Abstract
The aim of this paper is to study the dynamics of fishery resource with reserve area in the presence of bird predator. The aquatic region under investigation is divided into two zones: one free for fishing and another restricted for any kind of fishery. The criteria of biological and bionomic equilibrium of system are established. The points of local stability, global stability, and instability are obtained for the proposed model. An optimal harvesting policy is established using Pontryagin’s maximum principle. At last the theoretical results obtained are illustrated with the help of numerical simulation.
1. Introduction
Renewable resources like fishery, forestry, and oil exploration are important sources of food and materials which play an important role for survival and growth of biological population. Extensive and unregulated harvesting of marine fishes can lead to the depletion of several fish species. One potential solution of these problems is the creation of marine resources where fishing and other exploration activities are strictly prohibited. Sometimes, though marine reserves protect species inside the reserve area, they lead to increasing fish abundance in adjacent areas. So this aspect should also be considered to make effective use of reserves. Restrictions on gear and/or effort may also be considered as other ways to protect species from extinction.
Economic and biological aspects of renewable resources management have been considered by Clark [1]. Chaudhuri [2] studied the problem of combined harvesting of two competing fish species. Ganguly and Chaudhuri [3] proposed a model to study the regulation of single species fishery by taxation. Mesterton-Gibbons [4] also described a technique to find an optimal harvesting policy for Lotka-Voltra ecosystem to two interdependent populations. Kar and Misra [5] explained the possibilities of the existence of bionomic equilibrium in prey-predator model. Kar and Chaudhuri [6] derived the condition for global stability of system using a Lyapunov function. Kar [7] discussed the optimal harvesting policy using Pontryagin’s maximal principle of prey-predator system. Dubey et al. [8] proposed a dynamic model for single species fishery which depends partially on a logistically growing resource. They showed that both the equilibrium density of the fish population and the maximum sustainable yield increase as the resource biomass density increases. Dubey [9] proposed and analyzed the dynamics of a prey-predator model. The role of reserved zone is investigated and it is shown that the reserve zone has a stabilizing effect on predator-prey interactions. Zhang et al. [10] discussed the dynamics of prey-predator fishery model with harvesting of both prey and predator. Kar and Pahar [11] studied the dynamical behavior and harvesting problem of a prey-predator fishery.
From the literature discussed above and to the best of our knowledge, in the models considered by different authors [6, 7, 9–11] the predator is feeding in unreserved area only. In this paper, we have proposed a model in which predator birds are feeding in both reserved and unreserved areas. Moreover the predators are also being harvested from unreserved zone. The criteria of biological and bionomic equilibrium of system are established. The points of local stability, global stability, and instability are obtained for the proposed model. An optimal harvesting policy is also discussed using Pontryagin’s maximum principle. The paper is concluded with numerical simulation.
2. The Model
Consider a fishery resource system consisting of two zones: a free fishery and a reserve zone where fishing is not allowed. Each zone is supposed to be homogeneous. There is a bird predator feeding on both of them, that is, fishes of reserved as well as unreserved zones. It is assumed that the predator population is also harvested in unreserved zone. We suppose that the prey species migrate between the two zones randomly. The growth of prey in each zone in the absence of predator is assumed to be logistic. Keeping these in view, the model becomes
Here and are the respective biomass densities of the prey species inside the unreserved and reserved areas, respectively, at a time is the biomass density of predator at time ; and are migration rates from the unreserved area to reserved area and the reserved area to the unreserved area, respectively; and are the efforts applied to harvest the fish population and predator in unreserved zone, respectively; and are intrinsic growth rates of prey species inside the unreserved and reserved zones, respectively; and are the carrying capacities of prey species in the unreserved and reserved zones, respectively; and are the catchability coefficient of prey and predator in unreserved zone, respectively; is death rate of predator; and are the capturing rates and and are the conversion rates of prey in unreserved and reserved zones, respectively.
All the parameters are assumed to be positive. Here we observe that if there is no migration of fish population from the reserved area to the unreserved area (i.e., ) and , then . Similarly, if there is no migration of fish population from the unreserved area to reserved area (i.e., ) and , then . Hence, throughout our analysis, we assume that
In order to simplify the model proposed in (1), we assume that the capturing rates are the same from both reserves, that is, and conversion rates of prey in unreserved and reserved zones are the same, that is, . After incorporating these assumptions, the system (1) becomes where .
Lemma 1. All the solutions of the system (3) which initiate in are uniformly bounded.
Proof. Let and be a constant. Then By the theory of differential inequality [12], we have and for . This proves the lemma.
3. Existence of Equilibria
We find the steady states of (3) by equating the derivatives on the left hand sides to zero and solving the resulting algebraic equations. This gives three possible steady states, namely, , and .
At the population is extinct and this equilibrium point always exists.
Now, consider the equilibrium point , where the predator is not present. Here and are the positive solutions of This system (6) is already solved by Dubey et. al. [8] and local and global stability results for the system at are discussed there.
