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Research ArticleTheoretical Analysis of an Imprecise Prey-Predator Model withHarvesting and Optimal Control

Anjana Das and M Pal

Department of Applied Mathematics with Oceanology and Computer Programming Vidyasagar University MidnaporeWest Bengal-721101 India

Correspondence should be addressed to Anjana Das 1988anjagmailcom

Received 16 May 2018 Accepted 10 December 2018 Published 21 January 2019

Academic Editor K F C Yiu

Copyright copy 2019 Anjana Das and M Pal This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In our present paper we formulate and study a prey-predator system with imprecise values for the parameters We also considerharvesting for both the prey and predator speciesThenwe describe the complex dynamics of the proposedmodel system includingpositivity and uniform boundedness of the system and existence and stability criteria of various equilibrium points Also theexistence of bionomic equilibrium and optimal harvesting policy are thoroughly investigated Some numerical simulations havebeen presented in support of theoretical works Further the requirement of considering imprecise values for the set of modelparameters is also highlighted

1 Introduction

The eternal relationship between prey and predators is one ofthe major topics to be discussed in recent science Scientistsfrom various fields are currently engaged in finding outdifferent interactions among prey populations and predatorpopulations With the help of mathematical modeling onecan describe the strong and competitive relationship betweenthese two types of creatures However the discussion on thisfascinating topic with the help of some mathematical toolwas started during the first quarter of the twentieth centurythanks to the age-breaking works of Lotka [1] and Volterra[2] Influenced by those works researchers are still engagedin theoretical study of the ecological system with the help ofmathematicalmodeling In their book Kot [3] Britton (2003)described some ecological phenomena on various ecologicalinteractions including prey-predator interactions Smith [4]has demonstrated various aspects in theoretical ecology withthe help of some basic mathematical models May [5] hasalso considered and analyzed some other types of ecologicalsystems including prey-predator dynamics with the help ofsome sophisticated mathematical models which are com-paratively complex in nature Some other research works ontheoretical ecology including predator-prey dynamics can be

found like Cushing [6] Hadeler and Freedman [7] Chen etal [8] Kar [9 10] Kar et al [11] Chakraborty et al [12] andreferences therein

Harvesting is however a common and quite naturalphenomenon In fishery harvesting is used frequently as thebiological resources are mostly renewable resources On anexploited fishery system with interacting prey and predatorspecies researchers are considering harvesting on eitherprey species or predator species or harvesting on both preyand predator species Martin and Ruan [13] discussed thedynamics of harvesting of prey populations whereas in hisarticle Kar [9] describes phenomenonon selective harvestingon a prey-predator system Further harvesting in predatorspecies or both of prey and predator species can be foundin literature also (Kar and Pahari [14] Zhang and Zhang[15] Jana et al [16ndash18] Pal et al [19] Walters et al [20]Liu and Zhang [21] etc) From a bioeconomic point of viewharvesting on a species should be in a balance to both keepthe resource live and keep fishermen in profitable mode Inhis two books Clark [22 23] describes different harvestingpolicies in some realistic ecological systems with optimaloutcome

In this regard in our present article we consider aprey-predator type ecological system with harvesting on

HindawiJournal of OptimizationVolume 2019 Article ID 9512879 12 pageshttpsdoiorg10115520199512879

2 Journal of Optimization

both the species However till now most of the modelsare proposed by considering only precise set of parametersbut the natural world may not be precise every time Inmany situations at experimental field like birth and deathrate of different individuals of the same species interactionbetween two different species etc may be imprecise Forthis purpose introduction of fuzzy sets (Zadeh [24]) isnow considered as a revolutionary work However con-sideration of interval-valued parameters was due to thebroad application of fuzzy sets The imprecise parametersset may not always belong to the interval [0 1] but theymay belong to any interval of positive number Henceinterval-valued parameter set of an imprecise mathematicalmodel would be regardlessly better from a realistic point ofview

The rest of the paper is organized in the followingmanner In Section 2 we put some preliminaries on intervalvalue numbers In Section 3 on the basis of some realisticassumptions we formulate our predator-prey system andthen convert it to an imprecise parametric system whosedynamical behavior is thoroughly discussed in Section 4Section 5 is devoted to discussing the existence of bionomicequilibrium and optimal harvesting policies are studiedin Section 6 keeping harvesting parameter as the controlvariable In Section 7 we validate our theoretical resultsthrough some numerical simulation works and in the lastsection we present some key findings

2 Preliminaries

Here we give the definition of interval numbers with someoperations Weuse interval-valued function in lieu of intervalnumber

Definition 1 (interval number) Wedenote interval number119860as [119886 119886] and define it as 119860 = [119886 119886] = 119909 119886 le 119909 le 119886 119909 isin RwhereR is called the set of all real numbers and 119886 119886 are thelower and upper limits of the interval number respectively

A real number 119886 can also be used in form of intervalnumber as [119886 119886]

The basic operations between any two interval numbersare as follows

(i) [1198861 1198861] + [1198862 1198862] = [1198861 + 1198862 1198861 + 1198862](ii) [1198861 1198861] minus [1198862 1198862] = [1198861 minus 1198862 1198861 minus 1198862](iii) 119888[1198861 1198861] = [1198881198862 1198881198862] where 119888 is a real number

(iv) [1198861 1198861][1198862 1198862] = [min11988611198862 11988611198862 11988611198862 11988611198862max11988611198862 11988611198862 11988611198862 11988611198862]

(v) [1198861 1198861][1198862 1198862] = [1198861 1198861][11198862 11198862]Definition 2 (interval-valued function) For an interval [119886 119887]the interval-valued function can be created as 119891(119901) =119886(1minus119901)119887119901 for 119901 isin [0 1]

3 Predator-Prey Model with Harvesting withImprecise Parameter

31 CrispModel In this article we consider only two speciesnamely prey species and predator species Let 119909(119905) denote theprey biomass and 119910(119905) the predator class at any time 119905 Let theprey population grow logistically with intrinsic growth rate 119903and environmental carrying capacity 119896 Also let the predatorattack the prey in the predation rate 120572(gt 0) following themass action law Thus the differential equation for the preypopulation becomes119889119909 (119905)119889119905 = 119903119909 (1 minus 119909119896) minus 120572119909119910 (1)

Let119898(gt 0) be the conversion factor from the prey populationto the matured predator population 119889 be the natural deathrate and 120575 be the intraspecific competition rate for thepredator populations (due to Ruan et al [25]) Then thedifferential equation of the predator population 119910(119905) reducesto 119889119910 (119905)119889119905 = 119898120572119909119910 minus 119889119910 minus 1205751199102 (2)

