## Long-time behaviour of a stochastic prey–predator model.(English)Zbl 1075.60539

A stochastic version $dX_ {t} = (\alpha X_ {t} -\beta X_ {t}Y_ {t} - \mu X^ 2_ {t})\,dt + \sigma X_ {t}\,dW_ {t}, \quad dY_ {t} = (-\gamma Y_ {t} + \delta X_ {t}Y_ {t} - \nu Y^ 2_ {t})\,dt + \rho Y_ {t}\,dW_ {t}\tag{1}$ of the Lotka-Volterra system is studied, where $$\alpha$$, $$\beta$$, $$\gamma$$, $$\delta$$, $$\mu$$, $$\nu$$, $$\rho$$ and $$\sigma$$ are positive constants, and $$W$$ is a standard Wiener process. By setting $$X_ {t} = \exp (\xi _ {t})$$ and $$Y_ {t} = \exp (\eta _ {t})$$ the equations (1) are transformed to $d\xi _ {t} = (\alpha - \sigma ^ 2/2 - \mu e^ {\xi _ {t}} -\beta e^ {\eta _ {t}})\,dt + \sigma \,dW_ {t},\quad d\eta _ {t} = (-\gamma -\rho ^ 2/2 + \delta e^ {\xi _ {t}} - \nu e^ {\eta _ {t}})\,dt + \rho \,dW_ {t}.\tag{2}$ Let us set $$c_ 1 = \alpha - \sigma ^ 2/2$$, $$c_ 2 = \gamma + \rho ^ 2 /2$$. Let $$(\xi ,\eta )$$ be an arbitrary solution to (2). It is proven that if $$c_ 1>0$$ and $$\mu c_ 2 <\delta c_ 1$$, then there exists a unique invariant probability measure $$m^ *$$ for (2) and the distribution of $$(\xi _ {t},\eta _ {t})$$ converges to $$m^ *$$ as $$t\to \infty$$ in the total variation norm. If $$c_ 1>0$$ and $$\mu c_ 2 >\delta c_ 1$$, then $$\lim _ {t\to \infty } \eta _ {t} = -\infty$$ almost surely, while the law of $$\xi _ {t}$$ converges weakly to a measure having density $$C\exp (2c_ 1\sigma ^ {-2}x - 2\mu \sigma ^ {-2}e^ {x})$$. Finally, if $$c_ 1<0$$, then both $$\xi _ {t}$$ and $$\eta _ {t}$$ converge to $$-\infty$$ as $$t\to \infty$$ almost surely. In the course of proofs, it is shown that the laws of both $$(\xi _ {t},\eta _ {t})$$ and $$m^ *$$ have density with respect to two-dimensional Lebesgue measure, hence recent results on long-time behaviour of integral Markov semigroups [see e.g.K.Pichór and R.Rudnicki, J. Math. Anal. Appl. 249, 668–685 (2000; Zbl 0965.47026)] may be applied.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 47D07 Markov semigroups and applications to diffusion processes 60J60 Diffusion processes 92D25 Population dynamics (general)

Zbl 0965.47026
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### References:

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