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Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. (English) Zbl 1161.92048
Summary: We consider a non-autonomous stochastic Lotka-Volterra competitive system
\[ dx_i (t) = x_i(t) \Bigg[\bigg(b_i(t)-\sum_{j=1}^{n} a_{ij}(t)x_j(t)\bigg)\,dt+ \sigma_i(t) \,d B_i(t)\Bigg], \] where \(B_i(t)\) \((i=1 , 2,\dots, n)\) are independent standard Brownian motions. Some dynamical properties are discussed and 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.

92D40 Ecology
60J65 Brownian motion
34F05 Ordinary differential equations and systems with randomness
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