Here119898 119903 119896 120572 120575 119889 are all positive parametersNext if we consider that both the species are harvested

this is carried out on assuming the demand in the market ofboth species (prey and predator) Taking 119864 as the harvestingeffort for both species and 1199021 amp 1199022 as the catchability coef-ficient of the prey species and predator species respectivelythen our system reduces to119889119909 (119905)119889119905 = 119903119909 (1 minus 119909119896) minus 120572119909119910 minus 1199021119864119909119889119910 (119905)119889119905 = 119898120572119909119910 minus 119889119910 minus 1205751199102 minus 1199022119864119910 (3)

subject to the initial conditions119909 (0) ge 0119910 (0) ge 0 (4)

32 FuzzyModel The environment and other factors includ-ing temperature and food habits caused the parameters to beimprecise So they should be taken as interval number ratherthan a single value Let 119903 119889 120575 be the correspondinginterval numbers for 119903 119896 120572 119898 119889 120575 respectively Then theprey-predator model with combined harvesting effort 119864becomes 119889119909119889119905 = 119903119909 (1 minus 119896) minus 119909119910 minus 1199021119864119909119889119910119889119905 = 119909119910 minus 119889119910 minus 1205751199102 minus 1199022119864119910 (5)

where 119903 = [119903 119903] = [119896 119896] = [120572 120572] = [119898119898] 119889 = [119889 119889]120575 = [120575 120575]

Journal of Optimization 3

For the interval number [119903 119903] we consider the interval-valued function 119903119901 = (119903)(1minus119901)(119903)119901 for 119901 isin [0 1] Similarlytaking the other interval numbers in the sameway as functionform we get the model as

119889119909119889119905 = (119903)(1minus119901) (119903)119901 119909(1 minus 119909(119896)(1minus119901) (119896)119901)minus (120572)(1minus119901) (120572)119901 119909119910 minus 1199021119864119909119889119910119889119905 = (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 119909119910 minus (119889)(1minus119901) (119889)119901 119910minus (120575)(1minus119901) (120575)119901 1199102 minus 1199022119864119910

(6)


Here 119901 isin [0 1] where the value of 119901 depends on the under-lying environment

4 Dynamical Behavior

In this section we describe a thorough dynamical behaviorof the proposed model system To do so we first checkthe positivity of the solutions of crisp system and uniformboundedness of the solution of the same system Now it canalso be concluded that uniformboundedness and positivity inthe solutions also hold for the corresponding fuzzy systemsif these things hold in crisp system

41 Positivity First we consider the corresponding crispsystem in following form119889119909119909 = [119903 (1 minus 119909119896) minus 120572119910 minus 1199021119864]119889119905119889119910119910 = [119909 minus 119889 minus 120575119910 minus 1199022119864] 119889119905 (8)

Now on integration we have from above system of equa-tions

119909 (119905) = 119909 (0) exp([119903 (1 minus 119896) minus 119910 minus 1199021]119864) (ge 0) (9)

and119910 (119905) = 119910 (0) exp ([119909 minus 119889 minus 120575119910 minus 1199022119864]) (ge 0) (10)

Hence from above two expressions related to two statevariables will always be positive Thus the solution of cor-responding crisp problem will be nonnegative and so thesolution of the corresponding fuzzy system will also benonnegative

42 Uniform Boundedness In this section we now studythe uniform boundedness of the proposed imprecise systemNow from the first expression of system (5) we have119889119909119889119905 + 1199021119864119909 le 119903119909(1 minus 119896) (11)

Now by simple mathematics it can be concluded that 119903119909(1 minus119909) has maximum value 1199034 which is obtained for 119909 = 2Thus from above we have119889119909119889119905 + 1199021119864119909 le 1199034 (12)

Now Integrating both sides of the above inequality and thenapplying the theory of differential inequality due to ( seeBirkhoff and Rota [26]) we have

0 lt 119909 (119905) le 11990341199021119864 (1 minus 119890minus1199021119864119905) + 119909 (0) (13)

Now on letting 119905 997888rarr infin we have

0 lt 119909 (119905) le 11990341199021119864 + 1205981 (14)

Hence the biomass density of prey population 119909(119905) is uni-formly bounded with an upper and lower limit 11990341199021119864 + 1205981and 0 respectively

Nextwe are targeting to show that the biomass of predatorpopulation119910(119905) is uniformly bounded In this regard from (5)we have 119889119910119889119905 + 1199022119864119910 le ( 11990341199021119864 + 1205981)119910 minus 1205751199102 (15)

Thus similarly to the above it is to be claimed that the righthand side of the above expression has maximum value at119910 = (2120575)(11990341199021119864 + 1205981) and this maximum value is(()24120575)(11990341199021119864 + 1205981)2

Thus proceeding in the same way as prey populations andwith the help of Birkhoff and Rota [26] we can write0 lt 119910 (119905)

le ()241199022119864120575 ( 11990341199021119864 + 1205981)2 (1 minus 119890minus1199022119864119905) + 119910 (0) (16)


0 lt 119910 (119905) le ()241199022119864120575 ( 11990341199021119864 + 1205981)2 + 1205982 (17)

So the biomass density of predator populations 119910 is also uni-formly bounded with lower and upper bound respectively 0and (()241199022119864120575)(11990341199021119864 + 1205981)2 + 1205982

Hence the biomass density of both the population speciesis uniformly bounded


43 Existence of Equilibria The equilibrium points of thissystem are given below

(1) Trivial equilibrium 119864119879(0 0) (2) Axial equilibrium119864119860(11990910) [where1199091=(119896)(1minus119901)(119896)119901(1minus1199021119864(119903)(1minus119901)(119903)119901)] exists if (119903)(1minus119901)(119903)119901 gt 1199021119864(3) Interior equilibrium 119864119868(119909lowast 119910lowast) where

119909lowast = (119896)(1minus119901) (119896)119901 (120572)(1minus119901) (120572)119901 (119889)(1minus119901) (119889)119901 + (119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901 + (120572)(1minus119901) (120572)119901 1199022119864 minus (120575)(1minus119901) (120575)119901 1199021119864(119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901 + (119898)(1minus119901) (119898)119901 (119896)(1minus119901) (119896)119901 (120572)2(1minus119901) (120572)2119901119910lowast = (119898)(1minus119901) (119898)119901 (119896)(1minus119901) (119896)119901 (120572)(1minus119901) (120572)119901 (119903)(1minus119901) (119903)119901 minus 1199021119864 minus (119903)(1minus119901) (119903)119901 (119889)(1minus119901) (119889)119901 + 1199022119864(119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901 + (119898)(1minus119901) (119898)119901 (119896)(1minus119901) (119896)119901 (120572)2(1minus119901) (120572)2119901 (18)

The interior equilibrium existsif(120572)(1minus119901) (120572)119901 (119889)(1minus119901) (119889)119901+ (119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901 + (120572)(1minus119901) (120572)119901 1199022119864gt (120575)(1minus119901) (120575)119901 1199021119864

