Stochastic delay population dynamics.

*(English)*Zbl 1043.92028The authors consider a stochastically perturbed delay Lotka-Volterra model with additive noise of the form

$$dx\left(t\right)=diag({x}_{1}\left(t\right),...,{x}_{2}\left(t\right))[b+Ax\left(t\right)+Bx(t-\tau ))dt+\beta dw\left(t\right)]\xb7\phantom{\rule{2.em}{0ex}}\left(1\right)$$

Here $w\left(t\right)$ 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

$$\underset{t\to \infty}{lim\; sup}E\left|x\left(t\right)\right|\le K$$

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.

Reviewer: Evelyn Buckwar (Berlin)

##### MSC:

92D25 | Population dynamics (general) |

60H10 | Stochastic ordinary differential equations |

34K50 | Stochastic functional-differential equations |