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 equationdx(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 ∑i=1nxi(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 ∑i=1nxi(t) always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower-growth rate.