Applied Mathematics and Computation 217 (2011) 8778–8792

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Applied Mathematics and Computation

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

Optimal control of harvest and bifurcation of a prey–predator modelwith stage structure

Kunal Chakraborty a,⇑, Milon Chakraborty b, T.K. Kar b

a Department of Mathematics, MCKV Institute of Engineering, 243, G.T. Road (N), Liluah, Howrah 711204, West Bengal, Indiab Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711103, West Bengal, India

a r t i c l e i n f o a b s t r a c t

Keywords:Prey–predator fisheryStage structureDifferential algebraic systemSingularity induced bifurcationOptimal control

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.03.139

⇑ Corresponding author.E-mail addresses: kc_mckv@yahoo.co.in (K. Chak

This paper describes a prey–predator model with stage structure for prey. The adult preyand predator populations are harvested in the proposed system. The dynamic behaviorof the model system is discussed. It is observed that singularity induced bifurcation phe-nomenon is appeared when variation of the economic interest of harvesting is taken intoaccount. State feedback controller is incorporated to stabilize the model system in caseof positive economic interest. Harvesting of prey and predator population are used as con-trols to develop a dynamic framework to investigate the optimal utilization of the resource,sustainability properties of the stock and the resource rent earned from the resource. ThePontryagin’s maximum principle is used to characterize the optimal controls. Theoptimality system is derived and then solved numerically using an iterative method withRunge–Kutta fourth order scheme. Simulation results show that the optimal controlscheme can achieve sustainable ecosystem.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The field of renewable natural resources covers various areas such as, agriculture, fisheries and forestry. The understand-ing and management of this type of resource is a very complex problem. This complexity is mainly due to the issue of therequired sustainability of the underlying natural system. Fisheries management requires the integration of the biology andecology of fish resources with the socio-economic, resource user, and management institutional factors that affect the behav-ior of fishers and policy-makers. The fundamental objective of the fishery management is to ensure conservation of fisheryresource into the future and to provide a sustainable flow of benefits to human society. In order to control the stock, catchesand fishing effort of the fishery it is possible to derive some useful management procedure that can be imposed to the fisheryto get protection from overexploitation.

The problems of predator–prey systems in the presence of harvesting have been discussed by many authors, but mostresearch has focused attention on optimal exploitation, guided entirely by profits from harvesting. Stage-structuredpopulation models also have received great attention in recent years. Aiello and Freedman [20] studied a stage-structuredmodel of one species growth consisting of immature and mature members. Cui et al. [4] analyzed the effect of dispersalon the permanence of a stage-structured single-species population model without time delay. A significant amount ofresearch has been carried out based on different kinds of predator prey system with division of the predators into immatureand mature individuals like Kar and Pahari [15], Magnusson [6], Wang et. al. [19], Xu et al. [11], Gao et al. [13], Xu and Ma[10] etc. and references therein.

. All rights reserved.

raborty), tkar1117@gmail.com (T.K. Kar).

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8779

The structured models are able to respond the impacts of certain technical measures towards sustainability. They permit aqualitative description of the system since they take into account both features: the fish size and the time mechanism of repro-duction of the exploited stock. Freedman [3] pointed out that there are times when it is more natural to consider discrete mod-els than to consider continuous models. If the population numbers are small or if births and deaths all occur at discrete times,or within certain intervals of time, discrete model would indeed be more realistic. Therefore, seasonally reproducing speciesthat produce offspring during a short annual period have birth pulse dynamics usually described with discrete populationmodels. Gao and Chen [12] proposed an exploited single-species discrete model with stage structure for the dynamics in a fishpopulation for which births occur in a single pulse once per time period. They constructed bifurcation diagrams throughconsidering birth rate as the bifurcation parameter, and displayed the complex dynamic behaviors of the model system.

The books of Clark [1,2] and Mark Kot [8] are extremely relevant to the optimal control problems. Jerry and Rassi [9] exam-ined a structured fishing model, basically displaying the two stages of the ages of a fish population, which are in our case juve-nile and adults. They associated the maximization of the total discounted net revenues derived by the exploitation of the stockto this model. They also developed the exploitation strategy of the optimal control problem. Ding et al. [17] studied the controlproblem of maximizing the net benefit in the conservation of a single species with a fixed amount of resources. They investi-gated the existence of an optimal control and the uniqueness and characterization of the optimal control.

For the purpose of system modelling and analysis in different fields of applied sciences and engineering differential–algebraic equations (DAEs) is considered an essential tool. Naturally, differential or algebraic equations usually form themathematical model of the system. It is observed that a numerous number of research articles proposed the populationdynamics and analyzed the stability of the system in presence of harvesting effort and time delay but quite a few numberof articles investigated the dynamical behavior of bioeconomic models using DAEs. Kar and Chakraborty [14] proposed abiological economic model based on prey–predator dynamics and found singularity induced bifurcation (SIB) phenomenonand impulsive behavior of the model system at the interior equilibrium point using DAEs. They have designed an optimalcontroller to eliminate singularity induced bifurcation and impulsive behavior of the model system when positive economicinterest is taken into account.

