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Stochastic differential delay equations of population dynamics. (English) Zbl 1062.92055

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

$dx\left(t\right)/dt=\text{diag}\left({x}_{1}\left(t\right),...,{x}_{n}\left(t\right)\right)\left[b+Ax\left(t\right)+Bx\left(t-\tau \right)\right],\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $x$ and $b$ are $n$-dimensional vectors and $A$ and $B$ are $n×n$ 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