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Zbl 1075.60539
Rudnicki, Ryszard
Long-time behaviour of a stochastic prey--predator model. (English)
[J] Stochastic Processes Appl. 108, No. 1, 93-107 (2003). ISSN 0304-4149

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

MSC 2000:: *60H10 Stochastic ordinary differential equations
47D07 Markov semigroups of linear operators
60J60 Diffusion processes
92D25 Population dynamics

Keywords: stochastic Lotka-Volterra equations; Markov semigroups; invariant measure; asymptotic stability

Citations: Zbl 0965.47026

Cited in: Zbl 1204.37078

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