Here, we consider a simple harvested prey–predator model with stage structure for prey which is to be used to investigatethe comparative study of the resource stock and harvesting. The existence of SIB phenomenon is verified at the interior equi-librium of the system with zero economic profit and state feedback controller is designed to overcome SIB. The control prob-lem is formulated and the corresponding optimality system that characterizes the (continuous) optimal control solution isdescribed. We find a characterization for the optimal control and show numerical results for scenarios of different illustrativeparameter sets. The numerical results provide more realistic features of the system. The solution is performed using an iter-ative method with Runge–Kutta fourth order scheme of optimal control.

2. Model description and assumptions

We consider a prey–predator model with stage structure for prey and it is assumed that only adult prey and predatorpopulations are harvested. Let us assume x, y and z are respectively the size of juvenile prey, adult prey and predator pop-ulation at time t. It is considered that all these populations are growing in closed homogeneous environment. At any time t,the birth rate of the juvenile prey populations is assumed to be proportional to the density of existing adult prey populationwith proportionality constant r and the rate of transformation of the adult prey is proportional to the density of existingjuvenile prey with proportionality constant s. The predator population consumes the juvenile and adult prey populationsrespectively at the rate a and b and contributes to its growth rate axz and byz. Keeping these aspects in view, the dynamicsof the system may be governed by the following system of differential equations:

dxdt¼ ry� d1x� sx� cx2 � axz; ð1aÞ

dydt¼ sx� d2y� ky2 � byz; ð1bÞ

dzdt¼ axzþ byz� d3z� rz2; ð1cÞ

where d1(c), d2(k) and d3(r) are respectively the death rates (intra-specific coefficients) of the juvenile prey, adult prey andpredator populations.

Let us now consider two separate models. In the first model, only prey population is harvested however in the secondmodel, we consider the harvesting of predator population.

3. Model with prey harvesting

The functional form of harvest is generally considered using the phrase catch-per-unit-effort (CPUE) hypothesis [2] todescribe an assumption that catch per unit effort is proportional to the stock level. Therefore, harvesting function h1(t)can be written as:

h1 ¼ q1E1y; ð2Þ

8780 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

where q1 is the catchability co-efficients of adult prey population, E1 is the effort used to harvest the population. Let usextend our model by considering the following algebraic equation

ðp1q1y� c1ÞE1 �m1 ¼ 0; ð3Þ

where c1 is the constant fishing cost per unit effort, p1 is the constant price per unit biomass of landed fish of adult preypopulation and m1 is the total economic rent obtained from the fishery.

Hence, using (2) and (3) system (1) finally takes the form

dxdt¼ ry� d1x� sx� cx2 � axz; ð4aÞ

dydt¼ sx� d2y� ky2 � byz� q1E1y; ð4bÞ

dzdt¼ axzþ byz� d3z� rz2; ð4cÞ

ðp1q1y� c1ÞE1 �m1 ¼ 0; ð4dÞ

with initial conditions x(0) P 0, y(0) P 0 and z(0) P 0.

3.1. Qualitative analysis of differential algebraic model system

In this section, we mainly investigate the occurrence of singularity induced bifurcation and the effects of economic profiton the dynamics of the system (4). From the view of ecological management, it is sufficient to consider the interior equilib-rium point of the system (4), as it represents the coexistence of three species.

The differential algebraic system (4) can be expressed in the following way,

f ðX; E1;m1Þ ¼

f1ðX; E1;m1Þ

f2ðX; E1;m1Þ

f3ðX; E1;m1Þ

2664

3775 ¼

ry� d1x� sx� cx2 � axz

sx� d2y� ky2 � byz� q1E1y

axzþ byz� d3z� rz2

2664

3775;

gðX; E1;m1Þ ¼ ðp1q1y� c1ÞE1 �m1;

where X = (x,y,z)t.Let us now study the dynamic behavior of the system (4). The local stability of the interior equilibrium point,

P1ðx�; y�; z�; E�1Þ can be investigated using the SIB phenomena based on the assumption that the interior equilibrium pointexists. From system (4) we have the following matrix

M ¼ DXf � DE1 f ðDE1 gÞ�1DXg ¼� ry�

x� � cx� r �ax�

s � sx�y� � ky� þ p1q2

1y�E�1p1q1y��c1

�by�

az� bz� �rz�

2664

3775;

where DXf, etc., denotes the matrix of partial derivatives of the components of f with respect to X.To check the existence of the SIB phenomena, total economic rent m1 is assumed to be the bifurcation parameter. Con-

sequently we have the following theorem:

Theorem 1. The differential algebraic system (4) has a singularity induced bifurcation at the interior equilibrium pointP1ðx�; y�; z�; E�1Þ. When the bifurcation parameter m1 increases through zero the stability of the interior equilibrium pointP1ðx�; y�; z�; E�1Þ changes from stable to unstable.

Proof. It is evident that DE1 g ¼ p1q1y� c1 has a simple zero eigen value. Let us now define

DðX; E1;m1Þ ¼ DE1 g ¼ p1q1y� c1:

(i) From the existence of P1ðx�; y�; z�; E�1Þ it follows that

Trace½DE1 fadjðDE1 gÞDXg�P1¼ �p1q2

1y�E�1 – 0;

where Trace½DE1 fadjðDE1 gÞDXg�P1is the sum of the elements of the principal diagonal of the 3 � 3 order matrix

½DE1 fadjðDE1 gÞDXg� at the interior equilibrium point P1 of the system (4).

