Hopping Conduction and Bacteria: Transport in Disordered Reaction-Diffusion Systems

We report some basic results regarding transport in disordered reaction-diffusion systems with birth (A→2A), death (A→0), and binary competition (2A→A) processes. We consider a model in which the growth process is only allowed to take place in certain areas—“oases”—while the rest of space—the “desert”—is hostile to growth. In the limit of low oasis density, transport is mediated through rare “hopping” events, necessitating the inclusion of discreteness effects in the model. By first considering transport between two oases, we are able to derive an approximate expression for the average time taken for a population to traverse a disordered medium.