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Recurrence for stochastic predator-prey models. (English) Zbl 0926.34045

Martelli, Mario (ed.) et al., Differential equations and applications to biology and to industry. Proceedings of the Claremont international conference dedicated to the memory of Stavros Busenberg (1941–1993), Claremont, CA, USA, June 1–4, 1994. Singapore: World Scientific. 117-124 (1996).
The authors extend the well-known interaction predator-prey model to the Itô stochastic system \[ dX= X[f(X,Y)dt+ \sigma_1(X,Y)dW_1],\;dY= Y[g(X,Y)dt+ \sigma_2(X,Y)dW_2],\tag{\(*\)} \] where the functions \(f\), \(g\), \(\sigma_1\) and \(\sigma_2\) are \(C^1\)-functions and \(W_1\) and \(W_2\) are independent Brownian motions defined on \(\mathbb{R}^2_+\).
Under the assumptions \(\partial f/\partial y<0\) and \(\partial g/\partial x>0\), the authors investigate the influence of the random fluctuations on the drift of the system. The existence of a stable invariant distribution is equivalent to the positive recurrence for \((*)\). Some sufficiency conditions for positive recurrence are obtained.
For the entire collection see [Zbl 0901.00038].

MSC:

34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
92D50 Animal behavior
37H10 Generation, random and stochastic difference and differential equations
37N25 Dynamical systems in biology
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