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8781

(ii) It can be proved that

DXf DE1 f

DXg DE1 g

��������P1

¼

� ryx � cx r �ax 0

s � sxy � ky �by �q1y

az bz �rz 00 p1q1E1 0 0

���������

���������P1

¼ p1q21a

2x�y�z�E�1 þ p1q21crx�y�z�E�1 þ

p1q21rry�2z�E�1

x�– 0

(iii) It can also be shown that

DXf DE1 f Dm1 f

DXg DE1 g Dm1 g

DXD DE1D Dm1 D

��������������

P1

¼

� ryx � cx r �ax 0 0

s � sxy � ky �by �q1y 0

az bz �rz 0 00 p1q1E1 0 0 �10 p1q1 0 0 0

������������

������������P1

¼ p1q21a

2x�y�z� þ p1q21crx�y�z� þ p1q2

1rry�2z�

x�– 0:

It is observed from (i)–(iii) that all the conditions for singularity induced bifurcation are satisfied (see [16]). Hence the dif-ferential algebraic system (4) has a singularity induced bifurcation at the interior equilibrium point P1ðx�; y�; z�; E�1Þ for m1 = 0.

Again, it is noted that

M1 ¼ �Trace½DE1 fadjðDE1 gÞDXg�P� ¼ p1q21y�E�1 – 0:

M2 ¼ Dm1 D� DXD DE1Dð ÞDXf DE1 f

DXg DE1 g

!�1 Dm1 f

Dm1 g

!24

35

P1

¼ 1E�1:

Therefore, the existence of interior equilibrium point implies that

M1

M2¼ p1q2

1y�E�12> 0:

Hence, it can be concluded that when m1 increases through zero, one eigenvalue of the model system (4) moves from C� (setof all complex numbers with negative real part) to C+ (set of all complex numbers with positive real part) along the real axisby diverging through 1 (see [16]). Consequently the stability of the system (4) is influenced through this behavior i.e., thestability of the system at the interior equilibrium point P1ðx�; y�; z�; E�1Þ changes from stable to unstable. h

In consequence to the above theorem it is clear that the system (4) becomes unstable when the economic interest of theharvesting is considered to be positive. If we consider economic perspective of the fishery it is obvious that fishery agenciesare interested towards the positive economic rent earned from the fishery. It is also noted that an impulsive phenomenoncan occur through singularity induced bifurcation and it may lead to the collapse of the sustainable ecosystem of the fishery.

Therefore, it is necessary to reduce the impulsive phenomenon and to resume the sustainability of the ecosystem wheneconomic interest of the fishery is considered to be positive.

Thus, to stabilize the system (4) in case of positive economic interest a state feedback controller [7] can be designed of theform w1ðtÞ ¼ u1ðE1ðtÞ � E�1Þ, where u1 stands for net feedback gain.

After introducing the state feedback controller, we rewrite the system (4) as:

dxdt¼ ry� d1x� sx� cx2 � axz; ð5aÞ

dydt¼ sx� d2y� ky2 � byz� q1E1y; ð5bÞ

dzdt¼ axzþ byz� d3z� rz2; ð5cÞ

ðp1q1y� c1ÞE1 �m1 þ u1ðE1ðtÞ � E�1Þ ¼ 0: ð5dÞ

Consequently, we have the following theorem:

Theorem 2. The differential algebraic system (5) is stable at the interior equilibrium point, P1ðx�; y�; z�; E�1Þ, of the system (4) ifu1 > max(a1,a2,a3) where

8782 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

a1 ¼y�E�1p1q2

1sx�y� þ

ry�x� þ x�cþ y�kþ z�r

;

a2 ¼ry�2E�1p1q2

1x� þ x�y�cE�1p1q2

1 þ y�z�rE�1p1q21

x�z�a2 þ y�z�b2 þ sx�2cy� þ

ry�2kx� þ x�y�ckþ sx�z�r

y� þry�z�r

x� þ x�z�crþ y�z�kr;

a3 ¼x�y�z�a2E�1p1q2

1 þry�2z�rE�1p1q2

1x� þ x�y�z�crE�1p1q2

1

sx�2z�y� ða2 þ crÞ þ ðsx� þ ry�Þz�abþ ry�2z�

x� ðb2 þ krÞ þ x�y�z�ðb2cþ a2kþ ckrÞ

:

Proof. For the system (5), we can obtain the following Jacobian at the interior equilibrium point P1ðx�; y�; z�; E�1Þ, of the sys-tem (4),

N ¼� ry�

x� � cx� r �ax�

s � sx�y� � ky� þ p1q2

1y�E�1u1

�by�

az� bz� �rz�

2664

3775:

Therefore, the characteristic polynomial of the matrix N is given by

l3 þ r1ðX; E1Þl2 þ r2ðX; E1Þlþ r3ðX; E1Þ ¼ 0;

where

r1 ¼sx�

y�þ ry�

x�þ x�cþ y�kþ z�r� y�E�1p1q2

1

u1;

r2 ¼ x�z�a2 þ y�z�b2 þ sx�2cy�þ ry�2k

x�þ x�y�ckþ sx�z�r

y�þ ry�z�r

x�þ x�z�crþ y�z�kr� ry�2E�1p1q2

1

x�u1� x�y�cE�1p1q2

1

u1

� y�z�rE�1p1q21

u1;

r3 ¼sx�2z�a2

y�þ sx�z�abþ ry�z�abþ ry�2z�b2

x�þ x�y�z�b2cþ x�y�z�a2kþ sx�2z�cr

y�� x�y�z�a2E�1p1q2

1

u1þ ry�2z�kr

x�

þ x�y�z�ckr� ry�2z�rE�1p1q21

x�u1� x�y�z�crE�1p1q2

1

u1:

