if h < hMSY, then the fish population is

under exploitation. By constructing an appropriate Hamiltonian function and using

Pontryagin’s Maximum Principal, the optimal harvesting policy has been discussed. We also

found an optimal equilibrium solution. The idea contained in the chapter provides a better

understanding of the relative role of different factors; e.g., different predation rate of

phytoplankton and zooplankton by fish population and intensity of interference among

individual of predator.

_______________________________________________________________________________ *This content of this chapter has been published in Journal of Biological System,21(3): 1350023,2013.

Chapter 7

Modeling and Analysis of an SEIR Model with Different Types of Non-

linear Treatment Rates*

7.1 Introduction

The wide spread and frequent occurrence of many communicable diseases are intriguing

health care workers and policy makers all over the globe. In order to control or to eradicate a

disease, complete understanding of the dynamics of the disease progression is required. In

one way it can be achieved through mathematical modeling, which has been successfully

used to understand various aspects of many diseases and to suggest its control (Bailey (1975),

Anderson and May (1992), Hethcote (2000), Brauer and Castillo-Chavez (2001)). It has

thereby emerged as an important tool in epidemiology.

According to the International Epidemiological Association (IEA), epidemiology has

three main aims (Lowe and Kostrzewski (1973)):

1) to describe the distribution and magnitude of health and disease priblems in human

populations;

2) to identify aetiological factors (risk factors) in the pathogenesis of disease; and

3) to provide the data essential to the planning, implementation and evaluation of

services for the prevention, control and treatment of disease and to the setting up priorities

among those services.

Since the pioneer work of McKendrick (1925) and Kermack and McKendrick (1927), a

lot of work has been done to understand the infectious diseases. In these models, the entire

population is assumed to be divided into compartments of susceptible individuals (S),

infectious individuals (I) and removed or recovered individuals (R). Susceptible individuals

are those who are healthy and do not have any infection. Individuals who are infected with

the disease and are capable to transfer it to susceptible via contacts are called Infectious

individuals. As time progresses, infectious individuals loose the infectivity, and move to

removed or recovered compartment. These recovered individuals are immune to infectious

microbes and thus do not acquire the disease again.

Ch.-7 An SEIR Model

153

For certain diseases, on contact with infected individuals, susceptibles may not

immediately become infectious. Rather there is latency and after this latent period they

become infectious and can spread infection to susceptible individuals. These individuals who

are exposed to disease pathogen but are yet to become infectious are classified as Exposed

(E). The model along with exposed individuals corresponds to SEIR type (Liu et al. (1987),

Li and Muldowney (1995), Shulgin et al. (1998), Hethcote (2000), d'onofrio (2002), Li et al.

(2006), Bai and Zhou (2012)).

Epidemic models with exposed class have been well studied in literature (Liu et al.

(1987), Hethcote and Levin (1989), Greenhalgh (1997), Li et al. (2006)). Li et al. (2006)

analyzed the global stability of an SEIR epidemic model considering constant immigration

and infectious force in exposed, infected and recovered class. Li and Muldowney (1995)

considered the global dynamics of an SEIR model with a non-linear incident rate using the

geometric method. Liu et al. (1987) studied an epidemiological model with nonlinear

incidence rate instead of bilinear incidence. They investigated rich dynamics of the model

and established that the clearance of disease depends not only on parameters but also on its

initial values. They also showed that for some parametric values periodic solution may exist

through Hopf bifurcation. Greenhalgh (1997) studied dynamical properties of the model first

with constant infection rate and then with density dependent infection rate together with

density dependent death rate. It was observed that in case of constant contact, for disease

persistence, there is only one threshold parameter while in case of density dependent contact

there exists three threshold conditions. They also found Hopf birfucation in later case.

Korobeinikov (2004) established global stability for SEIR models with nonlinear incidence

by constructing a suitable Liapunov function.

Greenhalgh (1992) considered an SEIR model with density dependent death rate and

constant infection rate. They found that the system possess three equilibria: (i) when the

population becomes extinct, (ii) when the disease is eradicated and (iii) when the disease is

present within the population. They found that it is possible for this third equilibrium to exist

and be locally unstable. They found, numerically, the occurrence of cycles of disease

incidence with varying amplitude for different parametric values. Li. et al. (1999) established

the global stability of SEIR model with varying total population size and they further studied

an SEIR model with vertical transmission in a constant population and incidence term in the

form of bilinear mass action (Li et al. (2001)).

Ch.-7 An SEIR Model

154

Li et al. (2008) classified the individuals in infected class into the early and later stages

of infectives according to infection progression. The global stability of the disease-free

equilibrium and the local stability of the endemic equilibrium are established. Further, the

global stability of the endemic equilibrium is obtained when infection is not fatal.

Mukhopadhyay and Bhattacharyya (2008) considered distinct incidence rates for the exposed

and the infected populations in an SEIS model, which they further extended, to account for

the non-homogeneous mixing, by considering the effect of diffusion on different population

subclasses. The diffusive model is analyzed using matrix stability theory and conditions for

Turing bifurcation are derived. Acedo et al. (2010) obtained an exact global solution using

modal expansion infinite series for the classical SIRS model. They showed that for real initial

conditions the modal expansion series present a convergent behavior.

In recent years, classical epidemiological models with add-on of certain new factors such

as different treatment plans, quarantine, vaccination, bacteria, etc. have been investigated

(Bai and Zhou (2012), Hyman and Li (1998), Shulgin et al. (1998), Wu and Feng (2000),

Wang and Ruan (2004), Cui et al. (2008), Hu et al. (2008), Zhang and Suo (2010), Elbasha et

al. (2011), Shukla et al. (2011)) as treatment and isolation or quarantine are important ways

to control spread of the disease. Bai and Zhou (2012) studied global properties of an SEIRS

epidemic model with periodic vaccination and seasonal contact rate. Elbasha et al (2011)

studied the qualitative dynamics of an SIRS model with vaccination, with waning natural and

vaccine-induced immunity and found that model undergoes backward bifurcation depending

upon the protection level of the vaccine. Hu et al. (2008) considered an epidemic model with

standard incidence rate and treatment rate of infectious individuals. Here the treatment rate is

taken to be capacity dependent. Further, it is proportional to the number of infectives below

the capacity and a constant when the number of infectives exceeds the capacity. The

existence and stability of equilibria are found to be dependent on both basic reproduction

number and the capacity for treatment. For small capacity, a bistable endemic equilibria has

been determined.

