Analysis of a stochastic autonomous mutualism model

An autonomous Lotka–Volterra mutualism system with random perturbations is investigated. Under some simple conditions, it is shown that there is a decreasing sequence { Δ k } which has the property that if Δ 1 < 1 , then all the populations go to extinction (i.e.  lim t → + ∞ x i ( t ) = 0 , 1 ≤ i ≤ n ); if Δ k > 1 > Δ k + 1 , then lim t → + ∞ x j ( t ) = 0 , j = k + 1 , … , n , whilst the remaining k populations are stable in the mean (i.e., lim t → + ∞ t − 1 ∫ 0 t x i ( s ) d s = a positive constant , i = 1 , … , k ); if Δ n > 1 , then all the species are stable in the mean. Sufficient conditions for stochastic permanence and global asymptotic stability are also established.