  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.
