Environmental Brownian noise suppresses explosions in population dynamics

Mao, Xuerong and Marion, Glenn and Renshaw, Eric (2002) Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Processes and their Applications, 97 (1). pp. 95-110. ISSN 0304-4149

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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.