(19)

and(119903)(1minus119901) (119903)119901 (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901gt 119903 (119889)(1minus119901) (119889)119901 + 1199022119864+ (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 1199021119864(20)

hold if

119864 lt min (1198641 1198642) (21)

where

1198641= (120572)(1minus119901) (120572)119901 (119889)(1minus119901) (119889)119901 + (119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901(120575)(1minus119901) (120575)119901 1199021 minus (120572)(1minus119901) (120572)119901 1199022 (22)

and

1198642 = (119903)(1minus119901) (119903)119901 (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 minus (119889)(1minus119901) (119889)119901(119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 1199021 + (119903)(1minus119901) (119903)119901 1199022 (23)

44 Local Asymptotic Stability In this section we state andprove the local asymptotic stability criteria at different equi-librium points Also the corresponding conditions for whichthe system is stable at different equilibria are given below

Case 1 For trivial equilibrium the variational matrix at119864119879(0 0) is given by the following119881 (119864119879)= ((119903)(1minus119901) (119903)119901 minus 1199021119864 00 minus (119889)(1minus119901) (119889)119901 minus 1199022119864) (24)

Therefore the eigenvalues are given by 1205821 = (119903)(1minus119901)(119903)119901 minus1199021119864 1205822 = minus(119889)(1minus119901)(119889)119901 minus 1199022119864Here 1205822 lt 0 then 119864119879(0 0) is asymptotically stable if1205821 lt 0 ie if (119903)(1minus119901)(119903)119901 minus 1199021119864 lt 0 which implies 119864 gt (11199021)(119903)(1minus119901)(119903)119901

In the next theorem we state the stability criteria of trivialequilibrium point

Theorem 3 Trivial equilibrium point 119864119879(0 0) of the system islocally asymptotically stable if 119864 gt (11199021)(119903)(1minus119901)(119903)119901 holdsCase 2 At axial equilibrium 119864119860(1199091 0) the variational matrixis 119881 (119864119860) = (11986411986011 1198641198601211986411986021 11986411986022) (25)

where 11986411986011 = 1199021119864 minus (119903)(1minus119901) (119903)119901 11986411986012 = minus (120572)(1minus119901) (120572)119901 119909111986411986021 = 0 (26)

and 11986411986022 = (119896)(1minus119901)(119896)119901(119898)(1minus119901)(119898)119901(120572)(1minus119901)(120572)1199011 minus 1199021119864(119903)(1minus119901)(119903)119901 minus (119889)(1minus119901)(119889)119901 minus 1199022119864 Then the eigenvalues


of the characteristic equation of 119881(119864119860) are 1199021119864 minus(119903)(1minus119901)(119903)119901 and (119896)(1minus119901)(119896)119901(119898)(1minus119901)(119898)119901(120572)(1minus119901)(120572)1199011 minus1199021119864(119903)(1minus119901)(119903)119901 minus (119889)(1minus119901)(119889)119901 minus 1199022119864 The first oneof them is negative since (119903)(1minus119901)(119903)119901 gt 1199021119864 Now 119864119860is asymptotically stable if the second one is negativeie

(119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901sdot 1 minus 1199021119864(119903)(1minus119901) (119903)119901 minus (119889)(1minus119901) (119889)119901 minus 1199022119864 lt 0 (27)

which implies

119864 gt (119903)(1minus119901) (119903)119901 (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 minus (119889)(1minus119901) (119889)1199011199021 (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 + 1199022 (119903)(1minus119901) (119903)119901 (28)

In the next theorem we will state the local asymptoticstability criteria of the axial equilibrium or the predator freeequilibrium 119864119860(1199091 0) Theorem 4 e axial equilibrium 119864119860(1199091 0) is locally asymp-

totically stable if

11199021 (119903)(1minus119901) (119903)119901 gt 119864 gt (119903)(1minus119901) (119903)119901 (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 minus (119889)(1minus119901) (119889)1199011199021 (119896)(1minus119901) (119896)119901 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 + 1199022 (119903)(1minus119901) (119903)119901 (29)

In this condition the trivial equilibrium becomes unsta-ble

Case 3 The variational matrix for interior equilibrium119864119868(119909lowast 119910lowast) is written below

119881 (119864119868)= ( minus(119903)(1minus119901) (119903)119901(119896)(1minus119901) (119896)119901 119909lowast minus (120572)(1minus119901) (120572)119901 119909lowast

(119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 119910lowast minus (120575)(1minus119901) (120575)119901 119910lowast)(30)

The characteristic equation of 119881(119864119868) is given by

1205822 + 119878120582 +119872 = 0 (31)

where

119878 = (119903)(1minus119901) (119903)119901(119896)(1minus119901) (119896)119901 119909lowast + (120575)(1minus119901) (120575)119901 119910lowast (32)

and

119872 = (119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901(119896)(1minus119901) (119896)119901+ (119898)(1minus119901) (119898)119901 (120572)2(1minus119901) (120572)2119901119909lowast119910lowast (33)

Here 119878 gt 0 and 119872 gt 0 since 119909lowast gt 0 and 119910lowast gt 0 Then thevalues of 120582 are negative

Therefore The system is locally asymptotically stable at(119909lowast 119910lowast) and we state this criteria in the following theorem

Theorem 5 e interior equilibrium 119864119868(119909lowast 119910lowast) of the systemexists and is locally asymptotically stable if119864 lt min (1198641 1198642) (34)

where1198641= (120572)(1minus119901) (120572)119901 (119889)(1minus119901) (119889)119901 + (119903)(1minus119901) (119903)119901 (120575)(1minus119901) (120575)119901(120575)(1minus119901) (120575)119901 1199021 minus (120572)(1minus119901) (120572)119901 1199022 (35)

and



45 Global Stability Here we will discuss the global asymp-totic stability criteria of the system around its interior equi-librium point In next theorem we study the criteria

Theorem 6 e interior equilibrium 119864119868(119909lowast 119910lowast) of the systemis globally asymptotically stable provided it is locally asymptot-ically stable there

Proof A Lyapunov function is constructed here as follows

119881 (119909 119910) = int119909119909lowast

119909 minus 119909lowast119909 119889119909 + 119875int119910119910lowast

119910 minus 119910lowast119910 119889119910 (37)

where 119875 is suitable positive constant to be determined in thesubsequent steps

Taking derivative with respect to 119905 along the solutions ofthe system we have119889119881119889119905 = 119909 minus 119909lowast119909 119889119909119889119905 + 119875119910 minus 119910lowast119910 119889119910119889119905 (38)

