Expansion of cooperatively growing populations: Optimal migration rates and habitat network structures

Range expansion of species is driven by the interactions among individual- and population-level processes and the spatial pattern of habitats. In this work we study how cooperatively growing populations spread on networks representing the skeleton of complex landscapes. By separating the slow and fast variables of the expansion process, we are able to give analytical predictions for the critical conditions that divide the dynamic behaviors into different phases (extinction, localized survival, and global expansion). We observe a resonance phenomenon in how the critical condition depends on the expansion rate, indicating the existence of an optimal strategy for global expansion. We derive the conditions for such optimal migration in locally treelike graphs and numerically study other structured networks. Our results highlight the importance of both the underlying interaction pattern and migration rate of the expanding populations for range expansion. We also discuss potential applications of the results to biological control and conservation.

Figure 1

Phase diagram of the expansion dynamics with different $c$ and $\mu$. The symbols represent numerical solutions and stochastic simulations on random regular networks with $N=300$ and $k=6$ by averaging over 1000 independent realizations. The carrying capacity $C$ was set to 100 for computational efficiency. The dashed and solid lines are the solutions of Eqs. (7) and (9), respectively, which divide the panel into three different regions: I, extinction $U\left(\infty \right)=0$; II, localized survival $U\left(\infty \right)\approx 1$; and III, global expansion $U\left(\infty \right)=N$, where $U\left(t\right)={\sum }_{i=1}^N{u}_i\left(t\right)$.

Figure 2

Plot of $\langle U\left(\infty \right)\rangle$ as a function of the Allee threshold $c$. Illustrated data are from stochastic simulations on random regular networks with $N=300$, $k=6$, $\mu =0.05$, and different carrying capacities $C$.

Figure 3

Numerical calculations of $\langle U\left(\infty \right)\rangle$ for the range expansion on (a) Erdős-Rényi random networks with $N=800$ and $\langle k\rangle =6$ and (b) two-dimensional Delaunay triangulations [28, 29] with $N=800$. The data points are averaged over 3000 realizations. The dashed lines are the solutions of Eq. (9) with $k=6$, symbolizing the $c^{*}$ values on random regular networks (shown for comparison).

Figure 4

Probability $p$ of global expansion of the population as a function of the Allee threshold $c$ on Erdős-Rényi random networks with $N=1000$, $\langle k\rangle =8$, and $\mu =0.01$. The lines were obtained in terms of Eqs. (11) and (12).

Figure 5

Plot of $\langle U\left(\infty \right)\rangle$ as a function of the Allee threshold $c$. Shown data are the numerical solutions and stochastic simulations of our model on Erdős-Rényi random networks with $N=400$, $\langle k\rangle =6$, and different values of $c$ and $\mu$. Note that $\mu =0.138$ is the optimal migration rate calculated from ${\mu }_{\text{max}}=1/k-1/k^2$.

Figure 6

Critical Allee threshold $c^{*}$ versus $1/N$ for different $\mu$ on random regular networks. The line represents $c=1/N$. The symbols are the results from the numerical solutions and stochastic simulations. All the numerical results were obtained on random regular networks with $k=3$, averaging over 2000 independent realizations. Here $C=100$ and $\mu =10$ are used in the stochastic simulation.

Figure 7

Numerical solutions and stochastic simulations (symbols) versus estimations of Eq. (16) (lines) of the range expansion on Erdős-Rényi networks. (a) Critical threshold $c^{*}$ versus network size $N$ with $\mu =10$, $\lambda =4$, and $C=2000$. (b) Critical threshold $c^{*}$ versus average degree $\lambda$ with $\mu =10$, $N=300$, and $C=2000$.

Figure 8

Allee threshold $c^{*}$ as a function of $\mu$. The numerical solutions (symbols) are obtained on random regular networks with $N=300$ and different $k$. The thick lines are the solutions of Eq. (9).

Figure 9

Plot of (a) $c_{\text{max}}^{*}$ and (b) ${\mu }_{\text{max}}$ versus degree $k$. The circles represent the results from numerical solutions of Eq. (10). The squares are drawn from numerical solutions of Eq. (1) on random regular networks with $N=300$.

Figure 10

(a) Schematic illustration of the seed node's dynamics $f_s\left(u\right)$ with the same $c$ and different $\mu$ values. (b) Demonstrations of how the stable (solid lines) and unstable (dashed line) fixed points of $f_s\left(u\right)$ change with varying $\mu$. (c) Schematic illustration of the first-level nodes' dynamics $f_v\left(u\right)$ with different $S$. (d) Demonstrations of how the stable (solid lines) and unstable (dashed line) fixed points of $f_v\left(u\right)$ change with varying $\mu$. Note that $S=\mu /k$ here.

Figure 11

Numerical solutions of Eq. (10) for different $k$.

Figure 12

Illustrations of $f_r\left(u\right)$, $f_d\left(u\right)$, and $f_v\left(u\right)$. (a) The stable fixed point $u=1/k$ [with regard to $f_d\left(u\right)$] is smaller than $c$, so the global expansion is impossible. (b) The opposite of (a). (c) When $c=1/k$, an inappropriate $\mu$ makes the global expansion impossible. (d) The exact $\mu$ value for $f_v^{\prime }(1/k)=0$ when $c=1/k$ so that the global expansion can be realized.