Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

Abstract

The Fisher-KPP model, and generalisations thereof, is a simple reaction-diffusion models of biological invasion that assumes individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $λ$. Biologically-relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2\sqrt{λD}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically-relevant initial conditions lead to travelling waves that move with speed $c=2\sqrt{λD} > 0$. This means that, for biologically-relevant initial data, the Fisher-KPP model can not be used to study invasion with $c \ne 2\sqrt{λD}$, or retreating travelling waves with $c < 0$. Here, we reformulate the Fisher-KPP model as a moving boundary problem on $x < s(t)$, and we show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front, and can propagate with any wave speed, $-\infty < c < \infty$. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.