- Research Article
- Open Access
Optimal Harvest of a Stochastic Predator-Prey Model
Advances in Difference Equations volume 2011, Article number: 312465 (2011)
We firstly show the permanence of hybrid prey-predator system. Then, when both white and color noises are taken into account, we examine the asymptotic properties of stochastic prey-predator model with Markovian switching. Finally, the optimal harvest policy of stochastic prey-predator model perturbed by white noise is considered.
Population systems have long been an important theme in mathematical biology due to their universal existence and importance. As a result, interest in mathematical models for populations with interaction between species has been on the increase. Generally, many models in theoretical ecology take the classical Lotka-Volterra model of interacting species as a starting point as follows:
where , and . The Lotka-Volterra model (1.1) has been studied extensively by many authors. Specifically, the dynamics relationship between predators and their preys also is an important topic in both ecology and mathematical ecology. For two-species predator-prey model, the population model has the form
where , represent the prey and the predator populations at time , respectively, and , , are all positive constants.
Up to now, few work has been done with the following hybrid predator-prey model:
when a population model is discussed, one of most important and interesting themes is its permanence, which means that the population system will survive forever. In this paper, we show that the hybrid model (1.3) is permanent.
As a matter of fact, due to environmental fluctuations, parameters involved in population models are not absolute constants. Thus, it is important to reveal how environmental noises affect the population systems. There are various types of environmental noises (e.g., white noise and color noise) affect population system significantly. Recently, many authors have considered population systems perturbed by white noise (see e.g., [1–7]). The color noise can be illustrated as a switching between two or more regimes of environmental, which differ by factors such as nutrition or as rain falls [8, 9]. When both white noise and color noise are taken into account, there are many results on corresponding population systems [10–14]. Especially,  investigated a Lotka-Volterra system under regime switching, the existence of global positive solutions, stochastic permanence, and extinction were discussed, and the limit of the average in time of the sample path was estimated. Reference  considered competitive Lotka-Volterra model in random environments and obtained nice results. Here, we consider the stochastic predator-prey system under regime switching which reads
where is a Markov chain. Therefore, we aim to obtain its dynamical properties in more detail.
As we know, the optimal management of renewable resources, which has a direct relationship to sustainable development, is always a significant problem and focus. Many authors have studied the optimal harvest of its corresponding population model [15–20]. To the best of our knowledge, there is a very little amount of work has been done on the optimal harvest of stochastic predator-prey system. When the predator-prey model (1.2) is perturbed by white noise, we have the stochastic system as follows:
Suppose that the resource population described by the stochastic system (1.5) is subject to exploitation, under the harvesting effort , of , , respectively, the model of the harvested population has the form
In this paper, based on the arguments on model (1.4), we will obtain the the optimal harvest policy of stochastic predator-prey system (1.6).
The organization of the paper is as follows: we recall the fundamental theory about stochastic differential equation with Markovian switching in Section 2. We show that the hybrid system (1.3) is permanent in Section 3. Since stochastic predator-prey system (1.4) describes population dynamics, it is necessary for the solution of the system to be positive and not to explode to infinity in a finite time. Section 4 is devoted to the existence, uniqueness of global solution by comparison theorem, and its asymptotic properties. Based on the arguments of Section 4, in Section 5 predator-prey model perturbed by white noise (1.5) is considered, and the limit of the average in time of the sample path of the solution is obtain, moreover, optimal harvest policy of population model is derived. Finally, we close the paper with conclusions in Section 6. The important contributions of this paper are therefore clear.
2. Stochastic Differential Equation with Markovian Switching
Throughout this paper, unless otherwise specified, we let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets.) Let , , be a right-continuous Markov chain in the probability space tasking values in a finite state space with generator given by
where . Here, is the transition rate from to if , while . We assume that the Markov chain is independent of the Brownian motion. And almost every sample path of is a right-continuous step function with a finite number of simple jumps in any finite subinterval of .
We assume, as a standing hypothesis in following of the paper, that the Markov chain is irreducible. The algebraic interpretation of irreducibility is rank. Under this condition, the Markov chain has a unique stationary distribution which can be determined by solving the following linear equation:
Consider a stochastic differential equation with Markovian switching
on with initial value , where
For the existence and uniqueness of the solution, we should suppose that the coefficients of the above equation satisfy the local Lipschitz condition and the linear growth condition. That is, for each , there is an such that
for all , and those , with , and there is an such that
for all .
Let denote the family of all nonnegative functions on which are continuously twice differentiable in and once differentiable in . If , define an operator LV from to by
In particular, if is independent of , that is, , then
For convenience and simplicity in the following discussion, for any sequence , , we define
For any sequence , we define
And throughout the paper, we use to denote a positive constant whose exact value may be different in different appearances.
