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

TL;DR

This paper analytically confirms the existence of wavefront solutions in a degenerate reaction-diffusion system modeling bacterial growth, providing bounds on wave speeds and employing advanced mathematical techniques.

Contribution

It proves the existence of wavefronts in a degenerate reaction-diffusion model, addressing an open problem from prior numerical studies.

Findings

Existence of an infinite family of wavefronts parameterized by wave speed.

Derived upper and lower bounds for the wave speed threshold.

Applied analytical tools like shooting method and fixed-point theory.

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 Fr\'echet spaces, to establish existence, and the central manifold theorem to ascertain uniqueness.