A reaction-diffusion system with periodic front dynamics

A reaction-diffusion model motivated by Proteus mirabilis swarm colony deve lopment is presented and analyzed in this work. The principal variables are the concentrations of swarm cells and swimmer cells, which are multicellul ar and single-cell forms, respectively, of the Proteus mirabilis bacteria. The kinetic terms model the growth and division process of the swimmer cell s, as well as the formation and septation of swarm cells. In addition, a no nlinear diffusion is employed for the swarm cell migration that incorporate s several essential aspects of the concentration-dependent dynamics observe d in experiments. The model exhibits time-periodic colony evolution in which each period cons ists of two distinct phases: a swarming phase and a consolidation phase. Bo th 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 b eyond the colony's initial boundary. Gradually, this expansion slows down a nd stops, and then the consolidation phase begins, during which time the co lony boundary stays in place, but the swarm and swimmer concentrations insi de the colony boundary change significantly. Finally, when the swarm cell c oncentration 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 (g radual and abrupt) between them, using the method of matched asymptotic exp ansions and theory for parabolic partial differential equations. We show th at the dynamics of the diffusivity play a central role in determining the c olony 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 t oward the colony boundary. It grows sharply in amplitude and forms a spike when it reaches the boundary. Moreover, it is precisely this event that tri ggers swarming. Analysis of the diffusivity dynamics also leads to an under standing of the swarming phase dynamics and the gradual transition to the c onsolidation phase that follows it. These analyses show that the concentrat ions at the beginning of the two phases naturally repeat in a time-periodic manner. Finally, we present rigorous estimates for the inner and outer sol utions developed in the matched asymptotic analysis, and for their domains of validity.