Environmental Brownian noise suppresses explosions in population dynamics 3

7 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. 9 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 11 although the solution to the original ordinary di9erential equation may explode to in;nity in a ;nite time, with probability one that of the associated stochastic di9erential equation does not. 13 c © 2001 Published by Elsevier Science B.V.