Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation

In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system $ dx_i (t) = x_i(t)$[($b_i(t)$-$\sum_{j=1}^{n} a_{ij}(t)x_j(t))$$dt$$+ \sigma_i(t) d B_i(t)]$, where $B_i(t)$($i=1 ,\ 2,\cdots,\ n$) are independent standard Brownian motions. Some dynamical properties are discussed and the sufficient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated.

[1]  Persistence and periodic orbits for two-species non-autonomous diffusion lotka-volterra models , 2004 .

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

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

[4]  R. Khasminskii,et al.  Long term behavior of solutions of the Lotka-Volterra system under small random perturbations , 2001 .

[5]  Daqing Jiang,et al.  A note on nonautonomous logistic equation with random perturbation , 2005 .

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

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

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

[9]  Jifa Jiang,et al.  Average conditions for permanence and extinction in nonautonomous Lotka–Volterra system , 2004 .

[10]  X. Mao,et al.  A note on the LaSalle-type theorems for stochastic differential delay equations , 2002 .

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

[12]  Kondalsamy Gopalsamy,et al.  Global asymptotic stability in Volterra's population systems , 1984 .

[13]  Yanan Zhao,et al.  Existence, uniqueness, and global stability of positive solutions to the food-limited population model with random perturbation , 2005, Math. Comput. Model..

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

[15]  Alan C. Lazer,et al.  Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system , 2000 .

[16]  Zhidong Teng.  On the non-autonomous Lotka-Volterra N-species competing systems , 2000, Appl. Math. Comput..

[17]  Daqing Jiang,et al.  Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation☆ , 2008 .

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

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

[20]  Jifa Jiang,et al.  The permanence and global attractivity in a nonautonomous Lotka–Volterra system , 2004 .

[21]  Shair Ahmad,et al.  Extinction of species in nonautonomous Lotka-Volterra systems , 1999 .

[22]  Xuerong Mao,et al.  Stochastic differential equations and their applications , 1997 .

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

[24]  Donal O'Regan,et al.  Existence, uniqueness, stochastic persistence and global stability of positive solutions of the logistic equation with random perturbation , 2007 .

[25]  Werner Horsthemke,et al.  The Influence of External Real and White Noise on the LOTKA‐VOLTERRA Model , 1979 .

[26]  Alexandra Rodkina,et al.  Almost sure stability of some stochastic dynamical systems with memory , 2008 .