Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587

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Nonlinear Analysis: Hybrid Systems

journal homepage: www.elsevier.com/locate/nahs

Bifurcation analysis and control of a class of hybrid biologicaleconomic modelsI

Xue Zhang ∗, Qing-ling ZhangInstitute of Systems Science, Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, Liaoning,110004, PR China

a r t i c l e i n f o

Article history:Received 21 November 2007Accepted 28 April 2009

Keywords:Hybrid biological economic modelSaddle–node bifurcationSingular induced bifurcationNeimark–Sacker bifurcationState feedback control

a b s t r a c t

This paper systematically studies a hybrid predator–prey economic model, which isformulated by differential-difference-algebraic equations. It shows that thismodel exhibitstwo bifurcation phenomena at the intersampling instants. One is saddle–node bifurcation,and the other is singular induced bifurcation which indicates that economic profit maybring impulse at some critical value, i.e., rapid expansion of biological population interms of ecological implications. On the other hand, for the sampling instants, the systemundergoes Neimark–Sacker bifurcation at a critical value of economic profit, i.e., theincrease of economic profit destabilizes the system and generates a unique closed invariantcurve. Moreover, the state feedback controller is designed so that singular inducedbifurcation and Neimark–Sacker bifurcation can be eliminated and the population can bedriven to steady states by adjusting harvesting costs and the economic profit. At the sametime, by using Matlab software, numerical simulations illustrate the effectiveness of theresults obtained here.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

At present, mankind is facing the problems about shortage of resource and worsening environment. So there hasbeen rapidly growing interest in the analysis and modelling of biological systems. From the view of human need, theexploitation of biological resources and harvest of population are commonly practiced in the fields of fishery, wildlifeand forestry management. The predator–prey system plays an important and fundamental role among the relationshipsbetween the biological population.Many authors [1–5] have studied the dynamics of predator–preymodelswith harvesting,and obtained complex dynamic behavior, such as stability of equilibrium, Bogdanov–Takens bifurcation, Hopf bifurcation,limit cycle, heteroclinic bifurcation and so on. Quite a number of references [6–11] have discussed permanence, extinctionand periodic solution of predator–prey models. In addition, there is also a considerable amount of literature on discretedynamical systems, e.g., see [11–14] and references therein,modelling some specieswhose generations are non-overlappingby applying Euler scheme to differential equations. Most of these discussions on biological models are based on normalsystems governed by differential equations or difference equations.In daily life, economic profit is a very important factor for governments, merchants and even every citizen, so it

is necessary to research biological economic systems, which can be described by differential–algebraic equations ordifferential-difference-algebraic equations. At present, most of differential algebraic equations can be found in the generalpower systems [15,16], economic administration [17], robot system [18], mechanical engineering [19] and so on. There

I This work was supported by National Science Foundation of China (60574011).∗ Corresponding author. Tel.: +86 24 83689929.E-mail addresses: zhangxueer@gmail.com (X. Zhang), qlzhang@mail.neu.edu.cn (Q.-l. Zhang).

1751-570X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2009.04.009

X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587 579

are also several biological reports on differential algebraic equations [20–22]. Digital control systems of power grids [23]and biological systems, such as neural networks [24,25] and genetic networks [26], are typical models mixed with bothcontinuous and discrete time sequences which can mathematically be formulated as differential-difference-algebraicequations. In addition, a lot of fundamental analyzing methods for differential algebraic equations and differential-difference-algebraic equations have been presented, such as local stability [27,28], optimal control [29], singular inducedbifurcation [30] and so on. However, to the best of our knowledge, there are few reports on differential-difference-algebraicequations in biological fields. This paper mainly studies the stability and bifurcations of a new biological economic systemformulated by differential-difference-algebraic equations. In what follows, we introduce the new biological economicsystem.The basic model we consider is based on the following Lotka–Volterra predator–prey system with harvest

