Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure

We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average precolonization population size is small, thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate nonzero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasistationary probability distribution of population sizes in the precolonization state and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.

Figure 1

Solid line: a typical Monte Carlo realization of the random birth-death process $n\left(t\right)$ with the birth and death rates given by Eq. (3). The parameters are $B=2$, $r=2$, $N=30$, and $K=5000$. The initial condition is $n=2$. The dashed line shows the fixed point $n=n_3$ from Eq. (4). Inset: Blowup of $n\left(t\right)$ in the precolonization state. The solid line depicts the simulation, the dashed line shows the fixed point $n=n_1$ from Eq. (4).

Figure 2

The QSD ${\pi }_n$ as a function of $n$ for $B=2$, $r=2$, and $N=50$ in a log scale. The QSD includes two overlapping asymptotics: the bulk asymptotic, given by Eq. (10) with ${\pi }_0=e^{-r}$ (solid line), and the boundary layer asymptotic, given by Eq. (14) with the flux $J_c$ from Eq. (17) (dashed line).

Figure 3

The ratio of the MTCs as a function of $r$ for $B=2$ and $N=50$ in a log-log scale. Dashed line: $\tau /{\tau }^{\mathrm{exact}}$, where $\tau$ is given by Eq. (22) and ${\tau }^{\mathrm{exact}}$ is given by Eq. (26). Solid line: $\tau /{\tau }^{\mathrm{WKB}}$, where ${\tau }^{\mathrm{WKB}}$ is given by the asymptotic limit $r\ll N$ of Eq. (23) of Ref. [20]. Dashed-dotted line: Our theoretical prediction of $\tau /{\tau }^{\mathrm{WKB}}=1/\left(2\pi \gamma \right)$ in the limit of $\gamma \ll 1$, where ${\tau |}_{\gamma \ll 1}$ is given by Eq. (24a) and ${\tau }^{\mathrm{WKB}}{|}_{\gamma \ll 1}$ is the $\gamma \ll 1$ asymptotic of Eq. (23) of Ref. [20].

Figure 4

The MTC as a function of $N$ in a log scale for $B=5$ and $r=1$, $0.1$, $0.01$, and $0.001$ in (a), (b), (c), and (d), respectively. Solid line: $\tau$ from Eq. (22). Dashed line: ${\tau }^{\mathrm{exact}}$ from Eq. (26).