Hopping transport in hostile reaction-diffusion systems

We investigate transport in a disordered reaction-diffusion model consisting of particles which are allowed to diffuse, compete with one another $(2A\to A)$, give birth in small areas called “oases” $(A\to 2A)$, and die in the “desert” outside the oases $(A\to 0)$. This model has previously been used to study bacterial populations in the laboratory and is related to a model of plankton populations in the oceans. We first consider the nature of transport between two oases: In the limit of high growth rate, this is effectively a first passage process, and we are able to determine the first passage time probability density function in the limit of large oasis separation. This result is then used along with the theory of hopping conduction in doped semiconductors to estimate the time taken by a population to cross a large system.

Figure 1

(Color online) Main window: Plot showing $P_{\text{none}}(x,t)$ in $d=1$. The lines represent, from left to right, the function for $x=16$, 18, 20, 22, and 24. Inset: A blowup showing the early-time behavior of $P_{\text{none}}$.

Figure 2

(Color online) Binned FPT probabilities for $\nu =30$ from both the linear model with a source (red boxes) and Monte Carlo simulations of the nonlinear model (blue lines). The width of each bin is $200∕w$, where $w$ is the total hopping rate. The error bars on the simulation data represent sampling error.

Figure 3

(Color online) Graph showing the mean FPT as a function of $\nu$ for the linear model with a source (red diamonds) and the KMC simulation of the full model with competition (blue marks) for the second parameter set. The error bars represent a 95% confidence level for the simulation mean FPTs.

Figure 4

(Color online) Graph showing the mean FPT as a function of $\nu$ for the linear model with a source (red diamonds) and the KMC simulation of the full model with competition (blue marks) for the second parameter set with $g$ chosen to provide the best fit between the data sets.

Figure 5

(Color online) First passage times across a large system shown for seven different combinations of $\kappa$ and $R_{\max }$. Error bars are not shown since they are, in most cases, smaller than the symbol size. The lines represent best-fit lines for each $\kappa$, $R_{\max }$. The two lines with $\kappa R_{\max }=12.0$ lie nearly on top of one another, as one would expect; we have omitted every other data point for each of these runs for clarity. Note that the value of the slope [which is equal to $F\left(\kappa R_{\max }\right)$] increases as $\kappa R_{\max }$ increases.

Figure 6

(Color online) Cutoff growth rate $y_c$ as a function of death rate $z$ with $a=3.0$, $D=0.5$. Here ${\alpha }_{01}$ is the first zero of $J_0$. Note that in one and two dimensions, an arbitrarily small growth rate with $z=0$ will allow a stable population to take hold; in three dimensions, $y_c(z=0)={\pi }^{2}D∕4a^2$.

Figure 7

(Color online) Schematic of the contour integral which must be done to find $Y(R,a,t)$. The dashed lines represent contributions to the integral that vanish as they are moved further from the origin.