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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

$\begin{array}{cc}\hfill dx\left(t\right)& =x\left(t\right)\left(a-bx\left(t\right)-\frac{cy\left(t\right)}{my\left(t\right)+x\left(t\right)}\right)\phantom{\rule{0.166667em}{0ex}}dt+\alpha x\left(t\right)\phantom{\rule{0.166667em}{0ex}}d{B}_{1}\left(t\right),\phantom{\rule{1.em}{0ex}}x\left(0\right)={x}_{0}>0,\hfill \\ \hfill dy\left(t\right)& =y\left(t\right)\left(-d+\frac{fx\left(t\right)}{my\left(t\right)+x\left(t\right)}\right)\phantom{\rule{0.166667em}{0ex}}dt-\beta y\left(t\right)\phantom{\rule{0.166667em}{0ex}}d{B}_{2}\left(t\right),\phantom{\rule{1.em}{0ex}}y\left(0\right)={y}_{0}>0,\hfill \end{array}$

where ${B}_{1}$ and ${B}_{2}$ are independent Brownian motions, $a$, $b$, $c$, $d$, $f$, $m$, $\alpha$, $\beta$ are positive constants, and $x\left(t\right)$, $y\left(t\right)$ 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 $A\equiv a-\frac{{\alpha }^{2}}{2}-\frac{c}{m}>0$ and $B\equiv f-d-\frac{{\beta }^{2}}{2}>0$ , then

$\underset{t\to \infty }{lim inf}\phantom{\rule{4pt}{0ex}}{t}^{-1}{\int }_{0}^{t}y\left(s\right)/x\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$

is positive and ${lim}_{t\to \infty }{t}^{-1}{\int }_{0}^{t}x\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$ is finite and positive a.s.; if $A<0$, then ${lim}_{t\to \infty }x\left(t\right)=0$ and ${lim}_{t\to \infty }y\left(t\right)=0$ a.s.; and if $A>0$ and $B<0$, then ${lim}_{t\to \infty }y\left(t\right)=0$ and ${lim}_{t\to \infty }{t}^{-1}{\int }_{0}^{t}x\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$ is finite and positive a.s. Results of numerical simulations are presented to show that the populations exhibit this behavior.

MSC:
 92D40 Ecology 34F05 ODE with randomness 65C20 Models (numerical methods)
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