dxdt= rx

(1−

xK

)− axy− Ex,

dydt= −dy+ bxy− Ey,

where x and y represent the prey density and predator density at time t , respectively. r > 0, d > 0 are the intrinsic growthrate of prey and the death rate of predator in the absence of food, respectively. K > 0 is the carrying capacity of prey. a > 0and b > 0 measure the effect of the interaction of the two population. E represents harvesting effort. Ex and Ey indicatesthat the harvests for prey and predator population are proportional to their densities at time t .Combining the economic theory of fishery resource [31] proposed byGordon in 1954,we can obtain a biological economic

system expressed by differential algebraic equationdxdt= rx

(1−

xK

)− axy− Ex,

dydt= −dy+ bxy− Ey,

0 = E(pxx− cx)+ E(pyy− cy)−m,

(1)

where px > 0 and py > 0 are harvesting reward per unit harvesting effort for unit weight of prey and predator, respectively.cx > 0 and cy > 0 are harvesting cost per unit harvesting effort for prey and predator, respectively.m ≥ 0 is the economicprofit per unit harvesting effort. From amodelling view,more details of the algebraic equation can be found in the appendix.When the populations have non-overlapping generations, many authors [32,33] have argued that the discrete time

models governed by difference equations are more appropriate than the continuous ones. Suppose the predator populationhas no overlap between successive generations. Applying the Euler method to the predator equation of the system (1)with integral step size τ , we have the following biological economic system expressed by differential-difference-algebraicequation

dxdt= rx

(1−

xK

)− axy− Ex,

yn+1 = yn + τ (−dyn + bxnyn − Enyn),0 = E(pxx− cx)+ En(pyyn − cy)−m.

(2)

We nondimensionalize the system (2) with the following scaling

x =xK, y =

ayr, E =

Er, t = r t, τ = r τ ,

and then obtain the following biological economic system expressed by differential-difference-algebraic equationdxdt= x(1− x)− xyn − Ex,

yn+1 = yn + τ(−dyn + bxnyn − Enyn),0 = E(pxx− cx)+ En(pyyn − cy)−m,

(3)

where xn = x(nτ), En = E(nτ), yn are the value at the instant nτ and xn, yn, En are constantly held during nτ ≤ t < (n+1)τfor the system (3). The non-dimensional parameters are defined as

d =dr, b =

bKr, px = rK px, py =

r2

apy, cx = rcx, cy = rcy.

Remark 1. The system (3) above is called hybrid dynamical system [27], which is formulated by differential-difference-algebraic equations.

580 X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587

For simplicity, let(f (x(t), xn, yn, E(t), En,m)g(xn, yn, En,m)

h(x(t), xn, yn, E(t), En,m)

)=

( x(1− x)− xyn − Exyn + τ [−dyn + bxnyn − Enyn]E(pxx− cx)+ En(pyyn − cy)−m

).

Without confusion, t is occasionally dropped from the related variables, i.e., x(t) and E(t) are simply expressed by x and E,respectively.In this paper, we mainly discuss the effects of economic profit on the dynamics of the system (3) in the region R5

+=

{(x(t), xn, yn, E(t), En)|x(t) ≥ 0, xn ≥ 0, yn ≥ 0, E(t) ≥ 0, En ≥ 0}. In Section 2, we study the existence and stabilityof equilibrium and dynamics of the system (3). We give sufficient conditions of existence for saddle–node bifurcationand singular induced bifurcation at the intersampling instants in Section 3. For the sampling instants, Section 4 discussesNeimark–Sacker Bifurcation and proves that the increase of economic profit destabilizes the system and generates theunique closed invariant curve. Moreover, in Section 5, the state feedback controller has been designed so that singularinduced bifurcation and Neimark–Sacker bifurcation can be eliminated and the population can stay at steady states byadjusting the harvesting costs and the economic profit. Numerical simulations verify the effectiveness of mathematicalconclusions in Section 6. Finally, we give remarks to conclude this paper in Section 7.

2. Stability of equilibrium and dynamics of the system (3)

2.1. Existence and stability of equilibrium

The equilibrium of the system (3) satisfies the following equations{x(1− x− yn − E) = 0,yn + τ(−dyn + bxnyn − Enyn) = yn,E(pxx− cx)+ En(pyyn − cy)−m = 0.