Ghosh et al. (2004) proposed an SIS model for carrier-dependent infectious diseases

caused by direct contact of susceptibles with infectives as well as by carriers by assuming the

growth of both the human and the carrier populations logistic. They also assumed that the

density of carrier population increases with the increase in the cumulative density of

discharges by the human population into the environment. They found that if the growth of

Ch.-7 An SEIR Model

155

carrier population caused by conducive household discharges increases, the spread of the

infectious disease increases. Ghosh et al. (2006) investigated an SIS model for bacterial

infectious disease where the growth of human population is logistic. They concluded from the

analysis that the spread of the infectious disease increases when the growth of bacteria caused

by conductive environmental discharge due to human sources increases. Naresh et al. (2008)

proposed and analyzed a nonlinear mathematical model for the spread of carrier dependent

infectious diseases in a population with variable size structure and they studied the role of

vaccination. It is assumed here that the susceptible becomes infected by direct contact with

infectives and/or by the carrier population present in the environment. They obtained a

threshold condition, in terms of vaccine induced reproduction number which is, if less than

one, the disease dies out in the absence of carriers provided the vaccine efficacy is high

enough, and otherwise the infection persists in the population. The model also exhibits

backward bifurcation when the vaccine induced reproduction number is equal to one. They

have also shown that the constant immigration of susceptible makes the disease more

endemic. Naresh et al. (2009a) proposed and analysed a nonlinear mathematical model and

studied the spread of HIV/AIDS in a population of varying size with immigration of

infectives. They found that the disease is always persistent if the direct immigration of

infectives is allowed in the community. Further, in the absence of inflow of infectives, the

endemicity of the disease is found to be higher if pre-AIDS individuals also interact sexually

in comparison to the case when they do not take part in sexual interactions. Thus, if the direct

immigration of infectives is restricted, the spread of infection can be slowed down. Naresh et

al. (2009b) further developed an SIR model with nonlinear incidence rate and time delayed.

They established the existence and stability of the possible equilibria in terms of a certain

threshold conditions and basic reproduction number.

Gerberry and Milner (2009) proposed an SEIQR model for childhood disease

considering quarantined class (infectious individuals who are isolated from further contacts

for some time, usually the length of infection period). They found that the inclusion of

quarantined class in the classical SIR model for childhood disease can be responsible for self-

sustained oscillations. Recently, Liu et al. (2012) investigated the application of pulse

vaccination strategy to prevent and control some infectious diseases. This is described by

age-structured SIR model in which susceptible and recovered individuals are structured by

chronological age, while infected individuals are structured by infection age.

Ch.-7 An SEIR Model

156

It is known that the capacity of any community for treatment of a disease is limited,

Wang and Ruan (2004) considered a SIR epidemic model with constant removal rate (i.e., the

recovery from infected subpopulation per unit time) as given below:

, 0( ) .

0, 0

r Ih I

I

Then they performed stability analysis and showed that this model exhibits various

bifurcations. Zhang and Suo (2010) modified the removal rate to Holling type-II:

( ) , 0; , 01

Ih I I

I

.

They discussed the stability of equilibria and established that the model exhibits

Bogdanov-Takens bifurcation, Hopf bifurcation and Homoclinic bifurcation.

It may be noted that although various types of epidemiological models have been studied

and analyzed but SEIR model with different removal rates of Holling type-III and type IV are

yet to be studied. Here in this chapter, we consider an SEIR type model with these two

different removal rates to account for the different treatment capacity of the community. In

fact in Holling type-II, for any outbreak of the disease its treatment capacity is first very low

and then grows slowly with improvement of hospital’s condition and availability of effective

drugs etc. Further, when number of infected individuals is very large, the treatment capacity

reaches to its maximum due to limited treatment facilities. This condition pertains to newly

emergent diseases whose treatment is very limited.

In the present chapter, we have considered the following two types of removal rate

functions:

Case I- Holling type-III: 2

2( ) , 0, , 0.

1

Ih I I

I

Case II-Holling type-IV: 2

( ) , 0, , , 0I

h I I a bI

I ba

.

Holling type-III defines the condition in which removal rate first grows very fast initially

with increase in infective and then it grows slowly and finally settles down to maximum

Ch.-7 An SEIR Model

157

saturated value. After this any increase in infective will not affect the removal rate. This

condition pertains to a known disease which has re-emerged and has available treatment

modalities. Whereas in Holling type-IV, the removal/treatment rate initially grows with

growth of infective and reaches the maximum and then starts decaying. Such a situation may

arise due to limitation in availability of treatment for large number of infected individuals.

When supplies of treatment (medicine, immunization, etc.) are depleted, then inspite of high

number of infectives the available treatment becomes very low. This case may arise when

there is re-emergence and spread of disease in place of limited treatment facilities.

In this chapter, we have considered an SEIR epidemic model with Holling type-III

removal rate function. We have then discussed the positivity, boundedness and stability of the

model system. Further, the removal rate function is modified to Holling type-IV and

corresponding stability analysis is performed. A brief analysis is also presented when the

treatment rate function is of Holling type-II. Finally, numerical simulations are carried out to

support the analytical findings.


We consider an SEIR model with removal of infectious population via treatment,

assuming perpetual immunity of removed individual. As mentioned in the introduction we

shall consider two different removal rate functions (treatment functions). We assume that the

total population is divided into four compartments of Susceptible (S), Exposed (E), infectious

(I) and Recovered (R) classes. Let susceptibles be recruited in population at a constant rate A

and 0 be the natural death rate for the population in all classes. The susceptibles become

infected on contact with infected individuals. This interaction is considered to be of mass

action type. is rate at which susceptible is exposed to the infection. Upon infection the

susceptible individuals move to exposed class and only after latency they become infective

and move to infectious class. Let 1 be rate at which an individual leaves the exposed class

and becomes infective i.e.1

1

is the latency period. The infectious individuals are assumed to

leave via natural and disease related death as well as natural recovery. Let 2 be natural

recovery rate of the infectious individuals and hence 2

1

is the infectious period. Let 3 be

the death rate of the infectious individual due to infection. Removal of infectious individual is

Ch.-7 An SEIR Model

158

also made by treatment using removal rate function. Let ( )h I be recovery rate of the

infectious subpopulation through treatment which we consider of following two types:

Holling type-III and Holling type-IV. The schematic diagram of the interacting sub-

populations is shown in Fig. 7.1.

Fig. 7.1: Schematic diagram of the interacting populations.

Here we assume that after recovery the individuals become immunized and hence they

are no longer susceptible to it. This happens because acquired immune response leads to

development of immunological memory and therefore individual is not infected from the

same disease again and does not enter the susceptible population.

Keeping the above assumptions in mind, the mathematical model can be governed by the

following system of ordinary differential equations:

0

0 1

0 2 1 3

2 0

,

,

( ),

( ),

dSA S SI

dt

dEE E SI

dt

dII I E I h I

dt

dRI R h I

dt

(7.1)

(0) 0, (0) 0, (0) 0, (0) 0.S E I R

Ch.-7 An SEIR Model

159

Since removed class R does not have any effect on the dynamics of , S E and I class, we

shall study following reduced system:

0

0 1

0 2 1 3

,

,

( ),

dSA S SI

dt

dEE E SI

dt

dII I E I h I

dt

(7.2)

(0) 0, (0) 0, (0) 0.S E I

Note that the dynamical properties of R are completely determined once we know those

of , S E and I i.e. all the property of system (7.1) can be drawn from those of system (7.2).

