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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 429076, 12 pages
Dynamic Behaviors of a Discrete Two Species Predator-Prey System Incorporating Harvesting
Department of Mathematics, Minjiang University, Fujian, Fuzhou 350108, China
Received 2 May 2012; Revised 19 August 2012; Accepted 29 August 2012
Academic Editor: Jinde Cao
Copyright © 2012 Ting Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A discrete two species predator-prey captured system is studied. Firstly, a sufficient condition of a positive equilibrium point for this system is obtained. Secondly, we observe that the two nonnegative equilibriums of the system are unstable through the eigenvalue discriminant method, and the positive equilibrium point is asymptotically stable by Jury criterion. Lastly, we obtain the optimal capture strategy of the system from the maximum principle by constructing a discrete Hamiltonian function. To show the feasibility of the main results, a suitable example together with its numerical simulations is illustrated in the last part of the paper. The example with certain practical significance might give an optimal scheme of the greatest economic benefits for the captors.
With economic development, the rational development and management of the biological resources are directly related to its sustainable development. In this situation, more and more scholars considered the problems on ecological balance of biological systems. Also, the stability and permanence of biological systems are well studied recently [1–4]. According to the aim of people’s capture, we consider both the economic interest and the permanence of predator-prey system in this paper. The aim of this paper is to make a research on some suitable control of the ecological system in order to obtain the existence and development of the system.
In earlier stage, papers mainly considered the maximum sustainable yield in the field of optimal capture in order to guarantee the maximum of the capture yield, and the biological resources will not lose their reproduction capacity eventually . Recently, much more papers turn their attention to the optimal capture strategy [6–16]. Similarly, the optimal control theory is also a good way.
The predator-prey system is one of the typical ecological systems. For example, Leslie [6, 7] introduced the Leslie-Gower predator-prey systems as follows: where and are the density of prey species and the predator species at time , respectively; and are the intrinsic growth rate of prey and predator; is the density-dependent entry; is the number of eaten by per unit of time; represents the “carrying capacity” of the predator’s environment which is proportional to the number of prey. The Leslie-Gower system admits a unique equilibrium, and Korobeinikov  showed that the positive equilibrium is globally asymptotic stable. Recently, Zhang et al.  assumed that the predator and prey in the model have commercial importance, and they are subjected to constant effort harvesting. Let , denote the harvest of prey and predator, respectively. Zhang et al. formulated the system as follows: In addition, they discuss the stability and the optimal harvesting strategy.
Although many scholars considered the economic interest of the captured amount of the continuous system, the distribution with the fish is inhomogeneous and it is not possible to capture successively. Therefore, it is more reasonable to consider the discrete system. The research on the discrete captured model makes a great significance to improve the quality of people’s life. The continuous one model is of a continuous time of captured revenue ; we consider that the discrete model is divided into time segments of captured revenue. Therefore, it is more reasonable. On the one hand, during the exhausting of the fishing resources, people have to restrict the capture models into discrete ones to avoid the disappearance of those resources. On the other hand, we use the eigenvalue symbol of the coefficient matrix of the linear differential system to determine the stability of continuous models, while for the discrete models, the stability is determined by the eigenvalues of the coefficient matrix of the linear difference system by Lemma 2.2, Lemma 2.3 and Corollary 2.5. We raise the model as follows: where and are the population densities of prey and predator at time , respectively; denotes the intrinsic growth rate of prey and predator (or life factor); is the death rate of the predator ; and are the density-dependent entry; is the number of eaten by per unit of time; is the conversion rate (); and are the two parameters that measures the effort being spent by a harvesting agency (where , , , and are the catch-ability coefficients of the prey and predator species, and and denote the effort devoted to the harvesting), also , . All the parameters are assumed to be positive.
In the following context, we will consider the existence of the positive equilibrium, and, by applying the stability theory of linear difference equation, we obtain the stability of the positive equilibrium. We also get a theorem of the trivial solution asymptotic stability for the second-order constant coefficient linear homogeneous difference equations according to Jury criterion. We also use the relation between inhomogeneous difference equations and the corresponding homogeneous difference equations to prove the sufficient conditions of the positive equilibrium stability. Finally, by constructing a discrete Lyapunov function, we obtain the global asymptotic stability of the positive equilibrium. In order to consider economic benefit, we construct a discrete Hamiltonian function and use the maximum principle to get the optimal capture strategy of the system.
2. Stability of the Equilibrium Point of System (1.3)
2.1. The Equilibrium Point of System (1.3)
By simple calculations, we get that the system (1.3) has three possible nonnegative equilibriums: , , , and , where
Since , we have . So, we will only need to study the stability of the equilibria , , and of system (1.3). On these conditions, we have the following theorem.
