Travelling waves for a fast reaction limit of a discrete coagulation–fragmentation model with diffusion and proliferation

We study traveling wave solutions for a reaction–diffusion model, introduced in the article Calvez et al. (Regime switching on the propagation speed of travelling waves of some size-structured myxobacteriapopulation models, 2023), describing the spread of the social bacterium Myxococcus xanthus. This model describes the spatial dynamics of two different cluster sizes: isolated bacteria and paired bacteria. Two isolated bacteria can coagulate to form a cluster of two bacteria and conversely, a pair of bacteria can fragment into two isolated bacteria. Coagulation and fragmentation are assumed to occur at a certain rate denoted by k. In this article we study theoretically the limit of fast coagulation fragmentation corresponding mathematically to the limit when the value of the parameter k tends to \(+ \infty \) $+\infty$ . For this regime, we demonstrate the existence and uniqueness of a transition between pulled and pushed fronts for a certain critical ratio \(\theta ^\star \) ${\theta }^{\star }$ between the diffusion coefficient of isolated bacteria and the diffusion coefficient of paired bacteria. When the ratio is below \(\theta ^\star \) ${\theta }^{\star }$ , the critical front speed is constant and corresponds to the linear speed. Conversely, when the ratio is above the critical threshold, the critical spreading speed becomes strictly greater than the linear speed.