Hence we shall focus our study on the model system (7.2).

In the following, considering the saturated removal rate and supposing the removed

individuals with perpetual immunity, the SEIR epidemic models are formulated for two

different treatment rate functions. We study both the cases one by one.

Case I: Holling type-III:

We take 2

2( ) , 0, , 0.

1

Ih I I

I

It is easy to see that the treatment rate ( )h I is a continuously differentiable increasing

function of I and

) (0) 0,

) ( ) 0,

) lim ( ) ,I

i h

ii h I

iii h I

where

is the maximal treatment capacity of some community.

Thus in this case model (7.2) takes the form as:

Ch.-7 An SEIR Model

160

0

0 1

2

0 2 1 3 2

,

,

,1

dSA S SI

dt

dEE E SI

dt

dI II I E I

dt I

(7.3)

(0) 0, (0) 0 and (0) 0 .S E I

It is well known that the dynamics of a disease is closely related to the stability of the

equilibrium points of mathematical models. In classical epidemiological models there usually

exists two equilibrium points: a disease free equilibrium and a unique endemic equilibrium.

The stability of the disease free equilibrium is determined by a threshold parameter 0R ,

known as the basic reproductive number, i.e. the disease free equilibrium is asymptotically

stable if 0 1R , while it is unstable if 0 1R . Biologically, the disease can be cured if 0 1R ,

and it persists if 0 1R .

In the next section, we present the stability analysis of model (7.3).


First of all, we analyse model (7.3) which has been obtained from model (7.1) taking

( )h I as mentioned in Case I. In the following lemma, we establish a region of attraction for

model system (7.3). The proof of this lemma is similar to that of Freedman and So (1985),

Shukla and Dubey (1997) and hence is omitted.

Lemma 7.1

The set

0

{( , , ) : 0, 0, 0, }A

S E I S E I S E I

is a positively invariant region of system (7.3).

Ch.-7 An SEIR Model

161

The above lemma shows that all solutions of our model (7.3) are non-negative and

bounded. Thus, the model is biologically well-behaved.

In the following, we discuss the existence of equilibria. We note that model (7.3) has one

disease free equilibrium 0

0

,0,0A

P

and one endemic equilibrium * * * *( , , )P S E I . The

disease free equilibrium 0P always exists. For the existence of endemic equilibrium *P , we

note that * * *, , and S E I are the positive solutions of the following algebraic equations:

0

AS

I

, (7.4)

0 1 0( )( )

A IE

I

, (7.5)

2

0 2 3 1 2( ) + 0

1

II E

I

. (7.6)

Substituting the value of E from equation (7.5) in equation (7.6), we get a cubic

equation of the form:

3 2 0KI LI MI N , (7.7)

where

0 1 0 0 1

0 2 3 0 1

, , , ,

, .

K pq L pq A q M pq q N pq A

p q

We note that equation (7.7) has a real positive root *I if the following inequalities hold:

0

0

1 1Rp

, (7.8)

where 10

0

AR

pq

is the basic reproductive number.

Knowing the value of *I , the values of *S and *E can then be computed from equations

(7.4) and (7.5) respectively. This shows that the endemic equilibrium *P exists under

condition (7.8).

Remark:

If 0R exceeds the upper bound given by (7.8) for the existence of endemic equilibrium,

then there may exist more than one endemic equilibrium points. However, in this chapter

we are concentrating on the existence of unique equilibrium and its stability.

Ch.-7 An SEIR Model

162

Now we study the local and global stability behavior of the above two equilibria. To

study the local stability behavior of the equilibria of model system (7.3), we compute the

variational matrices corresponding to each equilibrium point.

Let 0M be the variational matrix corresponding 0P . Then we have

0

0

0

0

1

0

0

0

A

AM q

p

.

From the above matrix, we note that one eigenvalue of 0M is 0 of (which is negative)

and other two eigenvalues have negative real parts if only if 0 1R . Thus we can state the

following theorem.

Theorem 7.1

The disease free equilibrium point 0

0

,0,0A

P

is locally asymptotically stable if 0 1R

and is a saddle point with two dimensional stable manifold and one dimensional unstable

manifold if 0 1R .

Now the variational matrix *M corresponding to *P is given

* *

0

* * *

*

1 2*2

0

20

1

I S

M I q S

Ip

I

.

The characteristic equation of *M is given by

3 2

1 1 1 0A B C ,

where

Ch.-7 An SEIR Model

163

**

1 0 * 2

20

(1 )

IA I p q

I

,

*

* * *

1 0 0 1*2 2

2

(1 )

IB q I p q I S

I

,

*

* *

1 0 0 1*2 2

2

(1 )

IC q I p S

I

.

By the Routh-Hurwitz criteria, all eigenvalues of *M will have negative real parts if and

only if

1 1 1 1 10, 0 and A C A B C .

We note that 1A is always positive and thus we can state the following theorems:

Theorem 7.2

The endemic equilibrium * * * *( , , )P S E I is locally asymptotically stable iff the following

inequalities hold:

1 1 1 10 and C A B C . (7.9)

In the next theorem, we are able to find sufficient conditions for *P to be globally


Theorem 7.3


2 2

0A , (7.10)

2 2 *

1 2 0 1 1 2 0 ( )k A k k k q p k . (7.11)

where 1 2 ,k k and *k are

2

*

01 2 *

( )

2

q Ik

I

and

2 2

2 2 * *

0 0

2,

( )( )

Ak

I p k

where 2

* ** 0

* 2 2

0

.(1 )( )

Ik

I A

Ch.-7 An SEIR Model

164

Then *P is globally asymptotically stable with respect to all solutions initiating in the

interior of the positive octant .

Proof: Consider the following positive definite function about *P :

2 2 2* * *

1 2

1 1 1( )

2 2 2V t S S k E E k I I , (7.12)

where 1 2 and k k are positive constants to be chosen suitably later on. Differentiating V with

respect to time t along the solutions of model (7.3), a little algebraic manipulation yields:

* 2 * * * 2

11 12 22

* 2 * * * 2

11 13 33

* 2 * * * 2

22 23 33

1 1( ) ( )( ) ( )

2 2

1 1 ( ) + ( )( ) ( )

2 2

1 1 ( ) ( )( ) ( ) ,

2 2

dVa S S a S S E E a E E

dt

a S S a S S I I a I I

a E E a E E I I a I I

(7.13)

where

**

11 0 22 1 33 2 2 *2

*

12 1 13 23 1 2 1

( )0, 0, ,

(1 )(1 )

, , .

I Ia I a k q a k p

I I

a k I a S a k S k

Sufficient conditions for dV

dt to be negative definite are that the following inequalities

hold:

2

12 11 22a a a , (7.14)

2

13 11 33a a a , (7.15)

2

23 22 33a a a . (7.16)

If we choose 2

*

01 2 *

( )

2

q Ik

I

, then the condition (7.14) is automatically satisfied.