Now 1119909 119889119909119889119905 = (119903)(1minus119901) (119903)119901(1 minus 119909(119896)(1minus119901) (119896)119901)minus (120572)(1minus119901) (120572)119901 119910 minus 11990211198641119909 119889119909lowast119889119905 = (119903)(1minus119901) (119903)119901(1 minus 119909lowast(119896)(1minus119901) (119896)119901)minus (120572)(1minus119901) (120572)119901 119910lowast minus 11990211198641119910 119889119910119889119905 = (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 119909minus (119889)(1minus119901) (119889)119901 minus (120575)(1minus119901) (120575)119901 119910 minus 1199022119864

(39)

Then

119889119881119889119905 = (119909 minus 119909lowast) [[minus (119903)(1minus119901) (119903)119901(119896)(1minus119901) (119896)119901 (119909 minus 119909lowast)minus (120572)(1minus119901) (120572)119901 (119910 minus 119910lowast)]] + 119875 (119910 minus 119910lowast)sdot [(119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 (119909 minus 119909lowast)minus (120575)(1minus119901) (120575)119901 (119910 minus 119910lowast)] = minus (119903)(1minus119901) (119903)119901(119896)(1minus119901) (119896)119901 (119909minus 119909lowast)2 + (119875 (119898)(1minus119901) (119898)119901 minus 1) (120572)(1minus119901) (120572)119901 (119909minus 119909lowast) (119910 minus 119910lowast) minus 119875 (120575)(1minus119901) (120575)119901 (119910 minus 119910lowast)2

(40)

If we consider 119875 = 1119898 then 119889119881119889119905 reduces to the following119889119881119889119905 = minus (119903)(1minus119901) (119903)119901(119896)(1minus119901) (119896)119901 (119909 minus 119909lowast)2minus (120575)(1minus119901) (120575)119901(119898)(1minus119901) (119898)119901 (119910 minus 119910lowast)2 (41)

It is seen from the above that 119889119881119889119905 le 0That is the system is globally stable around its interior

equilibrium 119864119868(119909lowast 119910lowast)5 Bionomic Equilibrium

In this section we study the bionomic equilibrium of thecompetitive predator-prey model Here we consider thefollowing parameters (1) 119888 fishing cost per unit effort (2)1199011 price per unit biomass of the prey (3) 1199012 price per unitbiomass of the predator The net revenue at any time is givenby 119877 = (11990111199021119909 + 11990121199022119910 minus 119888) 119864 (42)

The interior equilibrium point of the system is on the linegiven below119909 ((119896)(1minus119901) (119896)119901 1199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901

+ (119903)(1minus119901) (119903)119901 1199022)+ 119910 ((119896)(1minus119901) (119896)119901 1199022 (120572)(1minus119901) (120572)119901minus (119896)(1minus119901) (119896)119901 1199021 (120575)(1minus119901) (120575)119901) = (119896)(1minus119901) (119896)119901sdot ((119903)(1minus119901) (119903)119901 1199022 + 1198891199021)

(43)

This biological equilibrium line meets x-axis at (119909119877 0) and y-axis at (0 119910119877) where119909

= (119896)(1minus119901) (119896)119901 ((119903)(1minus119901) (119903)119901 1199022 + (119889)(1minus119901) (119889)119901 1199021)(119896)(1minus119901) (119896)119901 1199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 + (119903)(1minus119901) (119903)119901 1199022 (44)

and119910= (119896)(1minus119901) (119896)119901 ((119903)(1minus119901) (119903)119901 1199022 + (119889)(1minus119901) (119889)119901 1199021)(119896)(1minus119901) (119896)119901 (1199022 (120572)(1minus119901) (120572)119901 minus 1199021 (120575)(1minus119901) (120575)119901) (45)

It is seen that always 119909 gt 0 but 119910 is feasible if 1199022(120572)(1minus119901)(120572)119901 gt1199021(120575)(1minus119901)(120575)119901The lsquozero-profit linersquo is given by119877 = (11990111199021119909 + 11990121199022119910 minus 119888)119864 = 0 (46)


Equation (6) together with the above condition represents thebionomic equilibrium of prey-predator harvesting system

For the points on the equilibrium line where (11990111199021119909 +11990121199022119910minus119888) lt 0 the fishery becomes useless Because it cannotproduce any positive economic revenue

These three cases may arise in bionomic equilibrium

Case 1 When fishing or harvesting of predator spe-cies is not possible then 119909119877 = 11988811990111199021 gives that119888 = (11990111199021(119896)(1minus119901)(119896)119901((119903)(1minus119901)(119903)1199011199022 + (119889)(1minus119901)(119889)1199011199021))((119896)(1minus119901)(119896)1199011199021(119898)(1minus119901)(119898)119901(120572)(1minus119901)(120572)119901 + (119903)(1minus119901)(119903)1199011199022)Case 2 When harvesting of prey is not possible then119910119877 = 11988811990121199022 gives that 119888 = (11990121199022(119896)(1minus119901)(119896)119901((119903)(1minus119901)(119903)1199011199022 +(119889)(1minus119901)(119889)1199011199021))((119896)(1minus119901)(119896)119901(1199022(120572)(1minus119901)(120572)119901 minus1199021(120575)(1minus119901)(120575)119901)) with 1199022(120572)(1minus119901)(120572)119901 gt 1199021(120575)(1minus119901)(120575)119901Case 3 When the bionomic equilibrium is at a point (119909119877 119910119877)where both 119909119877 gt 0 and 119910119877 gt 0 then the fishing of prey andpredator is possible Here

119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where

11990911 = (119896)(1minus119901) (119896)119901 11990121199022 ((119903)(1minus119901) (119903)119901 1199022 + (119889)(1minus119901)sdot (119889)119901 1199021) minus 119888 (119896)(1minus119901) (119896)119901 (1199022 (120572)(1minus119901) (120572)119901minus 1199021120575)

11990912 = 11990121199022 ((119896)(1minus119901) (119896)119901 1199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901+ (119903)(1minus119901) (119903)119901 1199022) minus (119896)(1minus119901) (119896)119901 11990111199021 (1199022 (120572)(1minus119901)sdot (120572)119901 minus 1199021 (120575)(1minus119901) (120575)119901) 11991011 = 119888 ((119896)(1minus119901) (119896)119901 1199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901+ (119903)(1minus119901) (119903)119901 1199022) minus (119896)(1minus119901) (119896)119901 11990121199022 (1199022120572minus 1199021 (120575)(1minus119901) (120575)119901)

(48)

Since 119909119877 gt 0 and 119910119877 gt 0 then the following two conditionshold (i) 119888 gt 1198881111988812 (49)

or 119888 lt 1198881111988812 (50)

where

11988811 = (119896)(1minus119901) (119896)119901 (1199012)2 (1199022)2 ((119903)(1minus119901) (119903)119901 1199022 + (119889)(1minus119901) (119889)119901 1199021)sdot ((119896)(1minus119901) (119896)119901 1199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 + (119903)(1minus119901) (119903)119901 1199022) 11988812 = (119896)(1minus119901) (119896)1199012 11990111199021 (1199022 (120572)(1minus119901) (120572)119901 minus 1199021 (120575)(1minus119901) (120575)119901)2

