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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.

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