Regime switching on the propagation speed of travelling waves of some size-structured myxobacteria population models

The spatial propagation of complex populations can depend on some structuring variables. In particular, recent developments in microscopy have revealed the impact of bacteria heterogeneity on the population motility. Biofilms of Myxococcus xanthus bacteria have been shown to be structured in clusters of various sizes, which remarkably, tend to move faster when they consist of a larger number of bacteria. We propose a minimal reaction-diffusion discrete-size structure model of a population of Myxococcus with two possible cluster sizes: isolated and paired bacteria. Numerical experiments show that this model exhibits travelling waves whose propagation speed depends on the increased motility of clusters, and the exchange rates between isolated bacteria and clusters. Notably, we present evidence of the existence of a characteristic threshold level θ*. on the ratio between cluster motility and isolated bacteria motility, which separates two distinct regimes of propagation speed. When the ratio is less or equal than θ*, the propagation speed of the population is constant with respect to the ratio. However, when the ratio is above θ*, the propagation speed increases. We also consider a generalised model with continuous-size structure, which also shows the same behaviour. We extend the model to include interactions with a resource population, which show qualitative behaviours in agreement to the biological experiments.