Long-time behaviour of a stochastic prey--predator model.
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.\textit{K.Pichór} and \textit{R.Rudnicki}, J. Math. Anal. Appl. 249, 668--685 (2000; Zbl 0965.47026)] may be applied.
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