Optimal Harvest of a Stochastic Predator-Prey Model
© Jingliang Lv and Ke Wang. 2011
Received: 12 January 2011
Accepted: 20 February 2011
Published: 17 March 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.
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.
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
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 .
3. Hybrid Predator-Prey Model
In this section, we mainly consider the permanence of the hybrid prey-predator system (1.3).
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.
Theorem 4.2 tells us that (1.4) has a unique global solution, which makes us to further discuss its properties.
Then, we give the following essential theorems which will be used.
So, we complete the proof.
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.
When the harvesting problems are considered, the corresponding average population level is derived below.
When the two species are both subjected to exploitation, it is important and necessary to discuss the corresponding maximum sustainable revenue.
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.
This research is supported by the national natural science foundation of China (no. 10701020)
- 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.MathSciNetView ArticleMATHGoogle Scholar
- 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)00539-0MathSciNetView ArticleMATHGoogle Scholar
- 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)90147-6MathSciNetView ArticleMATHGoogle Scholar
- 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.2005.11.064MathSciNetView ArticleMATHGoogle Scholar
- 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)00126-0MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.08.014MathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetView ArticleMATHGoogle Scholar
- Slatkin M: The dynamics og a population in a Markovian environment. Ecology 1978, 59: 249-256. 10.2307/1936370View ArticleGoogle Scholar
- 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.2004.02.001MathSciNetView ArticleMATHGoogle Scholar
- 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.2009.06.021MathSciNetView ArticleMATHGoogle Scholar
- Zhu C, Yin G: On hybrid competitive Lotka-Volterra ecosystems. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1370-e1379. 10.1016/j.na.2009.01.166MathSciNetView ArticleMATHGoogle Scholar
- 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.2009.03.066MathSciNetView ArticleMATHGoogle Scholar
- 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.2006.12.032MathSciNetView ArticleMATHGoogle Scholar
- 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.2009.02.010MathSciNetView ArticleMATHGoogle Scholar
- 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.Google Scholar
- 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)00029-1MathSciNetView ArticleMATHGoogle Scholar
- 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-XMathSciNetView ArticleMATHGoogle Scholar
- Alvarez LHR, Shepp LA: Optimal harvesting of stochastically fluctuating populations. Journal of Mathematical Biology 1998,37(2):155-177. 10.1007/s002850050124MathSciNetView ArticleMATHGoogle Scholar
- Alvarez LHR: Optimal harvesting under stochastic fluctuations and critical depensation. Mathematical Biosciences 1998,152(1):63-85. 10.1016/S0025-5564(98)10018-4MathSciNetView ArticleMATHGoogle Scholar
- 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.2010.04.002MathSciNetView ArticleMATHGoogle Scholar
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