Pattern Formation in Heterogeneous Reaction–Diffusion–Advection Systems with an Application to Population Dynamics

Heterogeneous reaction–diffusion–advection equations are proposed for studying pattern formation due to spatial heterogeneity. The equations contain a small parameter $\varepsilon $, indicating the ratio of the diffusion and advection rates and the reaction rate. Two-timing methods in the limit $\varepsilon \downarrow 0$ make it possible to reduce the original partial differential equation problem to the approximating ordinary differential equation problem, so that asymptotic states of solutions can be investigated. As an application to population dynamics, population models of the Lotka–Volterra type are considered for studying the effect of spatial heterogeneity on development of spatial and temporal distributions of individuals.