Extinction conditions for isolated populations affected by environmental stochasticity

We determine the critical patch size below which extinction occurs for populations living in one-dimensional habitats surrounded by completely hostile environments in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction-diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain bifurcation diagrams and relations for the mean population density at the stationary state, the critical patch size, and the mean number of individuals in the habitat. The effect of the noise differs, depending on whether it affects the net growth rate or the intraspecific competition term. Fluctuations in the net growth rate decrease the critical patch size, whereas fluctuations in the competition term do not change the critical patch size. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction-diffusion equations with multiplicative noise.