(51)

(ii) 119888 gt (119896)(1minus119901) (119896)119901 1199011119901211990211199022 ((119903)(1minus119901) (119903)119901 1199022 + (119889)(1minus119901) (119889)119901 1199021) (1199022 (120572)(1minus119901) (120572)119901 minus 1199021 (120575)(1minus119901) (120575)119901)11990121199022 ((119896)(1minus119901) (119896)119901 1199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 + (119903)(1minus119901) (119903)119901 1199022)2 (52)

or 119888 lt (119896)(1minus119901) (119896)119901 1199011119901211990211199022 ((119903)(1minus119901) (119903)119901 1199022 + (119889)(1minus119901) (119889)119901 1199021) (1199022 (120572)(1minus119901) (120572)119901 minus 1199021 (120575)(1minus119901) (120575)119901)11990121199022 (1198961199021 (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 + (119903)(1minus119901) (119903)119901 1199022)2 (53)

Then we conclude the bionomic equilibrium shorty in thefollowing theorem

Theorem 7 e bionomic equilibrium (119909119877 0) always exists(0 119910119877) exists when 1199022(120572)(1minus119901)(120572)119901 gt 1199021(120575)(1minus119901)(120575)119901 and(119909119877 119910119877) exists when conditions (49) (50) and (52) (53) holdsimultaneously

6 Optimal Harvesting Policy

Here both prey and predator populations are considered asfish populations The optimal net profit is obtained from

fishing We discuss in this section the optimal harvestingpolicy We consider the profit gained from harvesting tak-ing the cost as a quadratic function and focusing on theconservation of fish population The price assumed hereis inversely proportional to the available biomass of fish(prey and predator) ie if the biomass increases the pricedecreases (see Chakraborty et al (2011)) Let 119888 be the constantharvesting cost per unit effort and 1199011 and 1199012 be respectivelythe constant price per unit biomass of the prey and predatorNow our target is to get the maximum net revenues fromfishery Then the optimal control problem can be created inthe following way


119869 (119864) = int11990511199050

119890minus120590119905 [(1199011 minus V11199021119864119909) 1199021119864119909+ (1199012 minus V21199022119864119910) 1199022119864119910 minus 119888119864] 119889119905 (54)

subject to the system of differential equations (6) and theinitial conditions (7) V1 and V2 are economic constants and120590 is the instantaneous discount rate

Here the control 119864 is bounded in 0 le 119864 le 119864119898119886119909 and ourobject is to find an optimal control 119864119900 such that119869 (119864119900) = max

119864isin119880119869 (119864) (55)

where119880 is the control set defined by119880 = 119864 119864 119894119904 119898119890119886119904119906119903119886119887119897119890 119886119899119889 0 le 119864le 119864119898119886119909 119891119900119903 119886119897119897 119905 (56)

Here the convexity of the objective functional with respectto the control variable 119864 along with the compactness ofthe range values of the state variables can be combinedto give the existence of the optimal control 119864119900 Now theoptimal control can be found by using Pontryaginrsquosmaximumprinciple (Pontryagin et al (1962)) To optimize the objectivefunctional 119869(119864) we construct the Hamiltonian 119867 of thesystem as follows119867 = (1199011 minus V11199021119864119909) 1199021119864119909 minus (1199012 minus V21199022119864119910) 1199022119864119910 minus 119888119864

+ 1205821((119903)(1minus119901) (119903)119901 119909(1 minus 119909(119896)(1minus119901) (119896)119901)minus (120572)(1minus119901) (120572)119901 119909119910 minus 1199021119864119909)+ 1205822 ((119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 119909119910minus (119889)(1minus119901) (119889)119901 119910 minus (120575)(1minus119901) (120575)119901 1199102 minus 1199022119864119910)

(57)

Here the variables 1205821 and 1205822 are adjoint variables and thetransversality conditions are as follows120582119894 (1199051) = 0 119894 = 1 2 (58)

First we use the optimality condition 120597119867120597119864 = 0 to obtainthe optimal effort which is as follows

119864120590 = 11990111199021119909 minus 11990121199022119909 minus 119888 minus 11990211205821119909 minus 119902212058221199102 (V1119902211199092 + V2119902221199102) (59)

The adjoint equations are1198891205821119889119905 = 1205901205821 minus 120597119867120597119909 = minus1199021119864 (1199011 minus 2V111990221119864119909)+ ((120572)(1minus119901) (120572)119901 119910 + 1199021119864 minus (119903)(1minus119901) (119903)119901 (1 minus 2119909119870 )

+ 120590)1205821 minus (119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 11991012058221198891205822119889119905 = 1205901205822 minus 120597119867120597119910 = minus1199022119864 (1199012 minus 2V211990222119864119910)+ 119909 (120572)(1minus119901) (120572)119901 1205821minus ((119898)(1minus119901) (119898)119901 (120572)(1minus119901) (120572)119901 119909 minus (119889)(1minus119901) (119889)119901minus 2 (120575)(1minus119901) (120575)119901 119910 minus 1199022119864 minus 120590) 1205822

(60)

and therefore we have the following theorem regarding theoptimal value of the harvesting effort

Theorem 8 ere exists an optimal control 119864120590 correspondingto the optimal solutions for the state variables as 119909120590 and119910120590 suchthat this control 119864120590 optimizes the objective functional 119869 overthe region 119880Moreover there exist adjoint variables 1205821 and 1205822satisfying the first order differential equations given in (60) withthe transversality conditions given in (58) where at the optimalharvesting level the values of the state variables 119909 and 119910 arerespectively 119909120590 and 1199101205907 Numerical Simulation

In this section we analyze our mathematical model throughsome simulation worksThemain difference of our proposedmodel compared to other models of the same type is theconsideration of interval-valued parameters instead of fixed-valued parameters Inclusion of the parameter 119901 assumesthe value corresponding to the parameters of the system asan interval In this regard we first analyze the importanceof considering the parameter 119901 in Figure 1 For simulationpurpose we consider the following parametric values 119903 =[4 6] = [800 1000] = [050 075] = [06 08]119889 = [01 02] 120575 = [00001 00020] 1199021 = 12 1199022 = 0001119864 = 14 For different parametric values of 119901 (0 le 119901 le1) we have obtained various types of dynamical behaviorof the proposed prey-predator system From Figure 1 it canbe said that lower values of 119901 make the system unstable atthe interior equilibrium point whereas the higher values of119901 gradually make the system locally asymptotically stablearound the interior equilibrium point As the numerical valueof 119901 increases the instability solutions slowly become stable(unstable branches at 119901 = 07 are less than the number ofunstable branches at 119901 = 05 but still at 119901 = 07 the system isunstable but asymptotically stable at a higher value (119901 = 09))