(4)

By the analysis of roots for the system (4), we obtain the following theorem:

Theorem 1. (1) If px > cx + cy and (px − cx − cy)2 ≥ 4pxm, the system (3) has an equilibrium P1(x1, xn1, yn1, E1, En1), wherex1 satisfies the following equation

pxx2 − (px + cx + cy)x+m+ cx + cy = 0,

and x1 = xn1, yn1 = 0, E1 = En1 = 1− x1.(2) If harvesting effort 0 < E2 < (b− d)/(1+ b) satisfies the following equation

(px − py − bpy)E2 + [(px − py)d+ (py − cx − cy)b]E − bm = 0,

the system has a positive equilibrium P2(x2, xn2, yn2, E2, En2), where x2 = xn2 = (E2+d)/b, yn2 = 1−x2−E2, and En2 = E2.

The Jacobian matrix of the system (3) is

J =

eτA + (eτA − 1)A−1B (eτA − 1)A−1C∂gR∂xn

∂gR∂yn

, (5)

where A = fx − fEh−1E hx = 1− 2x− yn − E +pxxEpxx−cx

,

B = fxn − fEh−1E [hxn − hEn(hE + hEn)

−1(hx + hxn)] − fEn(hE + hEn)−1(hx + hxn)

= −pxxE(pyyn − cy)

(pxx− cx + pyyn − cy)(pxx− cx),

C = fyn − fEh−1E [hyn − hEn(hE + hEn)

−1hyn ] − fEn(hE + hEn)−1hyn

= −x+pyEnx

pxx− cx + pyyn − cy,

∂gR∂xn= gxn − gEn(hE + hEn)

−1(hx + hxn) = bτyn +τpxEyn

pxx− cx + pyyn − cy,

∂gR∂yn= gyn − gEn(hE + hEn)

−1hyn

= 1+ τ(−d+ bxn − En)+τpyEnyn

pxx− cx + pyyn − cy.

X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587 581

By using the Jacobian matrix J and the Theorem 2 in [27], the local stability of an equilibrium for all of instants is obtainedby the following theorem:

Theorem 2. Assume that P∗(x∗, x∗, y∗, E∗, E∗) is an equilibriumof the system (3) and satisfies (pxx∗−cx)(pxx∗−cx+pyy∗−cy) 6=0, 1− 2x∗ − y∗ − E∗ + pxx∗E∗/(pxx∗ − cx) 6= 0. If at P∗, the following conditions are satisfied

(1)∣∣∣∣(eτA + (eτA − 1)A−1B) ∂gR∂yn − (eτA − 1)A−1C ∂gR∂xn

∣∣∣∣ < 1,(2)

∣∣∣∣eτA + (eτA − 1)A−1B+ ∂gR∂yn∣∣∣∣ < 1+ (eτA + (eτA − 1)A−1B) ∂gR∂yn − (eτA − 1)A−1C ∂gR∂xn ,

P∗ is an asymptotically stable equilibrium.

2.2. Dynamics of the system (3)

Assume that P∗(x∗, x∗, y∗, E∗, E∗) is an equilibrium of the system (3) and satisfies (pxx∗− cx)(pxx∗− cx+ pyy∗− cy) 6= 0.Then, according to the implicit function theorem, there exists a unique C∞ mapping in open neighborhoods of P∗ such thatat sampling instants

En =m

pxxn − cx + pyyn − cy,

and at the intersampling instants

E =m(pxxn − cx)

(pxx− cx)(pxxn − cx + pyyn − cy).

Then we can obtain the following reduced system of the system (3)dxdt= fR(x, xn, yn) = x(1− x)− xyn −

mx(pxxn − cx)(pxx− cx)(pxxn − cx + pyyn − cy)

,

yn+1 = gR(xn, yn) = yn + τ[−d+ bxn −

mpxxn − cx + pyyn − cy

]yn.

Therefore, the dynamics of the system (3) is given as follows:(1) When t = 0,

E(0) = E0 =m

pxx0 − cx + pyy0 − cy,

for the given x0 and y0.(2) When nτ < t < (n+ 1)τ , for n = 0, 1, 2, . . . ,

dxdt= x(1− x)− xyn −

mx(pxxn − cx)(pxx− cx)(pxxn − cx + pyyn − cy)

,

E(t) =m(pxxn − cx)

(pxx− cx)(pxxn − cx + pyyn − cy).

