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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.

92D25 Population dynamics (general)
92D40 Ecology
34D23 Global stability of solutions to ordinary differential equations
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