Gard, Thomas C.; Ragsdale, Katherine B. 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]. Reviewer: D.Bobrowski (Poznań) 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 Keywords:Itô stochastic equation; predator-prey model; Lotka-Volterra drift term; Brownian motions; random fluctuations; stable invariant distribution PDFBibTeX XMLCite \textit{T. C. Gard} and \textit{K. B. Ragsdale}, in: Differential equations and applications to biology and to industry. Proceedings of the Claremont international conference dedicated to the memory of Starvros Busenberg (1941--1993), Claremont, CA, USA, June 1--4, 1994. Singapore: World Scientific. 117--124 (1996; Zbl 0926.34045)