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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=diag(x 1 (t),...,x n (t))b + A x ( t ) + B x ( t - τ ),(1)

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:
92D25Population dynamics (general)
34K50Stochastic functional-differential equations
60K99Special processes
34K60Qualitative investigation and simulation of models
34K25Asymptotic theory of functional-differential equations
60H20Stochastic integral equations
93D99Stability of control systems
92D40Ecology