Prevailing winds and spruce budworm outbreaks: a reaction-diffusion-advection model

Abstract

We extend the classical reaction-diffusion model for spatial population dynamics of spruce budworm on a finite domain with hostile boundary conditions by including an advection term representing biased unidirectional movement of individuals due to a prevailing wind. We use phase-plane techniques to establish existence of a critical value of advection speed that prevents outbreak solutions on any finite domain while possibly allowing an endemic solution. We obtain lower and upper bounds for this critical advection value in terms of biological parameters involved in the reaction term. We also perform numerical simulations to illustrate the effect of advection on the dependence of the domain size on the maximal population density of a steady state solution and on critical domain sizes for endemic and outbreak solutions. The results are also applicable to other ecological settings (rivers, climate change) where a logistically growing population is subject to predation by a generalist, diffusion and biased movement.