Stochastic delay Lotka–Volterra model.

*(English)*Zbl 1043.92034Summary: We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka–Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model
\[
\dot x(t)=\text{diag} (x_1(t),\dots,x_n(t))[b+Ax(t-\tau)]
\]
into the Itô form
\[
dx(t)=\text{diag}(x_1(t),\dots,x_n(t))[(b+Ax(t-\tau)) dt+\sigma x(t) dw(t)],
\]
and show that although the solution to the original delay equation may explode to infinity in finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix \(A\).

##### MSC:

92D40 | Ecology |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

92D25 | Population dynamics (general) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

##### Keywords:

Brownian motion; Stochastic differential delay equation; Explosion; Ultimate boundedness; Itô’s formula
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\textit{A. Bahar} and \textit{X. Mao}, J. Math. Anal. Appl. 292, No. 2, 364--380 (2004; Zbl 1043.92034)

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