Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

We consider a predator-prey model in which the preys disperse among patches () with stochastic perturbation. We show that there is a unique positive solution and find out the sufficient conditions for the extinction to the system with any given positive initial value. In addition, we investigate that there exists a stationary distribution for the system and it has ergodic property. Finally, we illustrate the dynamic behavior of the system with via numerical simulation.

[1]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[2]  Xuerong Mao,et al.  Stochastic differential delay equations of population dynamics , 2005 .

[3]  Daqing Jiang,et al.  Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation , 2009 .

[4]  H. I. Freedman,et al.  Global stability and predator dynamics in a model of prey dispersal in a patchy environment , 1989 .

[5]  D. Jiang,et al.  Existence, Uniqueness and Ergodicity of Positive Solution of Mutualism System with Stochastic Perturbation , 2010 .

[6]  Z. Teng,et al.  Boundedness and permanence in a class of periodic time-dependent predator–prey system with prey dispersal and predator density-independence , 2008 .

[7]  Michael Y. Li,et al.  A graph-theoretic approach to the method of global Lyapunov functions , 2008 .

[8]  Xuerong Mao,et al.  Stochastic Differential Equations With Markovian Switching , 2006 .

[9]  Xuerong Mao,et al.  Population dynamical behavior of Lotka-Volterra system under regime switching , 2009, J. Comput. Appl. Math..

[10]  Wang Wendi,et al.  Asymptotic Behavior of a Predator–Prey System with Diffusion and Delays , 1997 .

[11]  Michael Y. Li,et al.  Global-stability problem for coupled systems of differential equations on networks , 2010 .

[12]  Rui Xu,et al.  Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment , 2000 .

[13]  Y. K. Lin,et al.  Stochastic analysis of predator–prey type ecosystems , 2007 .

[14]  Yasuhiro Takeuchi,et al.  Global asymptotic behavior in single-species discrete diffusion systems , 1993 .

[15]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[16]  Xuerong Mao,et al.  Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation , 2009 .

[17]  Jingan Cui,et al.  The effect of dispersal on permanence in a predator-prey population growth model☆ , 2002 .

[18]  Zhidong Teng,et al.  Permanence and extinction of periodic predator-prey systems in a patchy environment with delay , 2003 .

[19]  Víctor Padrón,et al.  Environmentally induced dispersal under heterogeneous logistic growth. , 2006, Mathematical biosciences.

[20]  Y. Takeuchi,et al.  Predator-prey dynamics in models of prey dispersal in two-patch environments. , 1994, Mathematical biosciences.

[21]  Yasuhiro Takeuchi,et al.  Global stability and periodic orbits for two-patch predator-prey diffusion-delay models , 1987 .