*(English)*Zbl 1062.92055

The authors start with the deterministic $n$-dimensional delay Lotka-Volterra equation

where $x$ and $b$ are $n$-dimensional vectors and $A$ and $B$ are $n\times n$ matrices. Equation (1) can be seen as a basic model for the dynamical behaviour of a population of $n$ interacting species. The authors assume that the vector $b$, which represents the intrinsic growth rates of the $n$ species, is subject to noise. This gives rise to a stochastic delay Lotka-Volterra system with multiplicative noise. The drift and diffusion coefficients of this stochastic differential system are locally Lipschitz-continuous but do not satisfy a linear growth condition. In standard arguments the latter conditions ensures that a solution does not blow-up in finite time. Thus, the authors first consider several conditions that guarantee the global existence of a unique solution, which, in addition, stays positive almost surely.

Further, several asymptotic properties of the solutions are discussed. In particular, conditions for persistence with probability 1, asymptotic stability with probability 1 and stochastic ultimate boundedness are given. In the last section, an example of a 3-dimensional stochastic Lotka-Volterra food chain is considered and, as an illustration, specific conditions for asymptotic stability with probability 1 are given.

##### MSC:

92D25 | Population dynamics (general) |

34K50 | Stochastic functional-differential equations |

60K99 | Special processes |

34K60 | Qualitative investigation and simulation of models |

34K25 | Asymptotic theory of functional-differential equations |

60H20 | Stochastic integral equations |

93D99 | Stability of control systems |

92D40 | Ecology |