According to the Routh Hurwitz criterion it can be concluded that the system (5) is stable at the interior equilibrium pointP1ðx�; y�; z�; E�1Þ, of the system (4) if the net feedback gain u1 satisfies the following condition:

u1 > maxða1; a2; a3Þ;

where a1, a2 and a3 are as given in the theorem.Hence, it is possible to design a suitable controller function such that the singularity induced bifurcation can be

eliminated. Thus, the impulsive phenomenon of a sustainable ecosystem can also be removed. Again, the economic interestof fishery managers may be achieved using suitably designed state feedback controller. h

3.2. Numerical simulations of differential algebraic system

In this section, the existence of the analytically proved results are verified using some hypothetical value of the param-eters. For the purpose of simulation experiments we mainly use the software MATLAB 7.0 and MATHEMATICA 5.2. Let usconsider the differential algebraic model system (4) in the following form:

dxdt¼ 4:8y� 0:001x� 2:5x� 1:6xz� 0:2x2;

dydt¼ 2:5x� 0:002y� 0:3y2 � 1:2yz� 1:25E1y;

dzdt¼ 1:6xzþ 1:2yz� 0:003z� 0:4z2;

ð15� 1:25y� 0:2ÞE1 � 0:5 ¼ 0:

The existence of singularity induced bifurcation of the system (4) is clearly shown in the Table 1.

Table 1Existence of SIB phenomenon and state feedback controller to eliminate SIB.

Net revenue and state feedback gain Interior equilibrium of the system (4) Eigenvalues

m1 = �0.5 and u1 = 0 x⁄ = 0.00663, y⁄ = 0.00352,z� ¼ 0:0296; E�1 ¼ 3:73

�8.89692, �0.658129, �0.0128495

m1 = 0.5 and u1 = 0 x⁄ = 0.0323, y⁄ = 0.0188, z� ¼ 0:178; E�1 ¼ 3:25 �4.10338, �0.0713676, 6.3928m1 = 0.5 and u1 = 8 x⁄ = 0.0323, y⁄ = 0.0188, z� ¼ 0:178; E�1 ¼ 3:25 �6:98754; �0:00288808� 0:1196_ı,

�0:00288808þ 0:1196_ı

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8783

It is noted that when m1 = �0.5 the eigenvalues are �8.89692, �0.658129, �0.0128495 where as for m1 = 0.5 the eigen-values become �4.10338, �0.0713676, 6.3928. Therefore, it is clearly observed from the Table 1 that as m increases throughzero two eigenvalues of the characteristic polynomial of the system (4) remain same but one eigenvalue of the system (4)moves from C� to C+ along the real axis by diverging through 1. Hence, the stability of the system (4) at the interior equi-librium point ðx�; y�; z�; E�1Þ changes from stable to unstable.

To stabilize the system (4) in case of positive economic interest, let us consider a state feedback controller of the formw1(t) = u1(E1(t) � 3.25).

It is possible to evaluate the numerical value of net feedback gain using Theorem 2. Therefore, we computeu1 > max(0.199661,7.23087,2.21634) for the system (5). Let us take the numerical value of net feedback gain as u1 = 8. Itis clearly observed that when m1 increases through zero all the eigenvalues remain negative or complex with negative realparts, that is, the stability of the system (5) may be resumed at the interior equilibrium point ðx�; y�; z�; E�1Þ in case of positiveeconomic profit. Hence, singularity induced bifurcation can be eliminated from the system (5) at the interior equilibriumpoint ðx�; y�; z�; E�1Þ when net economic profit increases through zero and taken to be positive.

3.3. The optimal control problem

In commercial exploitation of renewable resources the fundamental problem from the economic point of view, is todetermine the optimal trade-off between present and future harvests. The emphasis of this section is on the profit-makingaspect of fisheries. It is a thorough study of the optimal harvesting policy and the profit earned by harvesting, focusing onquadratic costs and conservation of fish population by constraining the latter to always stay above a critical threshold. Theprime reason for using quadratic costs is that it allows us to derive an analytical expression for the optimal harvest; theresulting solution is different from the bang-bang solution which is usually obtained in the case of a linear cost function.It is assumed that price is a function which decreases with increasing biomass. Thus, to maximize the total discountednet revenues from the fishery, the optimal control problem can be formulated as

Jðh1Þ ¼Z tf

t0

e�dt ðp1 � v1h1Þh1 �c1h1

q1y

� �; ð6Þ

where c1 be the constant fishing cost per unit effort, p1 is the constant price per unit biomass of harvested population, v1 is aneconomic constant and d is the instantaneous annual discount rate.

The problem (6), subject to population Eqs. (4a)–(4c) and control constraints 0 6 h1 6 h1max, can be solved by applyingPontryagin’s maximum principle. The convexity of the objective function with respect to h1, the linearity of the differentialequations in the control and the compactness of the range values of the state variables can be combined to give the existenceof the optimal control.