If we choose

Ch.-7 An SEIR Model

165

2 2

2 2 * *

0 0

2,

( )( )

Ak

I p k

where * 2

* 0

*2 2 2

0

,(1 )( )

Ik

I A

then (7.10) (7.15). We also note that (7.11) (7.16). This shows that V is a Liapunov

function with respect to all solutions initiating in the interior of the positive octant ,

proving the theorem.

We note that if and A are small enough and is large enough, then conditions (7.10)

and (7.11) are satisfied. When decreases and increases, the treatment rate ( )h I

increases. This implies that by decreasing and by increasing , the treatment rate can be

increased and the system can be made globally asymptotically stable. It may also be noted

that by decreasing the recruitment rate A , the possibility of the system to be globally

asymptotically stable is more plausible.

The above analysis shows that as the treatment is increased, the endemic equilibrium

which was previously locally asymptotically stable will be globally asymptotically stable.

The local stability shows that the disease persists in the neighborhood of endemic

equilibrium. It does not give any information outside the neighborhood. The globally stability

shows that the disease persists in a finite region .

From eq. (7.6), it can easily be seen that as the treatment rate 2

2( )

1

Ih I

I

increases,

the equilibrium level of infectives decreases, and consequently from eqs. (7.4) and (7.5) it

follows that equilibrium level of susceptibles increases whereas the equilibrium level of

exposed individuals decreases. Thus, we conclude that higher level of treatment would cause

reduced level of infection in the population which in turn would lead to lesser number of

people getting exposed and more number of people susceptible.

Case II: Holling type-IV:

Now we shall take

2( ) , 0, , , 0

Ih I I a b

II b

a

.

We note that ( )h I is continuously differentiable function which satisfies:

Ch.-7 An SEIR Model

166

2

2 2 2 2

0 022

) (0) 0,

) ( ) 0 if and ( ) 0 if ,

) lim ( ) .a

i h

Ib

aii h I A ab h I A ab

II b

a

Iiii h I

I b

In this case we have the following system:

0

0 1

0 2 1 3 2

,

,

,

dSA S SI

dt

dEE E SI

dt

dI II I E I

IdtI b

a

(7.17)

(0) 0, (0) 0 and (0) 0S E I .

It can easily be verified that model system (7.17) has a disease free equilibrium

0

0

,0,0A

P

, which always exists and an unique endemic equilibrium ( , , )P S E I which

exists if the following inequalities hold:

11 mR ,

where 11

0

AR

q pb

is the basic reproduction number for model system (7.17) and

00

0 0

( )( )min ,m

p bp a

p pb b

.

By the Routh-Hurwitz criteria, the following results can be established immediately.

Ch.-7 An SEIR Model

167

Theorem 7.4

The disease free equilibrium 0P is locally asymptotically stable if 1 1R and is a saddle

point (with two dimensional stable manifold and one dimensional unstable manifold) if

1 1R .

Theorem 7.5

The endemic equilibrium point ( , , )P S E I is locally asymptotically stable iff the

following inequalities hold:

2 20, 0A C and 2 2 2A B C , (7.18)

where

2 0A p q I J ,

2 0 0 1B q p J I p J q I S ,

2 0 0 1C I q p J S ,

and

2

22

( ).

Ib

aJ h I

II b

a

Remark :

If 2I ab , then 2 0.A

We have the following result for the global stability of endemic equilibrium P .

Theorem 7.6


2 2

0A ab , (7.19)

0bI

A

, (7.20)

2 2

3 4 0 1 3 4 0( ) ( )k A k k k q p k , (7.21)

Ch.-7 An SEIR Model

168

where 3 4,k k and k are given below.

Then P is globally asymptotically stable with respect to all solutions initiating in the

interior of the positive octant .

Proof: We take the following positive definite function about the equilibrium point P

2 2 2

3 4

1 1 1( ) ( ) ( ) ( )

2 2 2V t S S k E E k I I , (7.22)

Under an analysis similar to the proof of Theorem 7.3 and by choosing

03 2

( )

2

q Ik

I

and

2 2

4 2

0 0

2

( )( )

Ak

I p k

and

0

2 2

2

0 0

AIb

ak

I A AI b b

a a

,

it can be cheeked that V is a Liapunov’s function under conditions (7.19)-(7.20) and hence

the theorem follows.

From the above theorem, we note that by decreasing the recruitment rate A of the

susceptible population, conditions (7.19) and (7.20) can be made satisfied. By increasing the

values of parameters a and , the treatment rate ( )h I can be increased and in such a case

condition (7.21) may be satisfied. This shows that by decreasing A and by increasing a and

, the system can be made globally asymptotically stable. Again as in the case of Holling

type-III recovery, it can be seen from the equilibrium analysis of model (7.17) the infected

individuals will settle down at a lower equilibrium level and susceptible individuals at higher

equilibrium level by increasing the treatment rate.

Remark:

Holling type-II functional response is a special case of Holling type IV. It may be noted

that if a is very large enough, then the treatment rate function 2

( )I

h II

I ba

reduces

to ( )1

I Ih I

I b I

, where

1

b and it has been studied without exposed class by

Zhang and Suo (2010). However, we shall mention briefly the results obtained for model

Ch.-7 An SEIR Model

169

(7.2) when ( )1

Ih I

I

. In such a case, model (7.2) has two equilibrium points,

namely, 0

0

,0,0A

P

and * * * *( , , )P B N E .

The equilibrium point 0

0

,0,0A

P

is

i) locally asymptotically stable if 2 1R , and

ii) unstable if 2 1R ,

where 12

0 ( )

AR

q p

is the basic reproduction number for model (7.2) in the case of

Holling type-II treatment rate.

The equilibrium point * * * *( , , )P B N E exists if the following inequalities hold:

2

0

1p

RP

.

Under an analysis similar to the previous cases, we can state the following results.

Theorem 7.7

The equilibrium point * * * *( , , )P B N E is locally asymptotically stable iff

0 0C and 0 0 0A B C ,

where

*

0 0 * 20

(1 )A p q I

I

,

* * *

0 0 0 1* 2 * 2(1 ) (1 )B q p I p q I S

I I

,

Ch.-7 An SEIR Model

170

* * 2 * *

0 0 1 1* 2(1 )C I q p S S I

I

.

Theorem 7.8

The equilibrium point * * * *( , , )P B N E is globally asymptotically stable if the condition

holds in :

2

1 2 1 1 2 *

0 (1 )

Al l l l q p

I

,

where

*

01 2 *2

( )

2

I ql

I

and

2 2

2

2 *

0 0 *

2

( )(1 )

Al

I pI

.