Now assume here that the interior equilibrium point exists and is a solution of Next, we discuss the local and global stability results at these equilibrium points.
4. Stability Analysis
The variational matrix of system (3) isAt the characteristic equation is , where Since Therefore, all eigenvalues are negative and hence is locally asymptotically stable. Let us now suppose that system (3) has a unique positive equilibrium . The variational matrix of (3) at is The characteristic equation of variational matrix of system (3) at is given by , where
According to Routh-Hurwitz criteria, the necessary and sufficient conditions for local stability of equilibrium point are , , and .
It is evident that and . Thus the stability of is determined by the sign of . By direct calculation, we obtain and hence is locally asymptotically stable.
Now we will discuss the global stability of the endemic equilibrium point of the system (3).
Theorem 2. The equilibrium point of system (3) is globally asymptotically stable if .
Proof. Let us consider the following Lyapunov function:
where and are positive constants to be chosen later on.
Differentiating with respect to time , we get
Choosing and , a little algebraic manipulation yields
Clearly if and only if .
Therefore, is globally asymptotically stable provided /.
5. Bionomic Equilibrium
In this section, we study the bionomical equilibrium of the model system (3). Let be the fishing cost per unit effort for prey species, let be the harvesting cost per unit effort for predator species, let be the price per unit biomass of the prey, and let be the price per unit biomass of the predator.
Therefore, the net economic revenue at any time is given by where and ; that is, and represent the net revenues for the prey and predator species, respectively.
The bionomical equilibrium is given by the following simultaneous equations: In order to determine the bionomic equilibrium, we now consider the following cases.
Case 1. If and , then the cost of harvesting is greater than the revenue for the predator and cost of harvesting of prey is less than revenue. Here the harvesting of predator will be stopped, that is, , and only the prey fishing remains operational.
We then have . Substituting into (20), we get . Now substituting and into (19), we get .
Now, substituting , , and into (18), we get
Case 2. If and , then the cost of fishing is more than the revenue and cost of harvesting of predator is less than revenue. Here harvesting of prey will be closed (i.e., ) and only predator harvesting will remain operational.
We then have . Substituting into (17) we get .
Now, substituting into (18), we get .
From these two equations and can be found. Substituting and into (20), we get
where , provided .
Case 3. If and , then the cost is greater than revenues for both the species and so the whole fishing and harvesting of predator will be closed.
Case 4. If and , then the revenues for both the species are being positive, and then the whole fishing and harvesting of predator will be in operation.
In this case, and .
Now substituting and into (18), (19), and (20), we get
Now,
Thus the nontrivial bionomic equilibrium point exists if conditions (25) hold.
6. Optimal Harvesting Policy
In this section, the optimal management of a fishery resource in the presence of predator is discussed. Here, our objective is to maximize the present value of a continuous time stream of revenues given by where denotes the instantaneous annual rate of discount. We intend to maximize (26) subject to the state equations (3) by invoking Pontryagin’s maximal principle (Clark [1]). The control variable is subjected to the constraints .
The Hamiltonian function for the problem is given by where , and are the adjoint variables.
The control variables and appear linearly in the Hamiltonian function .
Assuming that the control constraints are not binding, that is, the optimal solution does not occur at , we have singular control.
According to Pontryagin’s maximum principle Substitution and simplification yield Now, substituting and into (33) and using equilibrium equations, we get From (32), we get , whose solution is given by where and .
From (31), we get , whose solution is given by where .
From (29) and (36), we get the singular path Using , , and can be written as Thus (37) can be written as There exists a unique positive root of in the interval , if the following inequalities hold: For , we get from (34).
We then have Hence once the optimal equilibrium is determined, the optimal harvesting efforts and can be determined.
From (30), (35), and (36), we observe that do not vary with time in optimal equilibrium. Hence they remain bounded as .
7. Simulation
In order to investigate the dynamics of the system (3) with help of computer simulation, we choose the following set of values of parameters: in appropriate units with initial conditions .
From Figure 1, it is clear that the biomass density of prey species in unreserved area increases with respect to time and then decreases slightly and settles down at its equilibrium level.
From Figure 2, it is clear that biomass density of prey population in reserved zone increases sharply near to its carrying capacity and then settles down at its equilibrium level near the carrying capacity of this zone.
Figure 3 shows that biomass density of predator increases approximately linearly with respect to time and tries to settle at their equilibrium level. Figures 1, 2, and 3 also show that the endemic equilibrium point is locally asymptotically stable for the assumed values of parameters.
From Figure 4, we may conclude that the steady state is globally asymptomatically stable. All the solutions with different initial conditions and set of parameters in (44) satisfying the conditions of Theorem 2 converge to the same equilibrium point. Hence the theory established earlier is verified.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are grateful to the referee for critical review and useful suggestions that improved the paper.