(6)

(3) When t = (n+ 1)τ , for n = 0, 1, 2, . . . ,xn+1 = lim

t→(n+1)τx(t),

yn+1 = yn + τ[−d+ bxn −

mpxxn + pyyn − cx − cy

]yn,

En+1 =m

(pxxn+1 − cx + pyyn+1 − cy).

(7)

3. Local bifurcations at the intersampling instants

3.1. Saddle–node bifurcation

In the absence of predator, the system (3) reduces to the following differential–algebraic systemdxdt= f (x, E) = x(1− x)− Ex,

0 = g(x, E) = E(pxx− cx)−m.(8)

582 X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587

Theorem 3. If px > cx and m = msn = (px − cx)2/4px, there exists a saddle–node bifurcation for the equilibriumPsn((px + cx)/2px, (px − cx)/2px) of the system (8) at the intersampling instants.

Proof. When px > cx andm = msn = (px − cx)2/4px, the system (8) has an equilibrium Psn(pxcx/2px, (px − cx)/2px), whichsatisfies

Dx fR|(Psn,msn) = (Dx f − DE f (DE g)−1Dxg)|(Psn,msn) = 0,(

∂ fR∂m

)|(Psn,msn)

= (Dm f − DE f (DE g)−1Dmg)|(Psn,msn)

= −px + cxpx(px − cx)

,(∂2 fR∂x2

)|(Psn,msn)

=

(−2−

2mcxpx(pxx− cx)3

)|(Psn,msn)

= −2(px + cx)px − cx

.

According to the theory of generic local bifurcation in [28], the system (8) undergoes saddle–node bifurcation at theequilibrium Psn for the intersampling instants. This completes the proof. �

Remark 2. Form the property of the saddle–node bifurcation, it is clear that the number of equilibrium becomes two fromzero whenm decreases throughmsn = (px − cx)2/4px, i.e., two positive equilibria form < msn, one is stable

P (1)sn

((px + cx)+

√(px − cx)2 − 4pxm2px

,(px − cx)−

√(px − cx)2 − 4pxm2px

),

and the other

P (2)sn

((px + cx)−

√(px − cx)2 − 4pxm2px

,(px − cx)+

√(px − cx)2 − 4pxm2px

)is unstable, no equilibria when m > msn, and only one positive equilibrium when m = msn. Thus if the economic profitm < msn, the prey population and the harvesting effortwill stay at steady equilibrium P

(1)sn for the single population economic

system (8), which is favorable to the breeding of the prey population and permanent and constant harvesting for fishermen.

3.2. Singular induced bifurcation

When the economic profitm = msib = [(px− cx−bcx+dpx)py− cypx](bcx−dpx)/p2x , Psib(cx/px, cx/px, 1− (b+1)cx/px+d, bcx/px − d, bcx/px − d) is the equilibrium of the system (3) and satisfies

hE |Psib = (pxx− cx)|Psib = 0.

The following theorem shows that the system (3) undergoes singular induced bifurcation at the equilibrium Psib.

Theorem 4. If 0 < bcx − dpx < px − cx, the system (3) undergoes singular induced bifurcation at the equilibrium Psib whenbifurcation parameter m is increased through msib at the intersampling instants.

Proof. Define

δ = hE = pxx− cx.

It is easy to calculate that the system (3) at the equilibrium Psib satisfies

δ|(Psib,msib )= 0,

bE |(Psib,msib ) =(−trace(fEadj(hE)hx)

)|(Psib,msib )

= cx

(bcxpx− d

),∣∣∣∣ fx fE

hx hE

∣∣∣∣|(Psib,msib )

=

∣∣∣∣1− 2x− yn − E −xpxE pxx− cx

∣∣∣∣|(Psib,msib )

= cx

(bcxpx− d

),

∣∣∣∣∣ fx fE fmhx hE hmδx δE δm

∣∣∣∣∣|(Psib,msib )

=

∣∣∣∣∣1− 2x− yn − E −x 0pxE pxx− cx −1px 0 0

∣∣∣∣∣|(Psib,msib )

= cx,

X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587 583

and

c|(Psib,msib ) =(δm −

(δx, δE

) ( fx fEhx hE

)−1 (fmhm

) )|(Psib,msib )

=

(−(px, 0

) (1− 2x− yn − E −xpxE pxx− cx

)−1 (0−1

) )|(Psib,msib )

=px

bcx − dpx.

