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

The Fisher–Kolmogorov–Petrovsky–Piskunov (KPP) model, and generalizations thereof, involves simple reaction–diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity |$D$|⁠ , and logistic proliferation with rate |$\lambda $|⁠. For the Fisher–KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed |$c=2\sqrt {\lambda D}$|⁠. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher–KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed |$c=2\sqrt {\lambda D}> 0$|⁠. This means that, for biologically relevant initial data, the Fisher–KPP model cannot be used to study invasion with |$c \ne 2\sqrt {\lambda D}$|⁠ , or retreating travelling waves with |$c < 0$|⁠. Here, we reformulate the Fisher–KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher–KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, |$-\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. [ABSTRACT FROM AUTHOR]