An exact solution of a diffusive predator–prey system

Abstract

We consider a system of two nonlinear partial differential equations describing the spatio-temporal dynamics of a predator–prey community where the prey per capita growth rate is damped by the Allee effect. Using an appropriate change of variables, we obtain an exact solution of the system, which appears to be related to the issue of biological invasion. In the large-time limit, or for appropriate parameter values, this solution describes the propagation of a travelling population front. We show that the properties of the solution exhibit biologically reasonable dependence on the parameter values; in particular, it predicts that the travelling front of invasive species can be stopped or reversed owing to the impact of predation.