Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

Abstract

Abstract 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 generalization of the well‐known Fisher–KPP model, characterized by a leakage coefficient 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 which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting , which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting leads to retreating traveling 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 asymptotic analysis, leading to new insight into the possibilities of traveling waves, blow‐up, and extinction for this moving boundary problem. MATLAB software used to generate the results in this work is available on Github.