Population dynamics in spatially heterogeneous systems with drift: The generalized contact process

We investigate the time evolution and stationary states of a stochastic, spatially discrete, population model (contact process) with spatial heterogeneity and imposed drift (wind) in one and two dimensions. We consider in particular a situation in which space is divided into two regions: an oasis and a desert (low and high death rates). Carrying out computer simulations we find that the population in the (quasi) stationary state will be zero, localized, or delocalized, depending on the values of the drift and other parameters. The phase diagram is similar to that obtained by Nelson and coworkers from a deterministic, spatially continuous model of a bacterial population undergoing convection in a heterogeneous medium.

Figure 1

Critical values ${\lambda }_c\left(P_K\right)$ as a function of the jump probability $P_K$ for $q=0.5$. Results are extrapolated to $N=\infty$ (open circles) from system sizes of $N=50$, 100, 200, 300, 500. The average is taken over ${10}^3\text{–}{10}^4$ different realizations starting with random initial configurations at density.

Figure 2

(Color online) Critical value of ${\lambda }_C^N\left(q\right)$ for a homogeneous system as a function of drift $q$ in one dimension. $P_K=0.9$ and system sizes of $N=100$, 200, 250, 500 are used. The function that is used for the solid line is $1-\gamma (q-0.5)$ with $\gamma =0.13$.

Figure 3

Ratio of the critical values in the heterogeneous system to those in the homogeneous system, ${\lambda }_c\left(ϵ\right)∕{\lambda }_c\left(0\right)$. The parameters used for the simulation are $P_K=0$ and the results are extrapolated to $N=\infty$ (open symbols) from $N=50$, 100, 200, 500, 1000. The average is taken over ${10}^3-{10}^4$ different realizations. The function that is used for the solid line is $1-ϵ$.

Figure 4

Stationary density $\overline{\rho }\left(x\right)$ on an one dimensional ring of size $N=2000$ for $P_K=0$ and $\lambda =4$. ${\alpha }_I\left(x\right)$ and ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$ are used, respectively.

Figure 5

$\lambda \text{−}q$ phase diagram of the contact process with drift and spatial heterogeneity, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, in one dimension. Here $P_K=0.9$. The critical curve (solid line with filled circles), obtained by extrapolation to the infinite system from $N=50$, 100, 200, 500 (shown in the inset), separates the survival (delocalized or localized) from extinct phase. LP stands for the localized phase.

Figure 6

Drift-driven transition from the localized phase to the extinct phase. The parameters used for the simulation are $N=2000$, $\lambda =1.5$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, and $P_K=0.9$. Drift varies from $q=0.5$ where population is localized in the vicinity of the oasis to $q=0.598$ where the population becomes extinct.

Figure 7

Transition from the delocalized survival phase to the extinct phase. The parameters used for the simulation are system size of $N=2000$, $q=0.9$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, and $P_K=0.9$. Birth rate varies from $\lambda =2.4$ where population is extended everywhere to $\lambda =1.9$ where the population becomes extinct.

Figure 8

$\lambda \text{−}q$ phase diagram of the contact process with drift and spatial heterogeneity in two dimension. The parameters used for the simulation are $P_K=0.9$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$. The critical curve (solid line with filled circles) is obtained by extrapolation to the infinite system from $N^2={30}^2,{50}^2,{70}^2,{100}^2$ (shown in the inset). The oasis is a square of length $N∕5$. LP stands for the localized phase.

Figure 9

(Color online) Localized phase in two dimension for $\lambda =1.05$, $q=0.5$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, $P_K=0.9$, and $N^2={100}^2$.

Figure 10

(Color online) Delocalized phase in two dimension for $\lambda =1.75$, $q=0.7$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, $P_K=0.9$, and $N^2={100}^2$.

Figure 11

(Color online) Time evolution of population density ${\rho }_t\left(x\right)$ for $\lambda =1.5$, $q=0.59$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, $P_K=0.9$, and $N=2000$, i.e., for a weak drift in the localized phase. ${10}^4$ realizations starting with 1% of randomly distributed population are averaged at given Monte Carlo time steps.

Figure 12

(Color online) Time evolution of population density ${\rho }_t\left(x\right)$ for $\lambda =2.5$, $q=0.59$, ${\alpha }_{II}\left(x\right)$ with $ϵ=0.5$, $P_K=0.9$, and $N=2000$, i.e., for a weak drift in the delocalized phase. ${10}^4$ realizations starting with 1% of randomly distributed population are averaged at given Monte Carlo time steps.