Therefore, according to the Theorem 3 in [28], the system (3) undergoes singular induced bifurcation at the positive equi-librium Psib when the bifurcation parameterm increases throughmsib. Because of bE/c > 0, one eigenvalue of Amoves fromC− (the complex left half plane) to C+ (the complex right half plane) along the real axis by diverging through∞ when mincreases throughmsib. It brings an impulse (i.e., rapid) expansion of the population in terms of ecological implication. There-fore, the system (3) becomes unstable at the equilibrium Psib for the intersampling instants. This completes the proof. �

Remark 3. A hybrid system can be modelled by parameter dependent differential-difference-algebraic equation of thefollowing form{x = f (x(t), xn, yn, z(t), zn, α),

yn+1 = g(xn, yn, zn, α),0 = h(x(t), xn, yn, z(t), zn, α),

nτ ≤ t < (n+ 1)τ , n = 0, 1, . . . ,

where x, xn, yn, z and zn belong to the Euclidean spaces Rnx , Rnx , Rny , Rnz , Rnz , respectively. f : Rnx+nx+ny+nz+nz → Rnx ,g : Rnx+ny+nz → Rny and h : Rnx+nx+ny+nz+nz → Rnz are all assumed to be functions of class C∞. τ is a sampling interval.α ∈ R is a one-dimensional parameter. Local bifurcation analysis of hybrid system often results in four types of bifurcations:fold, flip, Neimark–Sacker and singularity induced bifurcations(SIB). The SIB is a new type of bifurcations and does not occurin ordinary differential equation system. It occurs when an equilibrium point crosses the following singular surface:

S := {(xn, yn, zn, α)|det[Dzh(x, xn, yn, z, zn, α)] = 0}.

At the SIB, the equilibrium point undergoes stability exchanges and one eigenvalue of the hybrid system becomesunbounded. One important implication of the occurrence of SIB is that it causes impulsive phenomena, which mightyield catastrophic consequences — for example, hard fault [34] of electrical circuits and rapid expansion of population forbiological system. More details of SIB can be found in [27,28].

4. Local bifurcations at the sampling instants

For simplicity, denote

J =(a11 a12a21 a22

),

where a11 = eτA + (eτA − 1)A−1B, a12 = (eτA − 1)A−1C , a21 = ∂gR/∂xn, a22 = ∂gR/∂yn.According to the Theorem 6 in [27], we obtain the following theorem:

Theorem 5. Assume that there exists an equilibrium Pns(xns, xns, yns, Ens, Ens) of the system (3) at m = mns, which satisfies(a11a22 − a12a21)|Pns = 1, ((a11 − a22)

2+ 4a12a21)|Pns < 0, (pxxns − cx)(pxxns − cx + pyyns − cy) 6= 0 and 1 − 2xns − yns −

Ens + pxxnsEns/(pxxns − cx) 6= 0. There exists the Neimark–Sacker bifurcation which generates a unique closed invariant curvefrom the equilibrium Pns, if at Pns

ddm

(√a11a22 − a12a21

)6= 0,(

12(a11 + a22)+ i

√4− (a11 + a22)2

)k6= 1, for k = 1, 2, 3, 4.

However, it is difficulty to analytically compute the bifurcation value mns, hence we will study Neimark–Sackerbifurcation numerically in Section 6.

