Changes in the Gradient Percolation Transition Caused by an Allee Effect

The establishment and spreading of biological populations depends crucially on population growth at low densities. The Allee effect is a problem in those populations where the per capita growth rate at low densities is reduced. We examine stochastic spatial models in which the reproduction rate changes across a gradient $g$ so that the population undergoes a 2D-percolation transition. Without the Allee effect, the transition is continuous and the width $w$ of the hull scales as in conventional (i.e., uncorrelated) gradient percolation, $w\propto g^{-0.57}$. However, with a strong Allee effect the transition is first order and $w\propto g^{-0.26}$.

Figure 1

(a) Sites in the spatial models are placed in the centers of the hexagons in a honeycomb lattice. Gray cells represent populated ($A$), white cells vacant ($\varnothing$) sites. In this example, the focal site in the center has $k=4$ populated neighbors. If this site is populated, it will die during a small time interval $dt$ with probability equal to $dt$. If, on the other hand, this site is vacant, it will become populated with probability $(k/6)b(x_f)dt$ in the GCP or $[\frac{1}{2}k(k-1)/15]b(x_f)dt$ in the GAP, where $b(x_f)$ is the local birth attempt rate. (b) Typical snapshot of the GCP, (c) of the GAP. Dark gray: The largest populated cluster. Light gray: All other populated sites. Black curve: Percolation hull. The mean hull position $\overline{x}$ and the width of the fluctuations $w$ are indicated at the bottom.

Figure 2

(a) To distinguish between a continuous and a first-order transition, we investigate squares of dimension $2w\times 2w$ centered at $\overline{x}$. We measure the fraction of sites $\rho$ that belong to the cluster covering the largest area within the square. (Typically, though not necessarily, this is the largest cluster on the entire lattice, shown in dark gray). (b) From data of several independent runs, we obtain the probability distribution for $\rho$, represented as a histogram. We show data for $g=5\times {10}^{-5}$. The smaller the gradient, the more weight is concentrated in the two peaks of the GAP distribution. (c) The distribution of cluster sizes $s$ in the stripe $|x-\overline{x}|<w$. We include clusters which are at least partially in this stripe, but exclude the system’s largest cluster. The dashed line $\propto s^{-\tau }$ is the tangent to the GCP distribution. The inset shows the data collapse for the GCP. The exponents are those of standard 2D percolation: $\tau =\frac{187}{91}$, $\sigma =\frac{36}{91}$, $\nu =\frac{4}{3}$ [16].

Figure 3

The hull width $w$ as a function of the gradient $g$. The lines are least-squares fits to the data. Error bars are smaller than the symbol sizes.

Figure 4

The probability $p_{\mathrm{lc}}$ that a site belongs to the largest cluster as a function of the site’s birth attempt rate $b$ in (a) the GCP, (b) the GAP for various gradients $g$. Insets: The functions collapse if the coordinates are rescaled. In both insets, the same nine data sets are shown as in the main panels.