3. Hybrid Predator-Prey Model
In this section, we mainly consider the permanence of the hybrid prey-predator system (1.3).
The solution of (1.3) obeys
It follows from the first equation that
Using the comparison theorem, we can derive the first inequality. Then, for any arbitrarily, there exists , such that for any
So, for any , we obtain
By comparison theorem, then
For is arbitrary, we therefore conclude
Assume that holds. Then, the solution to (1.3) satisfies
By virtue of the first equation of (1.3), we can get
Then, it follows from the assumption that there exists sufficiently small , such that . From Lemma 3.1, for above , there exists , such that for any . Thus,
By the comparison theorem, we have
On the other hand, the second equation of (1.3) implies that
Using the comparison theorem, similar to the proof of the first assertion, we directly obtain
Suppose that , , , , and hold. Then, the solution to (1.3) is permanent.
From Lemma 3.1 and Lemma 3.2, we immediately get
4. Stochastic Predator-Prey Model With Markovian Switching
In this section, we consider the stochastic differential equation with regime switching (1.4). If stochastic differential equation has a unique global (i.e., no explosion in a finite time) solution for any initial value, the coefficients of the equation are required to obey the linear growth condition and local Lipschitz condition. It is easy to see that the coefficients of (1.4) satisfy the local Lipschitz condition; therefore, there is a unique local solution on with initial value , , where is the explosion time.
And since our purpose is to reveal the effect of environmental noises, we impose the following hypothesis on intensities of environmental noises.
Assumption 4.1 ().
By virtue of comparison theorem, we will demonstrate that the local solution to (1.4) is global, which is motivated by 
Thus, is the unique solution of
with . By the comparison theorem, we get for a.s. It is easy to see that
with , has a unique solution
Obviously, , a.s.
So, we get , a.s., where
To sum up, we obtain
It can be easily verified that , , , all exist on , hence
There is a unique positive solution of (1.4) for any initial value , , and the solution has the properties
where , , , are defined as (4.1),(4.4),(4.6), and (4.9).
Theorem 4.2 tells us that (1.4) has a unique global solution, which makes us to further discuss its properties.
Now, we will investigate certain asymptotic limits of the population model (1.4). Referred to , it is not difficult to imply that
Then, we give the following essential theorems which will be used.
Let Assumption 4.1 hold. Then, the solution has the property
The proof is motivated by . By (4.12) and (4.13), then
Thus, it remains to show that
Note the quadratic variation of is , thus the strong law of large numbers for local martingales yields that
Therefore, for any , there exists some positive constant such that for any
Then, for any , we have
Moreover, it follows from (4.13) that for the above and , we get
By virtue of (4.6), we can derive for
Using the generalized Itô Lemma, we can conclude that
Denote and . It is obvious that . Hence,
So, we obtain
Note the fact that is arbitrary, we obtain that
By (4.12), we have a.s. Consequently,
So, we complete the proof.
Let Assumption 4.1 and hold. Then, has the property
From (4.12) and (4.13), we have
Now, we only show that . By virtue of (4.12), it remains to demonstrate that . From the proof of Theorem 4.3, we know that for any , there exists some positive constant such that for any
It follows from (4.34) that
For above and , we get for a.s. By the generalized Itô Lemma, then
Thus, it is easy to imply that
where and . By (4.9), we imply for
It follows from (4.40) that
For is arbitrary, we imply
Finally, we obtain
Let Assumption 4.1 and hold. Then, the solution to (1.4) obeys
Using the generalized Itô Lemma, we have
Thus, let , by the ergodic properties of Markov chains, we have
Hence, by virtue of the strong law of large numbers of local martingales, we get
The proof is complete.
5. Optimal Harvest Policy
When the harvesting problems of population resources is discussed, we aim to obtain the optimal harvesting effort and the corresponding maximum sustainable yield.
In the same way of Theorems 4.2–4.5, we can conclude the following results. Here, we do not list the corresponding proofs in detail, only show the main results.
Assume and hold. Then, (1.6) has a unique global solution for any initial value . In addition, the solution has the properties
When the harvesting problems are considered, the corresponding average population level is derived below.
Suppose that the conditions of Theorem 5.1 hold. If
then the solution to (1.6) obeys
When the two species are both subjected to exploitation, it is important and necessary to discuss the corresponding maximum sustainable revenue.
Let the conditions of Theorem 5.1 hold. Then, if , the optimal harvesting efforts of and , respectively, are
The optimal sustainable harvesting yield is
where , are the price of and .
Assume that , are the price of resources of and . Then, the maximum sustainable yield reads
Then, we can have
Therefore, there exists a unique extreme value point , where
That is, under the condition , we obtain (5.4) and (5.5). So, we obtain the optimal harvesting efforts of and .