Suppose h�1 is an optimal control with corresponding states x⁄, y⁄ and z⁄. We are seeking to derive optimal control h�1 suchthat

Jðh�1Þ ¼ maxfJðh1Þ : h1 2 Ug;

where U is the control set defined by

U ¼ fh1 : ½t0; tf � ! ½0;h1max�jh1 is Lebesgue measurableg:

The current value Hamiltonian [21, pp. 151–153] of this control problem is

H¼ ðp1�v1h1Þh1�c1h1

q1y

� �þ k1ðry�d1x� sx�cx2�axzÞþ k2ðsx�d2y� ky2�byz�h1Þþ k3ðaxzþbyz�d3z�rz2Þ;

where k1(t), k2(t) and k3(t) are the current value multipliers.Here we have used the current value Hamiltonian instead of the usual Hamiltonian. As the current value Hamiltonian

reduces the entire system autonomous therefore the optimal solution describing by the differential equations will beautonomous. It is to be noted that the autonomous differential equations are easier to solve compare to the nonautonomousdifferential equations.

The transversality conditions give ki(tf) = 0, i = 1,2,3.

8784 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

Now, it is possible to find the characterization of the optimal control h�1. On the set ftj0 < h�1ðtÞ < h1maxg, we have

@H@h1¼ p1 � 2h1v1 �

c1

q1y� k2 ¼ 0 at h�1ðtÞ:

This implies,

h�1 ¼p1q1y� � q1y�k2 � c1

2q1v1y�: ð7Þ

Now, the autonomous set of equations of the control problem (6) are

dk1

dt¼ dk1 �

@H@x¼ dk1 � ½ð�d1 � s� za� 2xcÞk1 þ sk2 þ zak3�; ð8Þ

dk2

dt¼ dk2 �

@H@y¼ dk2 �

c1h1

q1y2 þ rk1 þ ð�d2 � zb� 2ykÞk2 þ zbk3

� �; ð9Þ

dk3

dt¼ dk3 �

@H@z¼ dk3 � ½�xak1 � ybk2 þ k3ð�d3 þ xaþ yb� 2zrÞ�: ð10Þ

Therefore, we arrive to the following theorem:

Theorem 3. There exists an optimal control h�1 and corresponding solution x⁄, y⁄ and z⁄ that maximizes J(h1) over U. Furthermore,there exists adjoint functions, k1, k2 and k3 satisfying the Eqs. (8)–(10) with transversality conditions ki(tf) = 0, i = 1,2,3. Moreover,

the optimal control is given by h�1 ¼p1q1y��q1y�k2�c1

2q1v1y� .

3.4. Numerical simulation of optimal control problem

The numerical simulation of optimal control [5] under various parameter sets can be done using 4th order Runge–Kuttaforward–backward sweep method. In this method, the system state Eqs. (4a)–(4c) and their corresponding adjoint Eqs. (8)–(10) are simultaneously solved. Initially we make a guess for optimal control and then solve the system of state Eqs. (4a)–(4c)forward in time using the Runge–Kutta method with the initial conditions (x0, y0 and z0). Then, using the state values, theadjoint Eqs. (8)–(10) are solved backward in time using the Runge–Kutta method with the transversality conditions. At thispoint, the optimal control is updated using the values for the state and adjoint variables. The updated control replaces theinitial control and the process is repeated until the successive iterates of control values are sufficiently close. The conver-gence of such an iterative method is based on the work of Hackbush [18].

At first, we discretize the interval [t0, tn] at the points ti = t0 + ih(i = 0,1,2, . . . ,n) where h is the time step such that tn = tf.Now a combination of forward and backward difference approximation is used as follows:

xiþ1 � xi

h¼ ryi � d1xiþ1 � sxiþ1 � cx2

iþ1 � axiþ1zi;

yiþ1 � yi

h¼ sxiþ1 � d2yiþ1 � ky2

iþ1 � byiþ1zi � h1 i;

ziþ1 � zi

h¼ axiþ1ziþ1 þ byiþ1ziþ1 � d3ziþ1 � rz2

iþ1:

By using a similar technique, we approximate the time derivative of the adjoint variables by their first-order backward-difference and we use the appropriated scheme as follows:

kn�i1 � kn�i�1

1

h¼ dkn�i�1

1 � ½ð�d1 � s� ziþ1a� 2xiþ1cÞkn�i�11 þ skn�i

2 þ ziþ1akn�i3 �;

kn�i2 � kn�i�1

2

h¼ dkn�i�1

2 � c1h1 i

q1y2iþ1

þ rkn�i�11 þ ð�d2 � ziþ1b� 2yiþ1kÞkn�i�1

2 þ ziþ1bkn�i3

" #;

kn�i3 � kn�i�1

3

h¼ dkn�i�1

3 � ½�xiþ1akn�i�11 � yiþ1bkn�i�1

2 þ kn�i�13 ð�d3 þ xiþ1aþ yiþ1b� 2ziþ1rÞ�:

As the problem is not a case study, the real world data are not available for this model. We, therefore, take here some hypo-thetical data with the sole purpose of illustrating the results that we have established in the previous sections. Parametersand initial values used for the numerical simulations are as follows:

r ¼ 1:8; d1 ¼ 0:001; s ¼ 1:5; a ¼ 0:5; c ¼ 0:2; d2 ¼ 0:002; k ¼ 0:4; b ¼ 0:8; d3 ¼ 0:005; r ¼ 0:6;x0 ¼ 0:2; y0 ¼ 0:5; z0 ¼ 0:7; q1 ¼ 0:8; p1 ¼ 0:3; v1 ¼ 0:125; c1 ¼ 0:02; d ¼ 0:01; tf ¼ 200:

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8785

Figs. 1–3 illustrate the impact of harvesting on prey and predator biomass. It is clearly observed from the figures that theoptimal control (harvesting of adult prey population) can be suitably chosen such that the prey and predator populationsmay get protection from extinction and capable to reach their optimal stage. Again, it is evident from the figures that thepredator population is reached to its optimal stage in finite time not only due to the harvesting of adult prey populationbut also for the existence of the juvenile as well as adult prey populations. Figs. 4 and 5 respectively depict the variationof the harvesting function and the value function of the fishery with time. These figures also indicate that it is possible toachieve the commercial purpose of the fishery through implementing suitable regulatory harvesting policy to the fishery.

4. Model with predator harvesting

In this section, we mainly investigate the dynamical behavior of the system when harvesting of predator population istaken into account.

For this purpose, let us consider the differential algebraic model system as:

dxdt¼ ry� d1x� sx� cx2 � axz; ð11aÞ

dydt¼ sx� d2y� ky2 � byz; ð11bÞ

dzdt¼ axzþ byz� d3z� rz2 � q2E2z; ð11cÞ

ðp2q2z� c2ÞE2 �m2 ¼ 0; ð11dÞ

where q2 is the catchability co-efficients of predator population, E2 is the effort used to harvest the population, c2 is the con-stant fishing cost per unit effort, p2 is the constant price per unit biomass of landed fish of predator population and m2 is thetotal economic rent obtained from the fishery.

4.1. Qualitative analysis of differential algebraic model system

The differential algebraic system (11) can be expressed in the following way,

/ðX; E2;m2Þ ¼/1ðX; E2;m2Þ/2ðX; E2;m2Þ/3ðX; E2;m2Þ

264

375 ¼

ry� d1x� sx� cx2 � axzsx� d2y� ky2 � byz

axzþ byz� d3z� rz2 � q2E2z

264

375;

wðX; E2;m2Þ ¼ ðp2q2z� czÞE2 �m2;

where X = (x,y,z)t.

0 50 100 150 2000.2

0.25

0.3

0.35

0.4

0.45

0.5

Time

x (t)

Fig. 1. Variation of optimal juvenile prey biomass with time.

0 50 100 150 2000.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Time

y (t)

Fig. 2. Variation of optimal adult prey biomass with time.

0 50 100 150 2000.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Time

z (t)

Fig. 3. Variation of optimal predator biomass with time.

8786 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

It is assumed that P2ðx�; y�; z�; E�2Þ be the interior equilibrium of the system (11). The local stability of the interior equilib-rium point, P2ðx�; y�; z�; E�2Þ can be investigated using the SIB phenomena based on the assumption that the interior equilib-rium point exists. To check the existence of the SIB phenomena, total economic rent m2 is assumed to be the bifurcationparameter. Consequently we have the following theorem:

Theorem 4. The differential algebraic system (11) has a singularity induced bifurcation at the interior equilibrium pointP2ðx�; y�; z�; E�2Þ. When the bifurcation parameter m2 increases through zero the stability of the interior equilibrium pointP2ðx�; y�; z�; E�2Þ changes from stable to unstable.

Proof. The proof of the theorem is similar to the proof of Theorem 1. h

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

Time

Con

trol h

1

Fig. 4. Variation of optimal control of harvesting with time.

0 50 100 150 2000

2

4

6

8

10

12

14

Time

Valu

e Fu

nctio

n

Fig. 5. Variation of optimal value function of the fishery with time.

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8787

To stabilize the model system (11) in case of positive economic interest a state feedback controller, (Dai, [3]) can bedesigned of the form w2ðtÞ ¼ u2ðE2ðtÞ � E�2Þ,

where u2 stands for net feedback gain and the system becomes:

dxdt¼ ry� d1x� sx� cx2 � axz; ð12aÞ

dydt¼ sx� d2y� ky2 � byz; ð12bÞ

dzdt¼ axzþ byz� d3z� rz2 � q2E2z; ð12cÞ

ðp2q2z� c2ÞE2 �m2 þ u2ðE2ðtÞ � E�2Þ ¼ 0: ð12dÞ

8788 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

Consequently, we have the following theorem:

Theorem 5. The differential algebraic system (12) is stable at the interior equilibrium point, P2ðx�; y�; z�; E�2Þ, of the model system(11) if u2 > max(b1,b2,b3) where

b1 ¼z�E�2p2q2

2sx�y� þ

ry�

x� þ x�cþ y�kþ z�r;

b2 ¼sx�z�E�2p2q�2

y� þ ry�z�E�2p2q22

x� þ x�z�cE�2p2q22 þ y�z�kE�2p2q2

2

x�z�a2 þ y�z�b2 þ sx�2cy� þ

ry�2kx� þ x�y�ckþ sx�z�r

y� þry�z�r

x� þ x�z�crþ y�z�kr;

b3 ¼sx�2z�cE�2p2q2

2y� þ ry�2z�kE�2p2q2

2x� þ x�y�z�ckE�2p2q2

2

sx�2z�y� ða2 þ crÞ þ ðsx� þ ry�Þz�abþ ry�2z�

x� ðb2 þ krÞ þ x�y�z�ðb2cþ a2kþ ckrÞ

:

Proof. The proof of the theorem is similar to the proof of Theorem 2. h

4.2. The optimal control problem

To maximize the total discounted net revenues from the fishery, the optimal control problem can be formulated as

Jðh2Þ ¼Z tf

t0

e�dt ðp2 � v2h2Þh2 �c2h2

q2z

� �ð13Þ

where c2 be the constant fishing cost per unit effort, p2 is the constant price per unit biomass of harvested population, v2 is aneconomic constant and d is the instantaneous annual discount rate.