Theorem 2.1. is a non-negative equilibrium point if and only if , and is a positive equilibrium point.
2.2. Stability of the Positive Equilibrium and the Relevant Conclusions
Note that the nonlinear difference system, can be determined by the stability of the linear difference system
The characteristic equation of the coefficient matrix of the linear difference equation system (2.3) is
In view of (2.4), we construct the following Jury conditions table: , , , , , , , where until there are only three elements in the same row.
Theorem 2.4 (see , page 204, Jury conditions). All the zero solutions of the polynomial are in the complex plane of the unit circle if and only if where , , and are given by the Jury conditions table.
To get the second-order constant coefficient linear homogeneous difference equation satisfying Jury criterion in partial if , we get the following criterion form.
Corollary 2.5. Suppose . Then the zero solution of (2.3) is asymptotically stable if and only if , with .
2.3. Local Stability Analysis of the Equilibria
Theorem 2.6. Both of the equilibriums and are unstable equilibrium points. If with , then is a locally stability equilibrium point.
Proof. (1) For , the corresponding linear difference equation is
whose characteristic equation is . Thus the eigenvalues are , . By Lemma 2.2, it follows that is an unstable equilibrium point.
(2) For the non-negative equilibrium point , we make translation transformations, Substituting it into (1.3), getting that charged by , and linearizing, we get If the variational matrix of the system (1.3) is then . By Lemma 2.3, it implies that is an unstable equilibrium point.
(3) For the positive equilibrium point , we make translation transformations, Substituting it into (1.3), getting that charged by , and linearizing, we get Suppose that , , , and By Corollary of Theorem 2.4, it implies that if , , then also . By Corollary 2.5, it implies that if , , is a locally stability equilibrium point.
2.4. Global Stability
Theorem 2.7. Under the conditions of Theorem 2.6, if there are positive numbers and satisfying the following two inequalities, (i), (ii), then the positive equilibrium point of system (1.3) is globally stable.
Proof. We make translation transformations,
Substituting it into (1.3), getting that charged by , we have
where is an equilibrium point of (2.16). Make Taylor expanding the right side of (2.16) on the equilibrium point , we have
where and . If , then
For every , both of them are consistent (.) Then (2.17) can be rewritten as follows:
where , .
We have the following Lyapunov function:
By condition (i), (ii), and (2.19), we get the difference of Lyapunov function as follows: As if , then . If is great enough, then there exists a positive such that . So, if the interior equilibrium of system (2.16) is globally stable, then the interior equilibrium of system (1.3) is also globally stable.
3. Harvesting the Optimal Economic Benefit
The fishermen or the fishing companies must consider the cost effectiveness when harvesting all kinds of fish. It is necessary to consider not only the sale price, but also the injecting funds capture. If the largest capture intensity is , then . Suppose the cost is , , and suppose the prices of the two kinds of group are , . To obtain the optimal capture, we need to seek for the best efforts of the degrees , . Since the optimal balance point is , the goal function is given by
According to the discrete maximum principle, to seek optimal control , , we need the following Hamilton function: where ; is the instantaneous discount rate for periods; , are with ariables; , get maximum value , respectively. Consider the following equations: By (3.3) × − (3.4) × , we get By (3.3) × −(3.4) ×, we get So, That is, where .
Substitute by in the following two equations, we have
Suppose , that is, If , then By , , it implies Substitute (3.11) into (3.12), we have that is, From (3.12) and (3.14), it follows that the is the optimal equilibrium solution, whose best efforts of degrees are which is also the optimal equilibrium program. Thus the economic profits of the captured populations are completely determined by the discount rates , , .
4. Number Simulations
In this section, we consider some numerical simulations examples.
Example 4.1. Let , , , , , , , and into the system (1.3), then Through calculating, we have Let , , , we have In this case, the positive equilibrium point is . And the system (1.3) is globally asymptotically stable (Figure 1).
Example 4.2. Let , , , , , , , , , , and into (3.14) and (3.15), we have
Solve (3.14) and (3.15) by Maple, we get only one optimal equilibrium point:
, meeting the condition , , .
In the above values of parameters, we found that the optimal equilibrium point exists, and the corresponding optimal harvesting efforts are and (Figure 2).
The paper studies optimal capture problems of the predator-prey system. Firstly, we consider the existence and the stability of the positive equilibrium of system, and, by the maximum principle of a discrete model and Hamilton function, we obtain the optimal capture strategy that is under the condition . Finally, by applying numerical simulations, we show that system (1.3) is a globally stable positive equilibrium and an optimal harvesting policy. At last, we notice that there are corresponding results if the conditions switch to or , respectivelty.
This work is supported by the Technology Innovation Platform Project of Fujian Province (2009J1007).
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