5. State feedback control

Unstable fluctuations, bifurcations and the impulsive phenomena have always been regarded as unfavorable ones fromthe point of view of ecologicalmanagers. In order to plan harvesting strategies andmaintain the sustainable development of

584 X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587

system, it is necessary to take action to stabilize biological population. This paper proposes following state feedback controlmethod

dxdt= x(1− x)− xyn − Ex,

yn+1 = yn + τ [−dyn + bxnyn − Enyn],0 = E(pxx− cx)+ En(pyyn − cy)+ l1(E − E∗)+ l2(En − E∗n )−m,

(9)

where L = (l1, l2)T is the feedback gain, E∗ and E∗n are the coordinates of the equilibrium of the system (3).The Jacobian matrix of the system (9) is

J =

eτ A + (eτ A − 1)A−1B (eτ A − 1)A−1C∂ gR∂xn

∂ gR∂yn

, (10)

where A = 1− 2x− yn − E +pxxE

pxx−cx+l1,

B = −pxxE(pyyn − cy + l2)

(pxx− cx + pyyn − cy + l1 + l2)(pxx− cx + l1),

C = −x+pyEnx

pxx− cx + pyyn − cy + l1 + l2,

∂ gR∂xn= bτyn +

τpxEynpxx− cx + pyyn − cy + l1 + l2

,

∂ gR∂yn= 1+ τ(−d+ bxn − En)+

τpyEnynpxx− cx + pyyn − cy + l1 + l2

.

According to the theory of asymptotical stability [27], the following result can be obtained:

Theorem 6. If the gain matrix L = (l1, l2)T satisfies that the two eigenvalues of J at the equilibrium P∗ for the system(9) have moduli less than 1, P∗ is an asymptotically stable equilibrium at both sampling and intersampling instants.

In fact, the third equation of the system (9) can be rewritten as

E(pxx− cx)+ En(pyyn − cy)− m = 0,

where cx = cx − l1, cy = cy − l2 and m = m+ l1E∗ + l2E∗n .

Remark 4. The state feedback control method can be implemented by adjusting the harvesting costs and the economicprofit so that biological population can be controlled at steady states and those unfavorable phenomena will be eliminated.In our life, managers can take action, such as adjusting revenue, drawing out favorable policy to encourage fishery, abatingpollution and so on, in order to control the harvesting costs and economic profit.

6. Numerical simulation

To illustrate the above results obtained, let us consider the following particular cases.Case I: Saddle–node bifurcation at the intersampling instants.For the parameter values px = 3.7, and cx = 1, there exists a unique equilibrium P∗1 (0.6351, 0.3649) for the system

(8) when the economic profit is msn = 0.4926. For m < msn, the system (8) has two hyperbolic equilibria: take m = 0.45for example, two equilibria are P (1)1 (0.7424, 0.2576) and P

(2)1 (0.5279, 0.4721), whose eigenvalues are−0.3373 and 0.4396,

respectively, i.e., P (1)1 is stable and P(2)1 is unstable. Form > msn, the system (8) has no equilibria.

Case II: Singular induced bifurcation at the intersampling instants.Choose d = 0.5, b = 3, τ = 1, px = 3.7, cx = 1, py = 4.1, cy = 1.2,msib = 0.1609, then P∗2 (0.2703, 0.2703, 0.4189,

0.3108, 0.3108) is the positive equilibrium of the system (3). Due to fixing of xn = 0.2703, yn = 0.4189, En = 0.3108,the dynamics of the hybrid system (3) is reined by the system (6). By simply calculation, we obtain that the equilibriumand eigenvalue of the system (3) are P (1)2 (0.2701, 0.2703, 0.4189, 0.311, 0.3108) and −493.6009 for m = 0.1607 andthen become P (2)2 (0.2705, 0.2703, 0.4189, 0.3106, 0.3108) and 365.4519 for m = 0.1610, respectively. It is obvious thatthe eigenvalue moves from C− to C+ along the real axis by diverging through∞. From the point of biology, this meansthat economic profit may bring impulse at some critical value, i.e., rapid expansion of the density of prey population. Ifthe feedback gain is chosen as L = (0.5, 1)T, the eigenvalues become 0.3523 and 0.4776 for m = 0.1607 and m =0.1610, respectively. Therefore, impulsive phenomenon is eliminated by controlling the harvesting effort. That is, the preypopulation can stay at a steady state.

X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587 585

a b

c d

Fig. 1. Whenm = 0.102 > mns , a unique closed invariant curve from the equilibrium P∗3 occurs.

