The Speeds of Traveling Frontal Waves in Heterogeneous Environments

Abstract

Since Fisher’s pioneering work (Fisher 1937), many studies on traveling waves in a growing population have been performed. The model proposed by Fisher consists of a diffusion equation with a logistic growth term:

$${u_t} = d{u_{xx}} + \left( {\varepsilon - u} \right)u\quad for\quad x \in \left( { - \infty ,\infty } \right)$$

. Here u(x,t) denotes the population density at position x and time t, and d and ε are diffusivity and intrinsic growth rate, respectively. It has been shown from this equation that, starting from a localized distribution, the population evolves into a propagating wave of constant speed, $2\sqrt {\varepsilon d}$.