  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 $\lambda$. 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 can not 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 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.

Pattern Formation and Solitons;Populations and Evolution