In order to illustrate the results obtained in previous sections, we performed computer

simulations using MATLAB 6.1. We choose the following set of values of parameters (other

set of parameters may also exist)

0 1 2 37.0, 0.02, 0.2, =0.025, 0.03, 0.003, =0.3, =0.04A , (7.23)

with initial conditions (0) 210.0, (0) 10.0, (0) 8.0.S E I

For the above set of values of parameters, it can easily be seen that condition (7.8) for the

existence of the interior equilibrium * * * *( , , )P S E R is satisfied. Thus,

* * * *( , , )P S E R exists

and it is given by

* * *252.0905, 8.9009, 2.5893.S E I

It may further be noted that both the inequalities in eq. (7.9) are satisfied for the set of

values of parameters given in (7.23). This shows that the interior equilibrium *P is locally


Ch.-7 An SEIR Model

171

Now we present the simulation results of model (7.3) in figures (7.2-7.5), and the

comparative results of models (7.3) and (7.17) in figures (7.6-7.8). The behavior of ,S E and

I with respect to time t are plotted in Fig. (7.2) from which we note that the trajectories

approach to the endemic equilibrium *(252.0905,8.9009,2.5893)P . It may also be noted here

that conditions (7.10) and (7.11) in Theorem-7.3 are not satisfied for the set of values of

parameters chosen in (7.23). Since conditions (7.10)-(7.11) are sufficient (not necessary) for

*P to be globally asymptotically stable, no conclusion can be drawn at this stage regarding

global stability behavior of *P .

It may be noted from Fig. (7.2a) that the number of susceptible individuals increases

sharply with time till a steady state is acquired. This increase in susceptible individuals is

dependent on the recruitment rate (A) that is greater than the decline caused due to its natural

death rate and infection. In Fig. (7.2b) it is observed that the exposed individuals initially

decrease and then increase to get stabilized at a higher level. The initial decrease in the

number of exposed individuals is due to either natural death or transfer of the exposed

individuals to the infected class. Whereas later the increase in exposed individuals takes place

when there are large number of infected people in the population. Fig. (7.2c) shows a sharp

decline in number of infected individuals followed by a enhancement and settlement at an

elevated level. The decrease in the number of infected individuals is due to natural death,

disease induced death or natural recovery from the disease. Moreover many of these infected

individuals also respond to treatment. But some exposed individuals also enter the infected

class, thus the equilibrium value settles down at a relatively elevated level.

Now we choose another set of values of parameters as given below.

0 1 2 31.2, 0.05, 0.2, 0.003, 0.03,

0.03, =0.3, =0.00027

A

. (7.24)

With the above set of values of parameters, we note that the interior equilibrium

* * * *( , , )P S E I exists and it is given by

* * *14.9409, 1.8118, 1.0105.S E I

It may be noted here that the value of in eq. (7.24) is much more smaller than its

value in eq. (7.23). Thus, for the values of parameters chosen in eq. (7.24), the treatment rate

Ch.-7 An SEIR Model

172

is high in comparison to the treatment rate corresponding to the values of parameters chosen

in eq.(7.23).

Fig.7.2: 7.2a (Left): Susceptible individuals, 7.2b (Middle): Exposed individuals and 7.2c

(Right): Infected individuals approaching endemic equilibria *(252.0905,8.9009,2.5893).P

Fig. 7.3: Global stability of * 14.9409, 1.8118, 1.0105P . Trajectories initiating at

different initial values (IV) as mentioned in text approach and enter the infected steady state

*P .

Ch.-7 An SEIR Model

173

It can easily be seen that all conditions of Theorems- 7.2 and 7.3 are satisfied for the set

of values of parameters given in (7.24). This shows that *P is locally as well as globally

asymptotically stable. We also plotted trajectories starting with different initial values (IV).

Trajectories in green starts at IV1 = (210, 10, 6), red starts at IV2 = (150, 20, 5), blue starts at

IV3 = (250, 2, 20), black starts at IV4 = (25, 2, 20) and magenta starts at IV5 = (20, 2, 5). We

note that all the trajectories (see Fig.7.3) approach and enter * (14.9409,1.8118,1.0105)P .

We studied the effect of variation in infection rate for the set of values of parameters

given in eq. (7.23). The corresponding trajectories are plotted in Fig.(7.4a) and (7.4b). Fig.

(7.4a) shows that S decreases as increases. It seems that the point at which final

equilibrium attained depends on the rate of infection (infectivity). If is in a smaller range,

then S increases with respect to time and settles down at its equilibrium level for a fixed .

This implies that in a given population when the infection rate is less, then more people are

susceptible. This may be due to the fact that the recruitment rate is much higher than the

decline due to conversion to exposed class. When increases beyond a certain level,

initially S decreases and after certain time it starts increasing and then attains its steady state.

This could be happening because at high infectivity more people would get converted to

exposed class as compared to the people getting recruited in susceptible class. The decline in

susceptible class may take place because the infected individuals which has recovered would

also develop acquired immunity and thus no longer be susceptible to the re-infection by the

same causative agent. An initial decline in susceptible followed by a slight increase in type III

recovery may also be due to the fact that as the number of infected increases, the treatment

increases and subsequently the recovery rate also increases. This forces the number of

infected to decrease and therefore an increase in susceptible. Besides this, some new member

will also be recruited by birth to susceptible class and thus settles down at a slightly elevated

level.

From Fig. (7.4b), we note that I increases as increases. If is large enough, then I

initially increases rapidly and after showing an oscillatory behavior it obtains its equilibrium

level. This would happen because as the rate of infection increases there would be a sudden

increase in the number of infected people (because of conversion from susceptible). This

number is subsequently brought down due to effective treatment which controls the situation

and keeps the population in a steady state. This steady state is attained at a higher value in

Ch.-7 An SEIR Model

174

diseases with higher infectivity and limited treatment such as AIDS. In such cases a large

number of people may get infected.

If the rate of infection is small, I settles down at a lower value of steady state after

showing oscillatory behavior. When infectivity is less, then obviously lesser people will get

infected initially but gradually with time the number of infected people will built up till it

attains a steady state. Note that the final steady state depends upon both infectivity as well as

treatment. But in case of mild infections the number of people getting infected will remain

relatively low, thus settling at lower value. It may be noted that the legends for Fig.(7.4a) &

Fig.(7.4b) are same.

Fig. 7.4a and 7.4b: Effect of variation in α on the solution trajectories of susceptible and

infectious populations.

Fig. (7.5a) shows the behavior of infectives with respect to time t in the absence of any

treatment ( 0) . From this figure we note that infected individuals increase very fast, then

decrease and settle down at a relatively higher equilibrium level. This decrease may be due to

the recovery of infected individuals by the immune response of the body. Since only immune

response is not enough to control the infection, thus infected individuals settle down at a

relatively higher equilibrium level. Fig. (7.5b) shows the effect of treatment rate on the

infectious population I with respect to time t. This figure shows that I decreases as the

Ch.-7 An SEIR Model

175

treatment rate increases. It is also observed that the number of infectious populations

initially decreases for some time and with little increase it settles down at its equilibrium

level. This is due to the fact that as the disease is not getting totally eradicated it will persist

in the population but at a much lower level. This pattern observed here may be due to

treatment as well as immune response.

a)

b)

Fig. 7.5(a) and 7.5(b): Behavior of infectives without and with treatment

Ch.-7 An SEIR Model

176

The effect of treatment in the case of Holling type-III and IV recovery are shown in Figs.