Next we describe optimal control theory to simulatethe optimal control problem numerically We consider thesame parametric values as above and find the solutionof optimal control problem numerically For this purposewe solve the system of differential equations of the statevariables (6) and corresponding initial conditions (7) withthe help of forward Runge-Kutta forth order procedureAlso the differential equations of adjoint variables (50) andcorresponding transversality conditions (52) are solved withthe help of backward Runge-Kutta forth order procedure


p=0

xy

xy

xy

xy

xy

xy

p=005

p=02 p=07

p=09 p=1

minus202468

10121416

Popu

latio

ns

2000 4000 6000 8000 100000Time

minus202468

1012141618

Popu

latio

ns

2000 4000 6000 8000 100000Time

200 400 600 800 10000Time

minus5

0

5

10

15

20

Popu

latio

ns

minus5

0

5

10

15

20

Popu

latio

ns

2000 4000 6000 8000 100000Time

2000 4000 6000 8000 100000Time

minus5

0

5

10

15

20

Popu

latio

ns

2000 4000 6000 8000 100000Time

minus202468

1012141618

Popu

latio

ns

Figure 1 Stability of interior equilibrium depends on the numerical value of 119901 A higher value of 119901 makes the interior equilibrium a stableequilibrium whereas a lower value makes it unstable one

for the time interval [0 100] (see Jung et al [27] Lenhartand Workman [28] etc) Considering harvesting parameter119864 as the control variable in Figure 2 and in Figure 3 werespectively plot the changes of prey biomass with respect totime and those of predator biomass with respect to time bothin presence of control and in absence of control parameter Itis observed that when harvesting control is applied optimallythen the biomass of both prey species and predator species

diminishes which is in accord with our expectations Furtherin Figure 4 we plot the variation of control parameter (hereharvesting effort is the control parameter) and in Figure 5 weplot variations of adjoint variables It is also to be observedthat the level of optimal harvesting effort always belongs tothe range [010 045] Further according to the transversalityconditions both the adjoint variables 1205821 and 1205822 vanished atthe final time (see Figure 5)


With harvestingWithout harvesting

0

10

20

30

40

50

60Pr

ey

20 40 60 80 1000Time

Figure 2 Variation of prey biomass both with control and withoutcontrol cases


180

190

200

210

220

230

240

250

260

Pred

ator

20 40 60 80 1000Time

Figure 3 Variation of predator biomass both with control andwithout control cases

8 Discussions and Conclusions

The interactions between prey species and their predatorspecies is an important topic to be analyzed In presentera many experts are still analyzing the different aspectson this relationship For this purpose in our present paperwe formulate and analyze a mathematical model on prey-predator system with harvesting on both prey and predatorspecies Further the model system is improved with the con-sideration of system parameters assuming an interval valueinstead of considering a single value In reality due to variousuncertainty aspects in nature the parameters associated witha model system should not be considered a single value Butoften this scenario has been neglected although some recentworks considered these types of phenomena (see the works

20 40 60 80 1000Time

0

005

01

015

02

025

03

035

04

045

05

Har

vesti

ng eff

ort

Figure 4 Variation of harvesting effort with time

minus100

minus80

minus60

minus40

minus20

0

20

1

50 1000Time

minus50

0

50

100

150

200

250

300

350

400

2

50 1000Time

Figure 5 Variation of adjoint variables with time when harvestingcontrol is applied optimally

of Pal et al [19 29] Sharma and Samanta [30] Das and Pal[31] etc) Influenced by those works we also consider thatall the parameters associated with our system are of intervalvalue Further harvesting of both prey and predator speciesis considered with catch per unit biomass in unit time withharvesting effort 119864

The proposed model is analyzed for both crisp andinterval-valued parametric cases Different dynamical behav-ior of the system including uniform boundedness andexistence and feasibility criteria of all the equilibria andboth their local and global asymptotic stability criteria hasbeen described It is found that the system may possessthree equilibria namely the vanishing equilibrium point thepredator free equilibrium point and the interior equilibriumpoint Theoretical analysis shows that all of these threeequilibria may be conditionally locally asymptotically stabledepending on the numerical value of the harvesting param-eter 119864 The classical prey-predator model with harvestingeffort and without imprecise parametric space in general


enables the vanishing equilibriumor trivial equilibriumpointas an unstable equilibrium point but the consideration ofimprecise parametric space makes the trivial equilibrium aconditionally stable equilibrium This phenomenon wouldsurely describe the simultaneous extinction of a single speciesor both species although the crisp model failed to analyze it

Next we study explicitly the existence criteria of bionomicequilibrium considering 119864 as harvesting effort Further con-sidering harvesting effort as the control parameter we forman optimal control problemwith the objective of maximizingthe profit due to harvesting in a finite horizon of time andsolve that problem both theoretically and numerically Theobjective functional considered in optimal control problem isalso of both innovative and realistic type as we consider herethat the prices of biomass for both prey and predator speciesinversely depend upon their corresponding demands

Consideration of imprecise parameters set makes themodel more close to a realistic system which can be wellexplained with the help of Figure 1 It is shown that fordifferent values of the parameter 119901 associated with theimprecise values we obtain different nature of the coexistingequilibrium point As the numeric value of the associatedparameter 119901 increases the amount of unstable branches forboth the species reduces and ultimately becomes a stablesystem for a higher value of 119901(ge 09) As the nature ofthe interior or coexisting equilibrium is one of the mostimportant objects to study we may claim that the differentvalues of the imprecise parameter 119901 are able to make theproposed prey-predator system understandable and reflectthe real world problem

However in the present work we consider only a singleprey species interacting with a single predator species whichmakes the model a quite simple one For our future workwe preserve the option of considering more than one type ofprey species interacting with more than one type of predatorspecies with imprecise set of parameters Further due tounavailability of real world data to simulate our theoreticalworks we consider a hypothetical set for the parametersand obtain the result However we mainly aim to study thequalitative behavior of the system (not quantitative behavior)which would not be hampered at all due to the considerationof a simulated parametric set

Data Availability

The data used in the manuscript are hypothetical data

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A J Lotka Elements of Physical Biology Williams andWilkinsBaltimore New York 1925

[2] V Volterra ldquoVariazioni e fluttuazioni del numero diaindividuiin specie animali conviventirdquo Mem R Accad Naz Dei Linceivol 2 1926

[3] M Kot Elements of Mathematical Biology Cambridge Univer-sity Press Cambridge UK 2001

[4] J M Smith ldquoModels in Ecologyrdquo CUP Archive 1978[5] R M May Stability and Complexity in Model Ecosystems

