Dynamics of a stochastic Lotka–Volterra model perturbed by white noise

This paper continues the study of Mao et al. investigating two aspects of the equation dx(t) = diag x1(t ), . . . , xn(t) b + Ax(t) dt +σx(t)dW(t) , t 0. The first of these is to slightly improve results in [X. Mao, S. Sabais, E. Renshaw, Asymptotic behavior of stochastic Lotka–Volterra model, J. Math. Anal. 287 (2003) 141–156] concerning with the upper-growth rate of the total quantity n i=1 xi (t) of species by weakening hypotheses posed on the coefficients of the equation. The second aspect is to investigate the lower-growth rate of the positive solutions. By using Lyapunov function technique and using a changing time method, we prove that the total quantity n i=1 xi (t) always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower-growth rate. © 2005 Elsevier Inc. All rights reserved