Front stability of infinitely steep travelling waves in population biology

Abstract

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 reaction–diffusion models of biological invasion are extensions of the well–known Fisher–Kolmogorov–Petrovskii–Piskunov model that describes the spatiotemporal evolution of a 1D population density, u ( x , t ) , as a result of linear diffusion with flux J = − ∂ u / ∂ x , and logistic growth source term, S = u ( 1 − u ) . In 2020 Fadai introduced a new reaction–diffusion model of biological invasion with a nonlinear degenerate diffusive flux, J = − u ∂ u / ∂ x , and the model was formulated as a moving boundary problem on 0 < x < L ( t ) , with u ( L ( t ) , t ) = 0 and d L ( t ) / d t = − κ u ∂ u / ∂ x at x = L ( t ) (Fadai and Simpson 2020 J. Phys. A: Math. Theor. 53 095601). 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. In this work we consider the stability of the travelling wave solutions presented by Fadai. We provide general insight by first presenting two key extensions of Fadai’s model by considering: (i) generalised nonlinear degenerate diffusion with flux J = − u m ∂ u / ∂ x for some constant m > 0; and, (ii) solutions describing both biological invasion with d L ( t ) / d t > 0 , and biological recession with d L ( t ) / d t < 0 . After establishing the existence of travelling wave solutions for these two extensions, our main contribution is to 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 amplitude 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. Julia–based software, including level–set algorithms, is available on Github to replicate all results in this study.