Hyperbolic limit for a biological invasion

Abstract

In a spatially heterogeneous environment the propagation speed of a biological invasion varies in space. The traveling wave theory in a homogeneous case is not extended to a heterogeneous case. Taking a singular limit in a hyperbolic scale is a good way to study such a wave propagation with constant speed. The goal of this project is to understand the effect of biological diffusion on the wave speed in a spatial heterogeneous environment. For this purpose, we consider \begin{document}$ U_t = {\varepsilon}(\gamma(s)U)_{xx}+{1\over{\varepsilon}}U(1-{U/m(x)}), $\end{document} where $ m $ is a nonconstant carrying capacity, $ s = {U\over m} $ is a starvation measure and $ \gamma(s) = s^{\tilde{k}}, \tilde{k} \ge 1 $. The diffusion is a starvation driven diffusion. We show that the diffusion speed is constant even if $ m $ is nonconstant.