# zbMATH — the first resource for mathematics

Analysis of a stochastic autonomous mutualism model. (English) Zbl 1417.92141
Summary: An autonomous Lotka-Volterra mutualism system with random perturbations is investigated. Under some simple conditions, it is shown that there is a decreasing sequence $${\Delta_{k}}$$ which has the property that if $$\Delta_{1}$$, then all the populations go to extinction (i.e. $$\lim_{t\to +\infty }x_{i}(t)=0,1\leq i\leq n$$); if $$\Delta_{k}>1>\Delta_{k+1}$$, then $$\lim_{_{t\to +\infty }}x_{j}(t)=0, j=k+1,\ldots ,n$$, whilst the remaining $$k$$ populations are stable in the mean (i.e., $$\lim_{_{t\to +\infty }}t^{ - 1}\int _0^t x_{i}(s)ds=a$$ positive constant $$i=1,\ldots ,k$$); if $$\Delta_{n}>1$$, then all the species are stable in the mean. Sufficient conditions for stochastic permanence and global asymptotic stability are also established.

##### MSC:
 92D25 Population dynamics (general) 92D40 Ecology 34D23 Global stability of solutions to ordinary differential equations
Full Text: