A reaction-diffusion system with periodic front dynamics

A reaction-diffusion model motivated by Proteus mirabilis swarm colony development is presented and analyzed in this work. The principal variables are the concentrations of swarm cells and swimmer cells, which are multicellular and single-cell forms, respectively, of the Proteus mirabilis bacteria. The kinetic terms model the growth and division process of the swimmer cells, as well as the formation and septation of swarm cells. In addition, a nonlinear diffusion is employed for the swarm cell migration that incorporates several essential aspects of the concentration-dependent dynamics observed in experiments. The model exhibits time-periodic colony evolution in which each period consists of two distinct phases: a swarming phase and a consolidation phase. Both are very similar to those seen in experiments. During the swarming phase, the colony expands as swarm cells migrate outward to form a new terrace beyond the colony's initial boundary. Gradually, this expansion slows down and stops, and then the consolidation phase begins, during which time the colony boundary stays in place, but the swarm and swimmer concentrations inside the colony boundary change significantly. Finally, when the swarm cell concentration has reached a threshold, the consolidation phase ends abruptly and the next swarming phase begins, repeating the cycle. We analyze both of these phases, as well as the transitions and switches (gradual and abrupt) between them, using the method of matched asymptotic expansions and theory for parabolic partial differential equations. We show that the dynamics of the diffusivity play a central role in determining the colony evolution during the consolidation phase and in the occurrence of an abrupt transition to the subsequent swarming phase. In particular, we show that the diffusivity pro le forms a wave that propagates behind the front toward the colony boundary. It grows sharply in amplitude and forms a spike when it reaches the boundary. Moreover, it is precisely this event that triggers swarming. Analysis of the diffusivity dynamics also leads to an understanding of the swarming phase dynamics and the gradual transition to the consolidation phase that follows it. These analyses show that the concentrations at the beginning of the two phases naturally repeat in a time-periodic manner. Finally, we present rigorous estimates for the inner and outer solutions developed in the matched asymptotic analysis, and for their domains of validity.