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

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 κ$\kappa$ 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 which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting κ>0$\kappa > 0$, which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting κ<0$\kappa < 0$ 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.

Curvature dependences of wave propagation in reaction–diffusion models

“…Another situation of interest is the investigation of curvature-dependent front speeds in metastable systems with wave-pinning, such as the Allen–Cahn equation with Ffalse(ufalse)=ufalse(1−u2false) [49], where the dip-filling phase is absent. Finally, generalizations of our analyses to heterogeneous excitable media [50] and to Fisher–Stefan models representing reaction–diffusion systems coupled with moving boundary conditions [51,52] are also of interest, in both situations supporting travelling-wave solutions as well as dip-filling situations.…”

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

“…Indeed, solving Equations ( 1)-( 3) with κ < −1/(1 − u f ) does not lead to constant speed, constant shape travelling wave solutions. Instead, for these cases the time-dependent solutions appear to undergo blow-up, as explored in [41].…”

Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups

“…In [1], blow-up points to one-phase Stefan problems, however with Dirichlet boundary conditions, are treated and studied numerically. Let us finally mention the recent paper [31], which analyses the Fisher-Stefan model, a generalization of the well-known Fisher-KPP model, in the context of biological invasion, where the speed of the moving boundary is related to the flux of population there. By rescaling this problem, it can be compared to the supercooled Stefan problem.…”

“…By rescaling this problem, it can be compared to the supercooled Stefan problem. In this context, [31] provides new links between these models and a numerical scheme for the Fisher-Stefan model.…”