Fig. 2. Dynamics of the controlled system (9) with the feedback gain L = (3, 5)T .

Case III: Neimark–Sacker bifurcation at the sampling instants.Take d = 0.4, b = 2, τ = 0.1, px = 5.7, cx = 1.5, py = 3.5, cy = 0.9,mns = 0.1 for example, then the system (3)

has a unique positive equilibrium P∗3 (0.2414, 0.2414, 0.6759, 0.0827, 0.0827) and eigenvalues are λ± = 0.9986±0.0522i.For mns = 0.1, there are |λ±| = 1 and d|λ(mns)|/dm = 0.1375. Hence from the Theorem 5, there exists Neimark–Sackerbifurcation. Fig. 1 shows that there exists a unique closed invariant curve from the equilibrium P∗3 . If the feedback gain ischosen as L = (3, 5)T, the controlled system states stay at P∗3 after a short transient process, which is depicted in Fig. 2.

586 X. Zhang, Q.-l. Zhang / Nonlinear Analysis: Hybrid Systems 3 (2009) 578–587

7. Discussion

So far, considerable attention has been focused on continuous–time and discrete–time nonlinear systems in biology.However, to the best of our knowledge, the reports of differential–algebraic or differential-difference-algebraic biologicalsystems are few. This paper discusses a class of hybrid biological economic models, where the dynamics of prey populationis governed by differential equations, predator population is governed by difference equations and economic theory byalgebraic equations. This paper shows that this hybrid biological economic model exhibits the following three bifurcations.(1) Saddle–node bifurcation at the intersampling instants: if the economic profit m < msn, the prey population and theharvesting effort will stay at steady states. Thus it is favorable to the breeding of the prey population and permanent andconstant harvesting for fishermen by keeping the economic profit in some region.(2) Singular induced bifurcation at the intersampling instants: when the economic profit increases throughmsib, the densityof population will expand rapidly, which induce ecosystem unbalance and even biological disaster. Applying the statefeedback control method, which can be implemented by adjusting the harvesting costs and the economic profit, impulsivephenomenon will be eliminated and biological population can be controlled at steady states.(3) Neimark–Sacker bifurcation at the sampling instants: form > mns and |m−mns| small enough, the densities of prey andpredator population emerge periodical oscillation and a unique closed invariant curve occurs. After adjusting harvestingcosts and the economic profit, two populations stay at steady states.Therefore, the state feedback control method is efficient. In practice, we can regulate the harvesting costs and the

economic profit, such as adjusting revenue and drawing out favorable policy to encourage or improve fishery or abatingpollution, so that the population can be driven to steady states.

Acknowledgments

The authors gratefully thank the contribution of National ResearchOrganization and the anonymous authorswhoseworklargely constitutes this sample file.

Appendix

In 1954, Gordon [31] studied the effect of harvest effort on ecosystem from an economic perspective and proposed thefollowing economic theory:

Net Economic Revenue (NER) = Total Revenue (TR)− Total Cost (TC).

In the introductory section,wementioned thatm, pi and ci, i = x, y represent the economic profit, harvesting reward per unitharvesting effort for unitweight andharvesting cost per unit harvesting effort, respectively; x and y are the prey andpredatordensity, respectively; and E are harvesting effort for prey and predator population. Then NER = m, TR = pxEx + pyEy andTC = cxE + cyE. Substituting them into the economic theory equation mentioned above, the following algebraic equationcan be established

E(pxx− cx)+ E(pyy− cy) = m.

In fact, the dynamics of the predator–preymodel with constant harvesting or constant harvesting rate have been studiedby many references [1–3], where the mathematical models are described by ordinary differential equations. However, theharvesting (or harvesting rate) is not always constant and may vary with numerous factors, such as seasonality, revenue,market demand, harvesting cost and so on. Hence, it is more reasonable that the harvesting effort should be a variable fromreal world view. Based on the economic theory of Gordon, this paper regards both population(prey and predator) densitiesand the harvesting as variables, which are governed by differential equations and algebraic equations, and investigates theeffects of harvesting on the dynamics of the prey–predator system.

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