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Stochastic delay population dynamics. (English) Zbl 1043.92028

The authors consider a stochastically perturbed delay Lotka-Volterra model with additive noise of the form

dx(t)=diag(x 1 (t),...,x 2 (t))[b+Ax(t)+Bx(t-τ))dt+βdw(t)]·(1)

Here w(t) denotes Brownian motion. First they provide conditions guaranteeing that (1) has a unique global solution which remains positive almost surely. Then they consider ultimate boundedness in mean of the solution of (1), i.e., they give conditions such that

lim sup t E|x(t)|K

holds, where K is a constant. In the last section the authors discuss conditions implying that the population described by the solution of (1) becomes extinct with probability one.

MSC:
92D25Population dynamics (general)
60H10Stochastic ordinary differential equations
34K50Stochastic functional-differential equations