The problem (13), subject to population Eqs. (11a)–(11c) and control constraints 0 6 h2 6 h2max, can be solved by applyingPontryagin’s maximum principle. The convexity of the objective function with respect to h2, the linearity of the differentialequations in the control and the compactness of the range values of the state variables can be combined to give the existenceof the optimal control.

Suppose h�2 is an optimal control with corresponding states x⁄, y⁄ and z⁄. We are seeking to derive optimal control h�2 suchthat

Jðh�2Þ ¼ maxfJðh2Þ : h2 2 Vg;

where V is the control set defined by

V ¼ fh2 : ½t0; tf � ! ½0; h2max�jh2 is Lebesgue measurableg:

The current value Hamiltonian [21, pp. 151–153] of this control problem is

H¼ ðp2�v2h2Þh2�c2h2

q2z

� �þ k4ðry�d1x� sx�cx2�axzÞþ k5ðsx�d2y� ky2�byzÞþ k6ðaxzþbyz�d3z�rz2�h2Þ;

where k4(t), k5(t) and k6(t) are the current value multipliers.The transversality conditions give ki(tf) = 0, i = 4,5,6.Now, it is possible to find the characterization of the optimal control h�2. On the set ftj0 < h�2ðtÞ < h2maxg, we have

@H@h2¼ p2 � 2h2v2 �

c2

q2z� k6 ¼ 0 at h�2ðtÞ:

This implies,

h�2 ¼p2q2z� � q2z�k6 � c2

2q2v2z�: ð14Þ

Now, the autonomous set of equations of the control problem (13) are

dk4

dt¼ dk4 �

@H@x¼ dk4 � ½ð�d1 � s� za� 2xcÞk4 þ sk5 þ zak6�; ð15Þ

dk5

dt¼ dk5 �

@H@y¼ dk5 � ½rk4 þ ð�d2 � zb� 2ykÞk5 þ zbk6�; ð16Þ

dk6

dt¼ dk6 �

@H@z¼ dk6 �

c2h2

q2z2 � xak4 � ybk5 þ k6ð�d3 þ xaþ yb� 2zrÞ� �

: ð17Þ

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8789

Therefore, we arrive to the following theorem:

Theorem 6. There exists an optimal control h�2 and corresponding solution x⁄, y⁄ and z⁄ that maximizes J(h2) over V. Furthermore,there exists adjoint functions, k4, k5 and k6 satisfying the equations (15)–(17) with transversality conditions ki(tf) = 0, i = 4,5,6.Moreover, the optimal control is given by h�2 ¼

p2q2z��q2z�k6�c22q2v2z� .

4.3. Numerical simulation of optimal control problem

The numerical simulation of optimal control [5] under various parameter sets can be done using 4th order Runge–Kuttaforward–backward sweep method. In this method, the system state Eqs. (11a)–(11c) and their corresponding adjointEqs. (15)–(17) are simultaneously solved. Parameters and initial values used for the numerical simulations are as follows:

r ¼ 4:8; d1 ¼ 0:001; s ¼ 3:8; a ¼ 0:5; c ¼ 0:6; d2 ¼ 0:002; k ¼ 0:5; b ¼ 0:8; d3 ¼ 0:005;r ¼ 1:6; x0 ¼ 3; y0 ¼ 5; z0 ¼ 3; q1 ¼ 1:2; p1 ¼ 5; v1 ¼ 0:5; c1 ¼ 0:8; d ¼ 0:01; tf ¼ 100:

The impact of harvesting on prey and predator biomass is found in the Figs. 6–8. It is evident from the figures that the opti-mal control (harvesting of predator population) can be suitably chosen such that the prey and predator populations may getprotection from extinction and capable to reach their optimal stage. Again, it is ensured from the figures that the predatorpopulation may sustain and reach to its optimal stage (in finite time) in presence of harvesting if the existence of the juvenileas well as adult prey populations are suitably managed for a particular fishery. The Figs. 9 and 10 indicate the variation ofoptimal harvesting function and optimal value function of the fishery. Here, it is obvious that the optimal value functionincreases with the increasing harvesting function.

5. Discussions

The present paper seeks to find optimal exploitation strategies for a predator–prey system, but differs in two respectsfrom the previous studies. First, the ecosystem model is based on a more realistic specification of intra-specific competitionof each species. In particular, it includes search and handling of prey by predators and of predator by preys in order to derivethe ecosystem dynamics. Secondly, and more importantly, in the first part of the paper prey is harvested while predators areprotected and in the second part of the paper predator is harvested while preys are protected. More specifically, in contrastto other studies, we explicitly assess the existence of species, separately for prey and predator. Although, in the first part ofthe paper, the predator species is itself not harvested, incorporating its existence value is necessary because it is indirectlyaffected by the harvesting of its food. This paper is an endeavor to theoretically investigate the balance between exploitationand nature conservation. It also illustrates that several types of optimal harvesting solutions are possible and the solutionsdepend on economic parameters e.g. the maximum harvest rate, the discount rate, and the cost of fishing, as well as on eco-logical parameters such as intra-specific coefficient of each species.