(7.6a) and (7.6b), respectively. Fig. (7.6a) shows that as the treatment rate increases, the

number of infected individuals decreases and it tend to zero at high treatment rate ( 15) .

Fig. (7.6b) shows that I decreases as increases. If is high ( 15) in the case of type-IV

a)

b)

Fig. 7.6(a) and 7.6(b): 7.6(a) -Effect of different treatment on infectives with Holling

type-III and 7.6(b)-Holling type-IV removal rate function.

Ch.-7 An SEIR Model

177

recovery the infected individuals show an oscillatory behavior and finally it settles down at a

higher equilibrium level as compared to the type-III recovery. It may be noted here that in

case of type-III recovery (Fig.7.6a), the number of infective has drastically declined to almost

zero for intermediate and high treatment values whereas when the treatment is very less, then

the infected population stabilizes at a relatively high level. In contrast to this in type-IV

recovery (Fig. 7.6b), when the treatment is low or intermediate the infective have settled

down at a high value. But when the treatment is high, then the infective have stabilized at a

lower level but not negligible (unlike type-III recovery). This is because in Holling type III,

the treatment persists at saturation level and therefore the number of infective declines to

almost zero at intermediate and high levels of treatment but remains persistent at low level of

treatment. Whereas in Holing type-IV, the treatment itself declines (due to resource

limitation) after attaining a maximum value, and thus the infection is never cleaned from the

population.

The dynamics of infectious population with Holling type-III and IV treatment rates are

presented in Fig. (7.7a) and (7.7b) respectively. For Holling type-III treatment rate we chose

0.05 and then for Holling type-IV treatment rate we choos 15, 150, 75a b ,

keeping all other parameters same as in eq. (7.23). We plotted the solution trajectories

initiating from different initial points: S(0) = 250, E(0) = 10 and for I(0) we choose 8, 15 and

25 respectively.

It may be noted from Fig.7.7(a) that for smaller initial value of infectious populations

(I(0) = 8), the infectious individuals first decrease and then settle to steady state whereas for

I(0) = 25, infectious individuals first decrease and show some transient oscillation before

settling to its steady state. In order to show the initial pattern of infectious population clearly,

we plotted the trajectories for first twenty days in a separate window box. The initial

difference in behavior when the initial population size is high may be attributed to the fact

that in this condition because of higher population density more people will contact disease

leading to increased infection in population. But with time this gradually decays and settles

down at the equilibrium level. But for low initial population size, there is only a decline in the

infectives before reaching to its steady state in the case of Holling type-III treatment rate.

Further, in the case of Holling type-IV treatment rate (Fig. (7.7b)) it is observed that infected

population shows oscillations before settling to steady state. As noted from the window box

in Fig.(7.7b), if the number of initially infected individuals is small, then there is an increase

Ch.-7 An SEIR Model

178

in the infected population at first, whereas when initially infected population is more, then

infected population first decreases and then follow the pattern of transient oscillations and

finally settles down at its equilibrium level. Here the situation is slightly different from

Holling type-III treatment function. In the case of Holling type-IV treatment function, in both

small and large population sizes at initial time, there is an oscillatory behavior before settling

to equilibrium.

Fig. (7.7a) (Left) and Fig.(7.7b) (Right): Effect of removal rate functions on Infected

population (I) with Holling type-III (Left) and Holling type-IV (Right) removal rates for

different initial values. The small inset windows in both Fig. (7.7a) and (7.7b) represent

trajectories for initial period.

We also plotted the behavior of susceptible population, in case of Holling type-III and IV

treatment rates for the same set of parameter and initial values as above, in Fig. (7.8a) and

(7.8b) respectively. We note that in case of Holling type-III treatment rate the susceptible

population first decays and reaches to minima and then finally increases to reach and enter

the steady state value for all lower, intermediate and higher population density at initial time

(see Fig (7.8a)).

Ch.-7 An SEIR Model

179

Further in case of Holling type-IV treatment we note that before reaching to steady state

susceptible population shows transient oscillations for all the three initial values (see Fig

(7.8b)). It may further be noted that the behavior of susceptible population is reversed of the

infectious population in both Holling type-III and IV treatment functions. This emphasizes

the fact that susceptible population upon infection get converted to infectious population, and

thus increase in one leads to decrease in another and vice-versa. The susceptible population

first declines before attaining steady state for Holling type-III and IV treatment functions

whereas the infectious population first increases in the case of both Holling type-III and IV

treatment function.

Fig.(7.8a) (Left) and Fig. (7.8b) (Right): Effect of removal rate functions on Susceptible

population (S) with Holling type III (Left) and Holling type IV (Right) removal rates for

different initial values.

7.5 Conclusions

In this chapter, an SEIR mathematical model is proposed and analyzed. This model

consists of four variables, namely, susceptible, exposed, infectious and recovered individuals.

This chapter is, in fact, the generalization of the work (Zhang and Suo (2010)) by

introducing exposed class in the model and by taking two different types of treatment rates.

Ch.-7 An SEIR Model

180

Zhang and Suo (2010) studied an SIR model by taking the treatment rate in the form of

Holling type-II. We have considered two different types of treatment rates for the infected

individuals by taking in the form of Holling type-III and IV. In both the types of treatment

functions, the basic reproduction number 0R and 1R are reported with the characteristic: in

the case of Holling type-III treatment rate, the disease persists if 0

0

1 1Rp

and in the

case of Holling type-IV treatment rate, the disease persists if 11 mR . In both the cases,

the model has been analyzed using stability theory of ordinary differential equations. Criteria

for the disease free equilibrium point and the endemic equilibrium point to be locally and

globally asymptotically stable are obtained. It has been shown that in the case of Holling

type-III recovery the disease free equilibrium point is locally asymptotically stable if the

basic reproduction number 0 1R and unstable if

0 1.R Similar results are obtained in the

case of Holling type-IV recovery function. It has also been found that under certain

conditions, the endemic equilibrium can also be made locally as well as globally

asymptotically stable. The model of Zhang and Suo (2010) is further generalized in the case

of Holling type-II treatment function by taking into account the exposed class. Numerical

simulation has also been performed to investigate the dynamics of interacting subpopulations.

It has been observed that the number of infective as well as susceptible in all three cases

(Holling type-II, III and IV) depend upon the treatment function but do so in a variable

manner. In Holling type II, the treatment slowly increases, then attains its peak and finally

settles down at its saturation value. This case is applicable when availability of treatment is

poor including newly emergent diseases. In such a case a large proportion of population may

get infected as in the case of highly infectious diseases such as HIV.

