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

We report some basic results regarding transport in disordered reaction-diffusion systems with birth ($A\to 2A$), death ($A\to 0$), and binary competition ($2A\to A$) 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.

Figure 1

Comparison of theoretical mean FPT with Monte Carlo data. The error bars represent a 95% confidence interval. The parameters used were $w=1.0$, $y=0.25$, $z=0.1$, and $b=0.001$. Time is measured in units of $1/w$.

Figure 2

Comparison of theory with Monte Carlo data. Each box represents the probability $P(t_i)$ (from the linear theory) that a particular run hits site $n=25$ for the first time in the $i$th time bin. The points with error bars are from 5000 Monte Carlo simulations of the full nonlinear stochastic model. Time is measured as in Fig. 1, and the bins have a width of 25 units of time.