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Delay population dynamics and environmental noise. (English) Zbl 1093.60033

The paper presents in an integrated fashion fundamental results concerning $n$-dimensional stochastic delay differential equations of Lotka-Volterra type with logistic self-controls used to describe the dynamics of $n$ interacting populations subjected to environmental noise. Some previous results are reviewed and some new results are presented. The author considers a one-dimensional white noise additively affecting the per capita growth rates of populations in three different ways: a) with constant intensities (which may differ among populations); b) with intensities proportional to a weighted average of the $n$ interacting population sizes; c) with intensities proportional to a weighted average of the deviations between population sizes and their deterministic equilibria.

For a), conditions for non-explosion and positiveness of the solutions and for their ultimate boundedness (in mean) are obtained that hold both for deterministic and stochastic equations. However, large noise intensities may lead to extinction of an otherwise persistence population. For b), it is shown that any small amount of noise can suppress a potential population explosion, keeping the population sizes positive and even making the population ultimately bounded in probability. For c), noise intensity must be kept small if nice properties (non-explosion, persistence, and asymptotic stability) are to be kept.

##### MSC:
 60H10 Stochastic ordinary differential equations 92D25 Population dynamics (general) 34K50 Stochastic functional-differential equations