Environmental Brownian noise suppresses explosions in population dynamics

Xuerong Mao, Glenn Marion*, Eric Renshaw

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

803 Citations (Scopus)


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. 

Original languageEnglish
Pages (from-to)95-110
Number of pages16
JournalStochastic Processes and their Applications
Issue number1
Publication statusPublished - 2002


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


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