In Holling type-III, the treatment initially increases fast, then increases slowly, attains its

peak and is finally stabilized. For different initial values, the infected population shows

different patterns. This indicates that the dynamics of infected population depends on the

parameters as well as initial values. Here the quantum of infection (prevalence of disease as

well as the new infection) depends upon the maximum treatment capacity in the community.

The infectivity is high when the treatment is low and vice-versa. Our numerical experiment

also shows that in case of Holling type III treatment rate function, the susceptible individuals

first decrease and attain its minimum value and then starts increasing to enter the steady state.

Ch.-7 An SEIR Model

181

In the case of Holling type-IV, the treatment increases slowly, attains its peak and then

decreases to zero. Here the infected population initially increases and then shows transient

oscillations and finally it is stabilized at a much lower equilibrium level. This pertains to the

situation when the disease has been brought under control by an effective treatment. In case

of Holling type-IV treatment rate function, the susceptible individuals first show transient

oscillations and then settles down at its equilibrium level.

Conclusions

Mathematical modeling of interacting species is very challenging as the system is

complex as well as multifactorial. It involves different components which have non-linear

interaction. Hence, it is also challenging from mathematical point of view to analyse these

non-linear models. Therefore, in this thesis an attempt has been made to explore and analyse

models of interacting biological species for their long time behaviour.

In this thesis, we have developed some mathematical models. The qualitative and

quantitative analyses of these models have been carried out with the help of stability theory

of Ordinary Differential Equations and with the help of computer simulations.

In particular, we have considered :

i) interaction between renewable resource biomass and population taking into account

crowding effect. The resource biomass is harvested according to various types of

catch-rate function. Harvesting rate and taxations are suggested to use as a control

parameter to maintain the resource and the population at an optimal level,

ii) phytoplankton-zooplankton interaction model with self and cross diffusions in

aquatic environment,

iii) a food chain model of phytoplankton-zooplankton-fish, where fish population is

harvested according to catch-per-unit effort hypothesis. Three different types of

functional response (Holling type-II, III and IV) have been taken into account in

modeling the system, and

iv) an SEIR model with different types of treatment rates (Holling type-II, III and IV).

In each model system, boundedness of the system, local stability, instability and global

stability of all possible feasible equilibrium points are studied. Local bifurcation analysis has

also been carried out to understand the dynamics of the species. Optimal harvesting policy

has also been discussed in detail using the Pontryagin's Maximum Principle. Different

thresholds are obtained for the co-existence of resource biomass and biological population.

Some thresholds are also obtained on the harvesting effort and taxation. These thresholds can

be used by a regulatory agency so that the resource as well as the population can be

Conclusions

183

maintained at an optimal level. The maximum sustainable yield (MSY) has been obtained

that may be used as a guideline to harvest the resource biomass upto a maximum level

(MSY). Harvesting of resource beyond MSY will not only threaten the extinction of resource,

it will also decrease the yield.

In the case of an SEIR model, the disease free equilibrium and the endemic equilibrium

are analysed with different types of non-linear treatment rates. The thresholds on the basic

reproductive numbers are obtained to eradicate and to persist the disease.

In all cases, numerical simulations have been carried out to illustrate all theoretical

results. It is hoped that these models will stimulate researchers to pose relevant questions to

unravel the nature of the biological mechanism.

Specific Contributions

In this thesis, some mathematical models have been proposed and analysed to study the

long time behaviour of some biological species. In particular, we have considered the

following:

i) Chapters 2 to 4 deal with the interaction between renewable resource biomass and

population taking into account crowding effect. The resource biomass is harvested

according to various types of catch-rate function. In these chapters, the following

observations are made based on the models analysis:

a) It has been shown that if MSYh h , then the resource biomass will tend to zero,

and if MSYh h , then the resource biomass and population may be maintained

at desired level (where h = harvesting rate, MSYh = Maximum Sustainable

yield).

b) It has been shown that the harvesting effort E should be always kept less than

E to maintain the resource and the population at an optimal equilibrium

level, where E is optimal harvesting effort.

c) We have used tax on the per unit harvested resource biomass as a regulatory

instrument to derive the optimal tax trajectory. It has been shown that if

resource biomass, population and harvesting efforts all are kept along optimal

path, then resource biomass and population both can be maintained at an

appropriate level.

ii) In Chapter 5, phytoplankton-zooplankton interaction model with self and cross

diffusions in an aquatic environment is presented and the following observation are

made:

It has been found that critical wave length of the system plays an important role

in stabilizing the system. It has been observed that if the critical wave

length is very small so that the prey species moves towards the higher

concentration of the predator species and the predator species moves along its


185

own concentration gradient, then in such a case the positive equilibrium which

is stable without self and cross-diffusion becomes unstable in the presence of

self and cross-diffusions.

iii) Chapter 6 deals with a food chain model of phytoplankton-zooplankton-fish, where

fish population is harvested according to catch-per-unit effort hypothesis. Three

different types of functional response (Holling type-II, II and IV) have been used in

modeling the system.

In this chapter, some thresholds on the parameters are obtained that govern the

dynamics of the system. Our analysis shows that the model system has a chaotic

solution for a special set of parametric values as well as exhibits a bifurcation

pattern also. This chapter provides a better understanding of the relative role of

different factors; e.g., different predation rate of phytoplankton and zooplankton

by fish population and intensity of interference among individual of predator.

iv) In Chapter 7, an SEIR model with three different types of treatment rate (Holling

type-II, III and IV) is proposed and analysed. The following observations are made

in this chapter:

a) In Holling type-III, the treatment persists at saturation level and therefore the

number of infective declines to almost zero at intermediate and high levels of

treatment but remains persistent at low level of treatment. In Holing type-IV,

the treatment itself declines (due to resource limitation) after attaining a

maximum value, and thus the infection is never cleaned from the population.

In Holloing type-II, the treatment rate is low in the beginning due to shortage

of effective treatment techniques. Then, the treatment rate increases with the

improvement of Hospital's conditions including effective medicines and

consequently number of infectives decreases. If the number of infectives are

large enough, then the treatment will be saturated due to the limitation of

treatment capacity of the community.

b) It has been observed that the number of infective as well as susceptible in all

three cases (Holling type-II, III and IV) depend upon the treatment function but

do so in a variable manner. In Holling type - II, the treatment slowly increases,

then attains its peak and finally settles down at its saturation value. This case


186

is applicable when availability of treatment is poor including newly emergent

diseases. In such a case a large proportion of population may get infected as in

the case of highly infectious diseases such as HIV.

Future Scope of Work

In this thesis, some mathematical models are proposed to understand the interaction

between prey-predator, SEIR model and hence characteristic behaviour of those models are

analysed there using ordinary differential equation. In these models following aspects are

considered:

i) different types of harvesting rate to protect the resource biomass from over

exploitation taking into account the crowding effect also,

ii) the Rosenzweig-MacArther type model with self and cross diffusions, and

iii) an SEIR model taking treatment rates as Holling type-II, III and IV functional

responses.

