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 {Δk}{Δk} which has the property that if Δ1<1Δ1<1, then all the populations go to extinction (i.e. limt→+∞xi(t)=0,1≤i≤n); if Δk>1>Δk+1Δk>1>Δk+1, then limt→+∞xj(t)=0limt→+∞xj(t)=0, j=k+1,…,nj=k+1,…,n, whilst the remaining kk populations are stable in the mean (i.e., limt→+∞t−1∫0txi(s)ds=a positive constant, i=1,…,ki=1,…,k); if Δn>1Δn>1, then all the species are stable in the mean. Sufficient conditions for stochastic permanence and global asymptotic stability are also established.