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(x(1)(t),...,x(n)(t))[(b+Ax(t))dt+sigma x(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] conceming with the upper-growth rate of the total quantity Sigma(n)(i=1) x(i) (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 Sigma(n)(i=1) x(i) (t) always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower-growth rate. (c) 2005 Elsevier Inc. All rights reserved.