Travelling waves, blow-up and extinction in the Fisher-Stefan model

While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher-Stefan model, a generalisation of the well-known Fisher-KPP model, characterised by a leakage coefficient $kappa$ which relates the speed of the moving boundary to the flux of population there. This Fisher-Stefan model overcomes one of the well-known limitations of the Fisher-KPP model, since time-dependent solutions of the Fisher-Stefan model involve a well-defined front with compact support which is more natural in terms of mathematical modelling. Almost all of the existing analysis of the standard Fisher-Stefan model involves setting $kappa > 0$, which can lead to either invading travelling wave solutions or complete extinction of the population. Here, we demonstrate how setting $kappa < 0$ leads to retreating travelling waves and an interesting transition to finite-time blow-up. For certain initial conditions, population extinction is also observed. Our approach involves studying time-dependent solutions of the governing equations, phase plane and scaling analysis, leading to new insight into the possibilities of travelling waves, blow-up and extinction for this moving boundary problem. Matlab software used to generate the results in this work are available on Github.