Princeton University Press 2001[6] J M Cushing ldquoPeriodic two-predator one-prey interactions

and the time sharing of a resource nicherdquo SIAM Journal onApplied Mathematics vol 44 no 2 pp 392ndash410 1984

[7] K P Hadeler and H I Freedman ldquoPredator-prey populationswith parasitic infectionrdquo Journal of Mathematical Biology vol27 no 6 pp 609ndash631 1989

[8] F Chen L Chen and X Xie ldquoOn a Leslie-Gower predator-preymodel incorporating a prey refugerdquo Nonlinear Analysis RealWorld Applications vol 10 no 5 pp 2905ndash2908 2009

[9] T K Kar ldquoSelective harvesting in a prey-predator fishery withtime delayrdquoMathematical and Computer Modelling vol 38 no3-4 pp 449ndash458 2003

[10] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[11] T K Kar A Ghorai and S Jana ldquoDynamics of pest and itspredator model with disease in the pest and optimal use ofpesticiderdquo Journal of eoretical Biology vol 310 pp 187ndash1982012

[12] K Chakraborty S Jana and T K Kar ldquoEffort dynamics of adelay-induced prey-predator system with reserverdquo NonlinearDynamics vol 70 no 3 pp 1805ndash1829 2012

[13] A Martin and S Ruan ldquoPredator-prey models with delay andprey harvestingrdquo Journal of Mathematical Biology vol 43 no 3pp 247ndash267 2001

[14] T K Kar and U K Pahari ldquoNon-selective harvesting in prey-predator models with delayrdquo Communications in NonlinearScience and Numerical Simulation vol 11 no 4 pp 499ndash5092006

[15] Y Zhang and Q Zhang ldquoDynamical behavior in a delayedstage-structure population model with stochastic fluctuationand harvestingrdquo Nonlinear Dynamics vol 66 no 1-2 pp 231ndash245 2011

[16] S Jana M Chakraborty K Chakraborty and T K Kar ldquoGlobalstability and bifurcation of time delayed prey-predator systemincorporating prey refugerdquo Mathematics and Computers inSimulation vol 85 pp 57ndash77 2012

[17] S Jana S Guria U Das T K Kar and A Ghorai ldquoEffect ofharvesting and infection on predator in a prey-predator systemrdquoNonlinear Dynamics vol 81 no 1-2 pp 917ndash930 2015

[18] S Jana A Ghorai S Guria and T K Kar ldquoGlobal dynamicsof a predator weaker prey and stronger prey systemrdquo AppliedMathematics and Computation vol 250 pp 235ndash248 2015

[19] D Pal G S Mahapatra and G P Samanta ldquoStability andbionomic analysis of fuzzy prey-predator harvesting model inpresence of toxicity a dynamic approachrdquo Bulletin of Mathe-matical Biology vol 78 no 7 pp 1493ndash1519 2016

[20] C Walters V Christensen B Fulton A D M Smith and RHilborn ldquoPredictions from simple predator-prey theory aboutimpacts of harvesting forage fishesrdquo Ecological Modelling vol337 pp 272ndash280 2016

[21] J Liu and L Zhang ldquoBifurcation analysis in a prey-predatormodel with nonlinear predator harvestingrdquo Journal of eFranklin Institute vol 353 no 17 pp 4701ndash4714 2016

[22] C W Clark Bioeconomic modelling and fisheries managementJohn Wiley and Sons New York NY USA 1985


[23] C W Clark Mathematical Bio-Economics e Optimal Man-agement of Renewable Resources Pure andAppliedMathematics(New York) John Wiley amp Sons New York NY USA 2ndedition 1990

[24] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[25] S Ruan A Ardito P Ricciardi and D L DeAngelis ldquoCoexis-tence in competition models with density-dependent mortal-ityrdquo Comptes Rendus Biologies vol 330 no 12 pp 845ndash8542007

[26] G Birkhoff and G-C Rota Ordinary Differential EquationsGinn Boston 1982

[27] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems - Series B vol 2 no 4 pp 473ndash482 2002

[28] S M Lenhart and J T Workman Optimal Control Applied toBiological Models Mathematical and Computational BiologySeries Chapman amp HallCRC 2007

[29] D Pal G S Mahaptra and G P Samanta ldquoOptimal harvestingof prey-predator system with interval biological parameters abioeconomic modelrdquo Mathematical Biosciences vol 241 no 2pp 181ndash187 2013

[30] S Sharma and G P Samanta ldquoOptimal harvesting of a twospecies competition model with imprecise biological parame-tersrdquo Nonlinear Dynamics vol 77 no 4 pp 1101ndash1119 2014

[31] A Das and M Pal ldquoA mathematical study of an imprecise SIRepidemic model with treatment controlrdquo Applied Mathematicsand Computation vol 56 no 1-2 pp 477ndash500 2017

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both the species However till now most of the modelsare proposed by considering only precise set of parametersbut the natural world may not be precise every time Inmany situations at experimental field like birth and deathrate of different individuals of the same species interactionbetween two different species etc may be imprecise Forthis purpose introduction of fuzzy sets (Zadeh [24]) isnow considered as a revolutionary work However con-sideration of interval-valued parameters was due to thebroad application of fuzzy sets The imprecise parametersset may not always belong to the interval [0 1] but theymay belong to any interval of positive number Henceinterval-valued parameter set of an imprecise mathematicalmodel would be regardlessly better from a realistic point ofview

The rest of the paper is organized in the followingmanner In Section 2 we put some preliminaries on intervalvalue numbers In Section 3 on the basis of some realisticassumptions we formulate our predator-prey system andthen convert it to an imprecise parametric system whosedynamical behavior is thoroughly discussed in Section 4Section 5 is devoted to discussing the existence of bionomicequilibrium and optimal harvesting policies are studiedin Section 6 keeping harvesting parameter as the controlvariable In Section 7 we validate our theoretical resultsthrough some numerical simulation works and in the lastsection we present some key findings

2 Preliminaries

Here we give the definition of interval numbers with someoperations Weuse interval-valued function in lieu of intervalnumber

Definition 1 (interval number) Wedenote interval number119860as [119886 119886] and define it as 119860 = [119886 119886] = 119909 119886 le 119909 le 119886 119909 isin RwhereR is called the set of all real numbers and 119886 119886 are thelower and upper limits of the interval number respectively

A real number 119886 can also be used in form of intervalnumber as [119886 119886]

The basic operations between any two interval numbersare as follows

(i) [1198861 1198861] + [1198862 1198862] = [1198861 + 1198862 1198861 + 1198862](ii) [1198861 1198861] minus [1198862 1198862] = [1198861 minus 1198862 1198861 minus 1198862](iii) 119888[1198861 1198861] = [1198881198862 1198881198862] where 119888 is a real number

