The authors start with the deterministic -dimensional delay Lotka-Volterra equation
where and are -dimensional vectors and and are matrices. Equation (1) can be seen as a basic model for the dynamical behaviour of a population of interacting species. The authors assume that the vector , which represents the intrinsic growth rates of the 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.