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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 \[ dx(t)/dt= \text{diag}(x_1(t),\ldots,x_n(t))\big[b+ Ax(t)+ Bx(t-\tau)\big],\tag{1} \] 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 involving functional-differential equations
34K25 Asymptotic theory of functional-differential equations
60H20 Stochastic integral equations
93D99 Stability of control systems
92D40 Ecology
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