Invading and receding travelling waves of the Fisher-KPP equation with a mass-conserving, moving boundary

Abstract

The Fisher-KPP equation is the canonical reaction-diffusion equation used in the study of invasive phenomena in mathematical biology. While moving boundaries have already been included to introduce biologically realistic sharp interfaces, these flux-based conditions assume loss of mass as a byproduct of boundary movement. We consider a mass-conserving boundary condition at a free boundary of the Fisher-KPP equation to allow interface movement without population consumption. We use a combination of phase plane analysis, perturbation analysis, and numerical simulation to show that the model supports both invading and receding travelling wave solutions with distinct wave speeds and front densities. Prescribing the free boundary velocity as a linear function of the front density, we show the existence of multiple stable and unstable travelling wave solutions, as well as receding population blow-up. Our results indicate that in regions where multiple steady states exist, those with larger wave speeds are stable. This is corroborated by considering the parameter regime associated with the uniform steady state solution and performing linear stability analysis. The framework we develop is readily extendable to more general boundary velocity functions, offering a means of studying increasingly complex and biologically relevant boundary dynamics.