0 20 40 60 80 100

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

Time

x (t)

Fig. 6. Variation of optimal juvenile prey biomass with time.

0 20 40 60 80 1003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Time

y (t)

Fig. 7. Variation of optimal adult prey biomass with time.

0 20 40 60 80 1000.5

1

1.5

2

2.5

3

Time

z (t)

Fig. 8. Variation of optimal predator biomass with time.

8790 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

However, it may be concluded that if one species is harvested and another species is taken to be protected then in boththe cases there may induce an impulsive phenomenon which leads the system unstable when net economic revenue of thefishery is considered to be positive. But it is possible to design state feedback controller which stabilize the system wheneconomic rent is considered to be positive. The impact of present recession of the world fisheries can be analyzed usingthe results obtained in this paper. If the bioeconomic measures, ecological balance and sustainable development of the fish-ery are simultaneously taken under consideration then it may be recommended that the government or the regulatory agen-cies should encourage the fishermen towards size selected harvesting. It will not only recover certain rare species fromextinction which are economically valuable but also provide some corrective measures to overcome the present recession.

The inclusion of economic factors in fishery models is very important to assess the economic consequences of the fish-eries. To attain efficiency in the economic sense, we need to take into account the costs of fishing and revenues that weget from selling the harvested fish. The economic parameters as price per unit biomass of catch, fishing cost per unit effortand discount rate determine the stock level maximizing the present value of the flow of resource rent over time, therefore

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time

Con

trol h

2

Fig. 9. Variation of optimal control of harvesting with time.

0 20 40 60 80 1000

200

400

600

800

1000

1200

Time

Valu

e Fu

nctio

n

Fig. 10. Variation of optimal value function of the fishery with time.

K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792 8791

characterization of model system for those parameters should be analyzed to achieve the efficiency of the fishery ineconomic sense. Figs. 5 and 10 provide the impacts of the economic parameters on the harvested population as well ason the optimal control of harvest of the considered model system. As a result, the analysis presented here comes closerto reflecting the economic concerns and realities of the principal resource users in the fisheries, and as such our objectiveis to represent an advance over previous efforts to analyze and provide policy recommendations in the fisheriesmanagement.

6. Concluding remarks

The present work is divided in two parts. In the first part, we consider a model system where only adult prey population isharvested and come across several dynamical behavior of the system. It is observed that SIB phenomenon takes place whennet economic revenue of the fishery is increasing through zero to a positive quantity. In consequence to the aforesaid

8792 K. Chakraborty et al. / Applied Mathematics and Computation 217 (2011) 8778–8792

bifurcation, an impulsive phenomenon is occurred. Thus, the model system may exhibit cycle behavior and due to this cyclicnature some population exhibit periodic fluctuation in abundance, with periodic crashes and the system becomes unstable.The most important realistic feature of the paper is the state feedback controller which is designed to stabilize the modelsystem when the economic rent is taken to be positive. Numerical simulations are used to show that state feedbackcontroller can be designed to resume the stability of a model system in case of positive economic profit.

The optimality system is solved using an iterative method with a Runge–Kutta fourth order scheme. The state systemwith an initial guess is solved forward in time and then the adjoint system is solved backward in time. The controls areupdated at the end of each iteration using the formula for optimal controls. The iterations continue until convergence isachieved. Again, it is necessary to perform numerical simulations to accumulate information about the variability of resultsfrom each strategy of a model system and summarization of the obtained results. We have discussed an efficient numericalmethod based on optimal control to identify the best management policies to overcome the over exploitation of a fishery andachieve optimality in economic sense. The present work describes the use of harvesting as control to obtain strategies for thecontrol of a prey–predator system with stage structure. We find that it is possible to control the system in such a way thatthe system approaches a required state, using the harvesting as the control. The considered model system incorporates afully dynamic interaction between the harvesting and the perceived rent and this interaction is described using suitablenumerical simulations. The results we have obtained may be helpful for the fishery managers wishing to preserve economicoptimality through ensuring global sustainability.

In the second part of the paper we have described the model system with predator harvesting and similar type of resultsare obtained. However, it should be noted that we intend to present the comparative study of the considered model systemsand it is easy to compare the dynamical behavior of the model systems through our obtained results. Moreover, it is deemedimportant to undertake this type of study for the purpose of investigating the impact of harvesting so that sustainability ofthe ecosystem may be resumed through achieving the commercial purpose of the fishery. Hence, the obtained results of ourstudy are not only feasible to assess the biological, social and economic impacts of existing resource, but also provide anappropriate measures to maintain long run sustainability of the resource.

The entire study of the paper is mainly based on the deterministic framework. On the other hand it will be more realistic ifit is possible to consider the model system in the stochastic environment due to some ecological fluctuations and other fac-tors. Thus, a future research problem would be considered in stochastic environment. Again, to achieve the commercial pur-pose of the fishery it is also possible to determine optimal harvesting strategies using game theory.

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