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Zbl 1229.92076
Ji, Chunyan; Jiang, Daqing; Li, Xiaoyue
Qualitative analysis of a stochastic ratio-dependent predator-prey system.
(English)
[J] J. Comput. Appl. Math. 235, No. 5, 1326-1341 (2011). ISSN 0377-0427

This paper studies the stochastic predator-prey population model \align 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,\endalign 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.
[Melvin D. Lax (Long Beach)]
MSC 2000:
*92D40 Ecology
34F05 ODE with randomness
65C20 Models (numerical methods)

Keywords: Itô's formula; persistence in mean; extinction; stable in time average

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