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

Abstract

The Fisher-KPP model, and generalisations thereof, are simple reaction-diffusion models of biological invasion that assume individuals in the population undergo linear diffusion with diffusivity D, and logistic proliferation with rate λ. For the Fisher-KPP model, biologically-relevant initial conditions lead to long-time travelling wave solutions that move with speed c = 2 √ 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 = 2This means that, for biologically-relevant initial data, the Fisher-KPP model can not be used to study invasion with c 2 √ 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 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, −∞ < c < ∞. 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.