### Abstract

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 ̇ x(t) = diag(x_{1} (t),..., x_{n}(t))[b + Ax(t - τ)] into the Itô form dx(t) = diag(x_{1} (t),..., x_{n}(t) [(b + Ax(t - τ))dt + σ x(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a 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.

Language | English |
---|---|

Pages | 364-380 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 292 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Apr 2004 |

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### Keywords

- Brownian motion
- explosion
- Itô's formula
- stochastic differential delay equation
- ultimate boundedness

### Cite this

*Journal of Mathematical Analysis and Applications*,

*292*(2), 364-380. https://doi.org/10.1016/j.jmaa.2003.12.004

}

*Journal of Mathematical Analysis and Applications*, vol. 292, no. 2, pp. 364-380. https://doi.org/10.1016/j.jmaa.2003.12.004

**Stochastic delay Lotka-Volterra model.** / Bahar, Arifah; Mao, Xuerong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Stochastic delay Lotka-Volterra model

AU - Bahar, Arifah

AU - Mao, Xuerong

PY - 2004/4/15

Y1 - 2004/4/15

N2 - 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 ̇ x(t) = diag(x1 (t),..., xn(t))[b + Ax(t - τ)] into the Itô form dx(t) = diag(x1 (t),..., xn(t) [(b + Ax(t - τ))dt + σ x(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a 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.

AB - 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 ̇ x(t) = diag(x1 (t),..., xn(t))[b + Ax(t - τ)] into the Itô form dx(t) = diag(x1 (t),..., xn(t) [(b + Ax(t - τ))dt + σ x(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a 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.

KW - Brownian motion

KW - explosion

KW - Itô's formula

KW - stochastic differential delay equation

KW - ultimate boundedness

UR - http://www.scopus.com/inward/record.url?scp=1942508139&partnerID=8YFLogxK

UR - http://www.sciencedirect.com/science/article/pii/S0022247X03009089

U2 - 10.1016/j.jmaa.2003.12.004

DO - 10.1016/j.jmaa.2003.12.004

M3 - Article

VL - 292

SP - 364

EP - 380

JO - Journal of Mathematical Analysis and Applications

T2 - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -