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

Abstract This paper continues the study of Mao et al. investigating two aspects of the equation d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ ( b + A x ( t ) ) d t + σ x ( t ) d W ( 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 = 1 n 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 ∑ i = 1 n x i ( t ) always visits any neighborhood of the point 0 and we simultaneously give estimates for this lower-growth rate.