Substituting (5.4) and (5.5) into the representation of (5.7), we have the optimal sustainable yield
as desired. Therefore, we complete the proof.
The optimal management of renewable resources has a direct relationship to sustainable development. When population system is subject to exploitation, it is important and necessary to discuss the optimal harvesting effort and the corresponding maximum sustainable yield. Meanwhile, population systems are often subject to environmental noise. It is also necessary to reveal how the noise affects the population systems. Our work is an attempt to carry out the study of optimal harvest policy of population system in a stochastic setting. When both white noise and color noise are taken into account, we consider the limit of the average in time of the sample path of the stochastic model (1.4). Based on the arguments of (1.4), we discuss the corresponding stochastic system perturbed by white noise (1.5). We obtain the the optimal harvesting effort and the corresponding maximum sustainable yield.
Khasminskii RZ, Klebaner FC: Long term behavior of solutions of the Lotka-Volterra system under small random perturbations. The Annals of Applied Probability 2001,11(3):952-963.
Mao X, Sabanis S, Renshaw E: Asymptotic behaviour of the stochastic Lotka-Volterra model. Journal of Mathematical Analysis and Applications 2003,287(1):141-156. 10.1016/S0022-247X(03)00539-0
Soboleva TK, Pleasants AB: Population growth as a nonlinear stochastic process. Mathematical and Computer Modelling 2003,38(11-13):1437-1442. 10.1016/S0895-7177(03)90147-6
Du NH, Sam VH: Dynamics of a stochastic Lotka-Volterra model perturbed by white noise. Journal of Mathematical Analysis and Applications 2006,324(1):82-97. 10.1016/j.jmaa.2005.11.064
Mao X, Marion G, Renshaw E: Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes and Their Applications 2002,97(1):95-110. 10.1016/S0304-4149(01)00126-0
Jiang D, Shi N, Li X: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. Journal of Mathematical Analysis and Applications 2008,340(1):588-597. 10.1016/j.jmaa.2007.08.014
Li X, Mao X: Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems. Series A 2009,24(2):523-545.
Slatkin M: The dynamics og a population in a Markovian environment. Ecology 1978, 59: 249-256. 10.2307/1936370
Du NH, Kon R, Sato K, Takeuchi Y: Dynamical behavior of Lotka-Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise. Journal of Computational and Applied Mathematics 2004,170(2):399-422. 10.1016/j.cam.2004.02.001
Li X, Jiang D, Mao X: Population dynamical behavior of Lotka-Volterra system under regime switching. Journal of Computational and Applied Mathematics 2009,232(2):427-448. 10.1016/j.cam.2009.06.021
Zhu C, Yin G: On hybrid competitive Lotka-Volterra ecosystems. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1370-e1379. 10.1016/j.na.2009.01.166
Zhu C, Yin G: On competitive Lotka-Volterra model in random environments. Journal of Mathematical Analysis and Applications 2009,357(1):154-170. 10.1016/j.jmaa.2009.03.066
Luo Q, Mao X: Stochastic population dynamics under regime switching. Journal of Mathematical Analysis and Applications 2007,334(1):69-84. 10.1016/j.jmaa.2006.12.032
Luo Q, Mao X: Stochastic population dynamics under regime switching. II. Journal of Mathematical Analysis and Applications 2009,355(2):577-593. 10.1016/j.jmaa.2009.02.010
Clark CW: Mathematical Bioeconomics: The Optimal Management of Renewal Resources, Pure and Applied Mathematics. 2nd edition. John Wiley & Sons, New York, NY, USA; 1990:xiv+386.
Lungu EM, Øksendal B: Optimal harvesting from a population in a stochastic crowded environment. Mathematical Biosciences 1997,145(1):47-75. 10.1016/S0025-5564(97)00029-1
Fan M, Wang K: Optimal harvesting policy for single population with periodic coefficients. Mathematical Biosciences 1998,152(2):165-177. 10.1016/S0025-5564(98)10024-X
Alvarez LHR, Shepp LA: Optimal harvesting of stochastically fluctuating populations. Journal of Mathematical Biology 1998,37(2):155-177. 10.1007/s002850050124
Alvarez LHR: Optimal harvesting under stochastic fluctuations and critical depensation. Mathematical Biosciences 1998,152(1):63-85. 10.1016/S0025-5564(98)10018-4
Li W, Wang K: Optimal harvesting policy for general stochastic logistic population model. Journal of Mathematical Analysis and Applications 2010,368(2):420-428. 10.1016/j.jmaa.2010.04.002
This research is supported by the national natural science foundation of China (no. 10701020)