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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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