(iv) [1198861 1198861][1198862 1198862] = [min11988611198862 11988611198862 11988611198862 11988611198862max11988611198862 11988611198862 11988611198862 11988611198862]

(v) [1198861 1198861][1198862 1198862] = [1198861 1198861][11198862 11198862]Definition 2 (interval-valued function) For an interval [119886 119887]the interval-valued function can be created as 119891(119901) =119886(1minus119901)119887119901 for 119901 isin [0 1]

3 Predator-Prey Model with Harvesting withImprecise Parameter

31 CrispModel In this article we consider only two speciesnamely prey species and predator species Let 119909(119905) denote theprey biomass and 119910(119905) the predator class at any time 119905 Let theprey population grow logistically with intrinsic growth rate 119903and environmental carrying capacity 119896 Also let the predatorattack the prey in the predation rate 120572(gt 0) following themass action law Thus the differential equation for the preypopulation becomes119889119909 (119905)119889119905 = 119903119909 (1 minus 119909119896) minus 120572119909119910 (1)

Let119898(gt 0) be the conversion factor from the prey populationto the matured predator population 119889 be the natural deathrate and 120575 be the intraspecific competition rate for thepredator populations (due to Ruan et al [25]) Then thedifferential equation of the predator population 119910(119905) reducesto 119889119910 (119905)119889119905 = 119898120572119909119910 minus 119889119910 minus 1205751199102 (2)

Here119898 119903 119896 120572 120575 119889 are all positive parametersNext if we consider that both the species are harvested

this is carried out on assuming the demand in the market ofboth species (prey and predator) Taking 119864 as the harvestingeffort for both species and 1199021 amp 1199022 as the catchability coef-ficient of the prey species and predator species respectivelythen our system reduces to119889119909 (119905)119889119905 = 119903119909 (1 minus 119909119896) minus 120572119909119910 minus 1199021119864119909119889119910 (119905)119889119905 = 119898120572119909119910 minus 119889119910 minus 1205751199102 minus 1199022119864119910 (3)


32 FuzzyModel The environment and other factors includ-ing temperature and food habits caused the parameters to beimprecise So they should be taken as interval number ratherthan a single value Let 119903 119889 120575 be the correspondinginterval numbers for 119903 119896 120572 119898 119889 120575 respectively Then theprey-predator model with combined harvesting effort 119864becomes 119889119909119889119905 = 119903119909 (1 minus 119896) minus 119909119910 minus 1199021119864119909119889119910119889119905 = 119909119910 minus 119889119910 minus 1205751199102 minus 1199022119864119910 (5)

where 119903 = [119903 119903] = [119896 119896] = [120572 120572] = [119898119898] 119889 = [119889 119889]120575 = [120575 120575]




(6)













0 lt 119909 (119905) le 11990341199021119864 (1 minus 119890minus1199021119864119905) + 119909 (0) (13)


0 lt 119909 (119905) le 11990341199021119864 + 1205981 (14)





le ()241199022119864120575 ( 11990341199021119864 + 1205981)2 (1 minus 119890minus1199022119864119905) + 119910 (0) (16)


0 lt 119910 (119905) le ()241199022119864120575 ( 11990341199021119864 + 1205981)2 + 1205982 (17)








(19)


hold if

119864 lt min (1198641 1198642) (21)

where


and












which implies



totically stable if







1205822 + 119878120582 +119872 = 0 (31)

where


and






and






119881 (119909 119910) = int119909119909lowast


119910 minus 119910lowast119910 119889119910 (37)




(39)

Then


(40)







(43)










119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where



(48)


or 119888 lt 1198881111988812 (50)

where


(51)









119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






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(6)













0 lt 119909 (119905) le 11990341199021119864 (1 minus 119890minus1199021119864119905) + 119909 (0) (13)


0 lt 119909 (119905) le 11990341199021119864 + 1205981 (14)





le ()241199022119864120575 ( 11990341199021119864 + 1205981)2 (1 minus 119890minus1199022119864119905) + 119910 (0) (16)


0 lt 119910 (119905) le ()241199022119864120575 ( 11990341199021119864 + 1205981)2 + 1205982 (17)








(19)


hold if

119864 lt min (1198641 1198642) (21)

where


and












which implies



totically stable if







1205822 + 119878120582 +119872 = 0 (31)

where


and






and






119881 (119909 119910) = int119909119909lowast


119910 minus 119910lowast119910 119889119910 (37)




(39)

Then


(40)







(43)










119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where



(48)


or 119888 lt 1198881111988812 (50)

where


(51)









119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






p=0

xy

xy

xy

xy

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p=09 p=1

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(19)


hold if

119864 lt min (1198641 1198642) (21)

where


and












which implies



totically stable if







1205822 + 119878120582 +119872 = 0 (31)

where


and






and






119881 (119909 119910) = int119909119909lowast


119910 minus 119910lowast119910 119889119910 (37)




(39)

Then


(40)







(43)










119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where



(48)


or 119888 lt 1198881111988812 (50)

where


(51)









119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






p=0

xy

xy

xy

xy

xy

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p=005

p=02 p=07

p=09 p=1

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which implies



totically stable if







1205822 + 119878120582 +119872 = 0 (31)

where


and






and






119881 (119909 119910) = int119909119909lowast


119910 minus 119910lowast119910 119889119910 (37)




(39)

Then


(40)







(43)










119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where



(48)


or 119888 lt 1198881111988812 (50)

where


(51)









119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






p=0

xy

xy

xy

xy

xy

xy

p=005

p=02 p=07

p=09 p=1

minus202468

10121416

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119881 (119909 119910) = int119909119909lowast


119910 minus 119910lowast119910 119889119910 (37)




(39)

Then


(40)







(43)










119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where



(48)


or 119888 lt 1198881111988812 (50)

where


(51)









119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






p=0

xy

xy

xy

xy

xy

xy

p=005

p=02 p=07

p=09 p=1

minus202468

10121416

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Journal of












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Volume 2018













119909119877 = 1199091111990912 119910119877 = 1199101111990912 (47)

where



(48)


or 119888 lt 1198881111988812 (50)

where


(51)









119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






p=0

xy

xy

xy

xy

xy

xy

p=005

p=02 p=07

p=09 p=1

minus202468

10121416

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119869 (119864) = int11990511199050




119864isin119880119869 (119864) (55)




(57)






(60)






p=0

xy

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xy

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p=005

p=02 p=07

p=09 p=1

minus202468

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Journal of












Journal of







Volume 2018






Volume 2018









p=0

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Data Availability




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Journal of












Journal of







Volume 2018






Volume 2018

























Journal of












Journal of







Volume 2018






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theoretical analysis of an imprecise prey-predator model ......⋅b1− 1 ˝ ˛(1−)() −...

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