Wavefronts for a degenerate reaction-diffusion system with application to bacterial growth models

Abstract

We investigate wavefront solutions in a nonlinear system of two coupled reaction-diffusion equations with degenerate diffusivity: \[n_t = n_{xx} - nb, \quad b_t = [D nbb_x]_x + nb,\] where $t\geq0,$ $x\in\mathbb{R}$, and $D$ is a positive diffusion coefficient. This model, introduced by Kawasaki et al. (J. Theor. Biol. 188, 1997), describes the spatial-temporal dynamics of bacterial colonies $b=b(x,t)$ and nutrients $n=n(x,t)$ on agar plates. Kawasaki et al. provided numerical evidence for wavefronts, leaving the analytical confirmation of these solutions an open problem. We prove the existence of an infinite family of wavefronts parameterized by their wave speed, which varies on a closed positive half-line. We provide an upper bound for the threshold speed and a lower bound for it when $D$ is sufficiently large. The proofs are based on several analytical tools, including the shooting method and the fixed-point theory in Fréchet spaces, to establish existence, and the central manifold theorem to ascertain uniqueness.