Stationary distribution of stochastic population systems

In this paper we consider the stochastic differential equation (SDE) population model dx(t) = diag(x1(t), . . . , xn(t))[(b + Ax(t))dt + σdB(t)] for n interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004) [2] and Mao (2005) [6]) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.

[1]  Y. Kuang Delay Differential Equations: With Applications in Population Dynamics , 2012 .

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

[3]  K. Gopalsamy Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[4]  Xue-Zhong He,et al.  Persistence, Attractivity, and Delay in Facultative Mutualism , 1997 .

[5]  Qi Luo,et al.  Noise suppresses or expresses exponential growth , 2008, Syst. Control. Lett..

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

[7]  Feiqi Deng,et al.  Asymptotic properties of stochastic population dynamics , 2008 .

[8]  S. Zacks,et al.  Introduction to stochastic differential equations , 1988 .

[9]  Zhidong Teng,et al.  Some New Results of Nonautonomous Lotka–Volterra Competitive Systems with Delays☆☆☆ , 2000 .

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

[11]  X. Mao,et al.  Environmental Brownian noise suppresses explosions in population dynamics , 2002 .

[12]  X. Mao,et al.  Exponential Stability of Stochastic Di erential Equations , 1994 .

[13]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[14]  Thomas C. Gard Persistence in stochastic food web models , 1984 .

[15]  L. Rogers Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[16]  Xuerong Mao DELAY POPULATION DYNAMICS AND ENVIRONMENTAL NOISE , 2005 .

[17]  K. Sato,et al.  Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment , 2006 .

[18]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[19]  X. Mao,et al.  Stability of Stochastic Differential Equations With Respect to Semimartingales , 1991 .

[20]  T. Gard,et al.  Stability for multispecies population models in random environments , 1986 .

[21]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[22]  Xuerong Mao,et al.  STOCHASTIC DELAY POPULATION DYNAMICS , 2004 .

[23]  V. Kolmanovskii,et al.  Applied Theory of Functional Differential Equations , 1992 .

[24]  Xuerong Mao,et al.  Stochastic delay Lotka-Volterra model , 2004 .