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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

dx(t)=x(t)a - b x (t) - cy(t) my(t)+x(t)dt+αx(t)dB 1 (t),x(0)=x 0 >0,dy(t)=y(t)- d + fx(t) my(t)+x(t)dt-βy(t)dB 2 (t),y(0)=y 0 >0,

where B 1 and B 2 are independent Brownian motions, a, b, c, d, f, m, α, β 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 Aa-α 2 2-c m>0 and Bf-d-β 2 2>0 , then

lim inf t t -1 0 t y(s)/x(s)ds

is positive and lim t t -1 0 t x(s)ds is finite and positive a.s.; if A<0, then lim t x(t)=0 and lim t y(t)=0 a.s.; and if A>0 and B<0, then lim t y(t)=0 and lim t t -1 0 t x(s)ds is finite and positive a.s. Results of numerical simulations are presented to show that the populations exhibit this behavior.

MSC:
92D40Ecology
34F05ODE with randomness
65C20Models (numerical methods)
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