It may be pointed out here that the functional response plays an important role on the

dynamics of the system. Again in a biological phenomena, the present dynamics and the

present rate of change of the state variables depend not only on the present state of the

process but also on the history of phenomenon and on the past values of the state variables.

Thus a differential equation with time delay exhibits a more realistic dynamics of the system.

Keeping the above aspect in view, we propose the following future plan that can be carried

out:

The functional responses considered in the models in the thesis can be modified by

Beddington-De-Angelis type and Crowley-Martin type functional responses and

then the dynamics of the system can be investigated,

The effect of time delay can be incorporated in the model and the behaviour of the

system can be studied,

The impulse harvesting rate can be taken in the formulation of modeling and then

the behaviour can be examined,

A ratio-dependent prey-predator model with Holling type-II functional response is

studied by Liu and Yan (2011), and with delay and constant harvesting is studied

by Misra and Dubey (2010). Keeping these two works in mind, we propose to

study the dynamics of a ratio-dependent prey-predator model with delay and

different types of harvesting rates.

Future Work

188

A predator-prey model with Holling type-II functional response and with a

constant prey-refuge has been studied by Chena et al. (2010). The above model is

again generalized (Ji and Wu (2010)) by taking into account the effect of constant

rate of prey harvesting. Keeping these two works in mind, we would like to

propose a model to formulate a more general and realistic predator-prey model by

taking into account the different types of functional responses and different types

of realistic harvesting rate functions with taxation as a control parameter.

The harvesting of a prey-predator model with stage-structure has been studied

recently by Chakraborty et al. (2011), Heng and Wang (2013) with Holling tye-II

functional response. We propose to study a resource dependent prey-predator

model with stage-structure and with optimal harvesting of prey and predator.

We hope that the above future directions of research will focus on some important

problems in the emerging areas of mathematical ecology.

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prey system with cross diffusion'' Non-linear Anal.: RWA. 2010, 11(1): 372-390.

[206] Zhang L., Wang W. , Xue Y. , Jin Z. ''Complex dynamics of Holling-type IV

predator-prey model'' J. Stat. Mech. 2009, P07007.

[207] Zhang X., Chen L., Neumann A. U. ''The stage structure predator-prey model and

optimal harvesting policy'' Math. Biosci. 2000, 168(2): 201-210.

[208] Zhang Z., Suo S. ''Qualitative analysis of a SIR epidemic model with saturated

treatment rate'' J. Appl. Math. Comput. 2010, 34: 177-194.

205

List of Publication

From Thesis Work

Published/Accepted

1. Dubey B., Patra A. ''A mathematical model for optimal management and utilization

of a renewable resource by population'', Journal of Mathematics. Article ID 613706,

Vol. 2013, 9pages, 2013.

2. Dubey B., Patra A. '' Optimal management of a renewable resource utilized by a pop-

lation with taxation as a control variable'', Nonlinear Analysis: Modelling and Control

18(1): 37-52, 2013.

3. Dubey B., Patra A. Srivastava P. K., Dubey U. S. ''Modeling and analysis of an SEIR

model with different types of non-linear treatment rates'', Journal of Biological Syst-

ems, 21(3): pp. 1350023 (25 pages) 2013.

4. Dubey B., Patra A. Upadhyay R. K. ''Dynamics of phytoplankton, zooplankton and

fishery resource model'', Application and Applied Mathematics, 2013.

Communicated

1. Dubey B., Patra A., Upadhyay R. K., Thakur N. K. ''A Predator-Prey Interaction

Model with Self and Cross-Diffusion in an aquatic system'', Journal of Biological

Systems.

2. Dubey B., Patra A., Sahani S. K. ''Modelling the Dynamics of a Renewable Resource

under harvesting with taxation as a control variable '', The Scientific World Journal.

Paper Presented in National/ International Conference

1. Patra A., Dubey B. ''A mathematical model for optimal management and utilization of

a renewable resource by population'', National Conference on Modeling,

Computational Fluid Dynamics & Operations Research, Birla Institute of Technology

& Science, Pilani, 4-5 Feb, 2012.

2. Patra A., Dubey B. '' Dynamics of a renewable resource model under optimal

harvesting'', 100th

Indian Science Congress, Calcutta University, Kolkata, 3-7 January,

2013.

206

Brief Biography of the Supervisor

Dr. Balram Dubey is Professor in the Department of Mathematics, Birla Institute of

Technology & Science, Pilani, Pilani Campus, Rajasthan. He stood first rank of merit at

Intermediate of Science (I. Sc.) in Bhagalpur University and third rank of merit at B. Sc. pass

course in Bhagalpur University. He was first rank merit at B.Sc.(Hons) in Bhagalpur

University and has broken all the previous records of that University. He has completed his

M.Sc. degree in Mathematics from Indian Institute of Technology, Kanpur. He obtained his

Ph.D. degree in the year 1994 from IIT, Kanpur. He was Research Associate in IIT, Kanpur

from 1994-1995. After that Dr. Dubey joined as a Lecturer in Department of Mathematics,

Tezpur University (central). Then he joined IIT, Kanpur as a visiting faculty from 2000-2002.

During this time he was awarded ''Best Teaching Award'' for tutorship in MATH102 in 2001.

After that he joined as an Assistant Professor in Department of Mathematics in Birla Institute

of Technology & Science, Pilani, Pilani campus in 2002. He became Associate Professor in

August 2010 and Professor in February 2013. His research interests are in the area of

Mathematical Modeling of Ecological, Biological, Environmental and Engineering Systems.

As a results of his research accomplishment, he has published more than 60 research articles

in national and international journals of repute. He is also the author of the book

''Introductory Linear Algebra'', Asian Books Pvt. Ltd. 2007.

207

Brief Biography of the Candidate

Atasi Patra obtained her B.Sc.(Hons) in Mathematics in 1998 from Vidyasagar

University, Midnapore, West Bengal and M. Sc. in Mathematics in 2000 from Jadavpur

University, Kolkata. Then she served as a Assistant Teacher (Govt.) at Purbasukutia

Gopalkrishna Balika Vidyalay, Panskura, WB for six years from 2002 to 2008. She qualified

SLET (State Level Eligibility Test in Lectureship, West Bengal) in 2003 and also West

Bengal School Service Commission Examinations twice for assistant teachership in 2001 and

2003. Later she moved to Birla Institute of Technology & Science, Pilani in order to pursue

her Doctor of Philosophy (Ph.D.) in 2009. She has published 4 research papers in

International Journals and presented 2 papers in International and National Conferences. She

has been awarded women scientist fellowship by Department of Science and Technology

(DST), New Delhi under WOMEN SCIENTISTS SCHEME (WOS-A) in February, 2012.

Presently she has been working towards the completion of abovementioned project under

WOS-A scheme.

some mathematical model s dynamics of biological...

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