Extinction dynamics of a discrete population in an oasis

Understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology. The first theoretical approach to the problem dates back to the 1950s with the Kierstead, Slobodkin, and Skellam (KiSS) model, namely a continuous reaction-diffusion equation for a population growing on a patch of finite size $L$ surrounded by a deadly environment with infinite mortality, i.e., an oasis in a desert. The main outcome of the model is that only patches above a critical size allow for population persistence. Here we introduce an individual-based analog of the KiSS model to investigate the effects of discreteness and demographic stochasticity. In particular, we study the average time to extinction both above and below the critical patch size of the continuous model and investigate the quasistationary distribution of the number of individuals for patch sizes above the critical threshold.

Figure 1

Density field $\theta \left(x\right)$ at stationarity for the continuous KiSS model (black solid curve) compared with the local density of $B$ particles, averaged over $n_r=10$ realizations, in the discrete model (empty circles). The main parameters are as follows: $N={10}^3,R=0.1,L=6$. The (blue) dashed curve represents the local density of $B$ particles obtained with a different model with fixed $A$ particles (see the Appendix for details).

Figure 2

Survival probability and biomass as a function of time for two values of the patch size above the critical value $L_c=\pi$: $L=4$ (a) and $L=3.15$ (b), with $N=400$ particles. Top panels show the survival probability $P_s\left(t\right)$, see text. Bottom panels show the time evolution of the continuous $\left[{\mathcal{B}}_C\left(t\right)\right]$ and discrete $\left[\mathcal{B}\right(t\left)\right]$ biomasses, the error bars on the latter display the standard deviation, ${\sigma }_{\mathcal{B}}\left(t\right)$. The (blue) dotted curves denote the minimum and maximum values of $N_B\left(t\right)/N$ observed in all $n_r=1000$ realizations. The inset in (b) zooms the framed area in the main figure to better show that $\mathcal{B}\left(t\right)>{\mathcal{B}}_C^{\infty }$ at long times.

Figure 3

Limiting biomass $B_C^{\infty }$ and average biomass in the quasistationary state, ${\mathcal{B}}_{\mathrm{QS}}$, as a function of $L-L_c$ for different values of $N$ as in the legend. The solid curve displays $0.75{\lambda }_1$ that corresponds to the behavior in Eq. (11). The factor $0.75$ cannot be caught by the argument exposed in the main text. Inset: Standard deviation of the biomass (10) in the quasistationary regime, ${\sigma }_{{\mathcal{B}}_{\mathrm{QS}}}$, as a function of $N$ for two values $(L=3.5$ and $L=7)$ of the patch size. The number of realizations used for the averages ranges from 500 to $10000$ depending on the value of $N$.

Figure 4

Quasistationary distribution for $L-L_C=0.3$ and three values of $N$ as in legend. The solid curves display the Gaussian distribution obtained using the mean and standard deviation values measured from data. We observe a fairly good agreement at least for $N$ large enough. When $L\to L_C$ and/or $N$ is not large enough the agreement ceases to be so good, as one can observe from the left tail at $N=500$.

Figure 5

Mean extinction time, $T_e$, vs population size, $N$, for different patch sizes $L=L_c+\delta L$. Panel (a) from bottom to top $\delta L=-0.3,-0.1,-0.02,0,0.02,0.1,0.3$, encompassing the extinction (empty red symbols) and persistence (blue filled symbols) regions, as well as the critical point (black half-filled symbols) of the continuous model. The continuous red lines display the prediction (12) with $b^{\prime }$ fitted from data; the dashed blue lines correspond to the exponential behavior (14) with $a$ fitted. For $L=L_c$ we found a power-law behavior, $T_e\sim N^{\gamma }$, with best fitting exponent $\gamma =0.565$ (see text). Panels (b) and (c) display a zoom of the main figure, in appropriate semilog scale, of both the logarithmic (12) and the exponential (14) regime, respectively. We emphasize that the slope in (b) is not fitted but obtained from the first eigenvalue. Depending on the value of $N$ the number of realizations ranges from 500 to 5000.

Figure 6

Mean extinction time $T_e$ as a function of the patch size $L$ for $N=40$ (red circles), $N=100$ (orange triangles), and $N=1000$ (blue squares). In the inset $T_e/logN$ versus $L$ is compared to the asymptotic prediction (15) (solid black curve). The vertical dashed line marks the critical value $L_c=\pi$. The number of realizations used for the averages ranges from 500 to 5000 depending on the value of $N$.

Figure 7

Average extinction time vs $N$ for $L=L_c$ at varying $R$ as from the legend and considering nonmotile $A$ particles. The power-law behavior $T_e\sim N^{0.565}$, identifying the extinction-survival transition, is robust for variations of the interaction distance around the reference value $R=0.1$; for much larger values $\left(R=\pi /4\right)$ the critical behavior is considerably altered. The same happens if one considers a model in which particles of type $A$ do not move (filled diamonds).