A stochastic version
of the Lotka-Volterra system is studied, where , , , , , , and are positive constants, and is a standard Wiener process. By setting and the equations (1) are transformed to
Let us set , . Let be an arbitrary solution to (2). It is proven that if and , then there exists a unique invariant probability measure for (2) and the distribution of converges to as in the total variation norm. If and , then almost surely, while the law of converges weakly to a measure having density . Finally, if , then both and converge to as almost surely. In the course of proofs, it is shown that the laws of both and 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.