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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.

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