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Qualitative analysis of a stochastic ratio-dependent predator-prey system. (English) Zbl 1229.92076
This paper studies the stochastic predator-prey population model \begin{aligned} dx(t) &= x(t)\Biggl(a- bx(t)- {cy(t)\over my(t)+ x(t)}\Biggr)\, dt+\alpha x(t)\,dB_1(t),\quad x(0)= x_0> 0,\\ dy(t) &= y(t)\Biggl(- d+{fx(t)\over my(t)+ x(t)}\Biggr)\,dt-\beta y(t)\,dB_2(t),\quad y(0)= y_0> 0,\end{aligned} where $$B_1$$ and $$B_2$$ are independent Brownian motions, $$a$$, $$b$$, $$c$$, $$d$$, $$f$$, $$m$$, $$\alpha$$, $$\beta$$ are positive constants, and $$x(t)$$, $$y(t)$$ represent the populations of prey and predators, respectively. It is proved that the system has a unique positive solution whose mean is uniformly bounded. If $$A\equiv a-{\alpha^2\over 2}-{c\over m}> 0$$ and $$B\equiv f-d-{\beta^2\over 2}> 0$$ , then $\liminf_{t\to\infty}\;t^{-1}\int^t_0 y(s)/x(s)\,ds$ is positive and $$\lim_{t\to\infty} t^{-1}\int^t_0 x(s)\,ds$$ is finite and positive a.s.; if $$A< 0$$, then $$\lim_{t\to\infty} x(t)= 0$$ and $$\lim_{t\to\infty} y(t)= 0$$ a.s.; and if $$A> 0$$ and $$B< 0$$, then $$\lim_{t\to\infty}y(t)= 0$$ and $$\lim_{t\to\infty} t^{-1} \int^t_0 x(s)\,ds$$ is finite and positive a.s. Results of numerical simulations are presented to show that the populations exhibit this behavior.

##### MSC:
 92D40 Ecology 34F05 Ordinary differential equations and systems with randomness 65C20 Probabilistic models, generic numerical methods in probability and statistics
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