### Abstract

Population systems are often subject to environmental noise, and our aim is to show that (surprisingly) the presence of even a tiny amount can suppress a potential population explosion. To prove this intrinsically interesting result, we stochastically perturb the multivariate deterministic system ẋ(t) = f(x(t)) into the Itô form dx(t) = f(x(t))dt + g(x(t))dw(t), and show that although the solution to the original ordinary differential equation may explode to infinity in a finite time, with probability one that of the associated stochastic differential equation does not.

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

Pages | 95-110 |

Number of pages | 16 |

Journal | Stochastic Processes and their Applications |

Volume | 97 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2002 |

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### Keywords

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

### Cite this

}

*Stochastic Processes and their Applications*, vol. 97, no. 1, pp. 95-110. https://doi.org/10.1016/S0304-4149(01)00126-0

**Environmental Brownian noise suppresses explosions in population dynamics.** / Mao, Xuerong; Marion, Glenn; Renshaw, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Environmental Brownian noise suppresses explosions in population dynamics

AU - Mao, Xuerong

AU - Marion, Glenn

AU - Renshaw, Eric

PY - 2002

Y1 - 2002

N2 - Population systems are often subject to environmental noise, and our aim is to show that (surprisingly) the presence of even a tiny amount can suppress a potential population explosion. To prove this intrinsically interesting result, we stochastically perturb the multivariate deterministic system ẋ(t) = f(x(t)) into the Itô form dx(t) = f(x(t))dt + g(x(t))dw(t), and show that although the solution to the original ordinary differential equation may explode to infinity in a finite time, with probability one that of the associated stochastic differential equation does not.

AB - Population systems are often subject to environmental noise, and our aim is to show that (surprisingly) the presence of even a tiny amount can suppress a potential population explosion. To prove this intrinsically interesting result, we stochastically perturb the multivariate deterministic system ẋ(t) = f(x(t)) into the Itô form dx(t) = f(x(t))dt + g(x(t))dw(t), and show that although the solution to the original ordinary differential equation may explode to infinity in a finite time, with probability one that of the associated stochastic differential equation does not.

KW - boundedness

KW - Brownian motion

KW - explosion

KW - Itô's formula

KW - stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=0242563961&partnerID=8YFLogxK

U2 - 10.1016/S0304-4149(01)00126-0

DO - 10.1016/S0304-4149(01)00126-0

M3 - Article

VL - 97

SP - 95

EP - 110

JO - Stochastic Processes and their Applications

T2 - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 1

ER -