Front stability of infinitely steep travelling waves in population biology

Abstract: Reaction-diffusion models are often used to describe biological invasion, where populations of individuals that undergo random motility and proliferation lead to moving fronts. Many models of biological invasion are extensions of the Fisher-KPP model that describes the evolution of a 1D population density as a result of linear diffusion and logistic growth. In 2020 Fadai introduced a new model of biological invasion that was formulated as a moving boundary problem with a nonlinear degenerate diffusive flux. Fadai's model leads to travelling wave solutions with infinitely steep, well-defined fronts at the moving boundary, and the model has the mathematical advantage of being analytically tractable in certain parameter limits. We aim to provide general insight by first presenting two key extensions by considering: (i) generalised nonlinear degenerate diffusion with flux; and, (ii) solutions describing both biological invasion, and biological recession. We establish the existence of travelling wave solutions for these two extensions, and then consider stability of the travelling wave solutions by introducing a lateral perturbation of the travelling wavefront. Full 2D time-dependent level-set numerical solutions indicate that invasive travelling waves are stable to small lateral perturbations, whereas receding travelling waves are unstable. These preliminary numerical observations are corroborated through a linear stability analysis that gives more formal insight into short time growth/decay of wavefront perturbation amplitude.