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Stationary distribution of stochastic population systems. (English) Zbl 1387.60107

Summary: We consider the stochastic differential equation (SDE) population model \(dx(t)=\mathrm{diag}(x_1(t),\dots,x_n(t))[(b+Ax(t))dt +\sigma dB(t)]\) for \(n\) interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. [A. Bahar and X. Mao, Int. J. Pure Appl. Math. 11, No. 4, 377–400 (2004; Zbl 1043.92028);X. Mao, Stoch. Dyn. 5, No. 2, 149–162 (2005; Zbl 1093.60033)]) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
92D25 Population dynamics (general)
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