Population dynamics in an intermittent refuge

Population dynamics is constrained by the environment, which needs to obey certain conditions to support population growth. We consider a standard model for the evolution of a single species population density, which includes reproduction, competition for resources, and spatial spreading, while subject to an external harmful effect. The habitat is spatially heterogeneous, there existing a refuge where the population can be protected. Temporal variability is introduced by the intermittent character of the refuge. This scenario can apply to a wide range of situations, from a laboratory setting where bacteria can be protected by a blinking mask from ultraviolet radiation, to large-scale ecosystems, like a marine reserve where there can be seasonal fishing prohibitions. Using analytical and numerical tools, we investigate the asymptotic behavior of the total population as a function of the size and characteristic time scales of the refuge. We obtain expressions for the minimal size required for population survival, in the slow and fast time scale limits.

Figure 1

Pictorial representation of a one-dimensional habitat subject to an external harmful effect (downwards arrows) with a refuge (thick segment) of size $L$ in the inactive (a) and active (b) states. In the active state, the refuge is able to block the harmful effect.

Figure 2

Asymptotic population distribution when the refuge is static, for two different refuge sizes $L$ indicated. In this case, $L_c\simeq 0.73$, for our choice of the parameter values defined in Sec. 2. The gray vertical lines indicate the boundaries of the refuge for each refuge size.

Figure 3

Temporal evolution of the total population size $N$, for different values of the period $\tau$, fixed average rate $\lambda =0.1$, and size $L=1.28$ (for the values of the parameters used, $L_c=1.295$). The dotted and dashed lines represent the slow and fast limit approximation (for details, see Sec. 3a).

Figure 4

Average population growth rate $\langle \mathrm{\Phi }\rangle$, computed over entire cycles, as a function of protocol period $\tau$ for $L=1.28$ and $\lambda =0.1$. The dashed and dotted lines represent the rates at the slow and fast limits (for details, see Sec. 3a).

Figure 5

Temporal evolution of the growth rate $\mathrm{\Phi }$, for different time scales $\tau$, with $L=1.28$ and $\lambda =0.1$. The dotted line denotes the maximal value ${\mathrm{\Phi }}_0$.

Figure 6

Population growth rate ${\mathrm{\Phi }}_0$ vs refuge size $L$ (black circles). The solid line represents the ansatz given by Eq. (3.3), and the dotted line represents the growth rate in the limit case of harsh conditions outside the refuge, explicitly given by Eq. (B1), and the dashed line the linear approximation given by Eq. (A3). In the inset, we show that the fitting parameter $L^{*}$ in the ansatz (3.3) follows Eq. (2.3).

Figure 7

Theoretical predictions for the critical size $L_c$ in the slow and fast protocol limits, given by Eqs. (3.4) and (3.2), respectively, together with numerical data for different $\tau$. The striped region (black) between the curves represents the variability of $L_c$ with $\tau$. The inset shows $L_c$ vs $\tau$ for $\lambda =0.16$ and the slow limit approximation (blue dotted line).

Figure 8

Velocity $V(x=L/2)$ vs population density $u(x=L/2)$ at the refuge boundary, for different values of $\tau$ indicated, with $\lambda =0.1$ and refuge size $L=2$ (hence, $L>L_c$). The single dot represents the limiting case $\tau \to 0$.

Figure 9

Temporal evolution of the total population $N$, the scaled flux $J/D$, the populations inside $N_{\text{in}}$ and outside $N_{\text{out}}$ the refuge (after resetting the population inside the refuge at $t=50$), and their difference. In this case, the refuge size is $L=2>L_c$. After the transient, the population achieves the recovery state, growing with rate ${\mathrm{\Phi }}_0$.