Optimal dispersal in ecological dynamics with Allee effect in metapopulations

Abstract

We introduce a minimal agent-based model to understand the effects of the interplay between dispersal and geometric constraints in metapopulation dynamics under the Allee Effect. The model, which does not impose nonlinear birth and death rates, is studied both analytically and numerically. Our results indicate the existence of a survival-extinction boundary with monotonic behavior for weak spatial constraints and a nonmonotonic behavior for strong spatial constraints so that there is an optimal dispersal that maximizes the survival probability. Such optimal dispersal has empirical support from recent experiments with engineered bacteria.

Fig 1. Time series (in mcs) for the total number of of agents for D = {0.03, 0.05, 0.07, 0.09, 0.14, 0.19} with L = 10, N = 10⁴ L, n_s = 1 and k = 2.

Each color corresponds to one sample. The symbols were obtained from Monte Carlo simulations and the lines from Eqs (3) and (4).

Fig 2. Stationary density of agents a^∞ vs mortality rate α with D = 0.2, k = 2, L = 10, N = 10⁴ L, n_s = 1.

The symbols come from the Monte Carlo Simulations and the lines come from the numerical integration of Eqs (3) and (4).

Fig 3. Phase diagram α v D for n_s = 1, 2, …, 6 sources, L = 10, N = 10⁴ L and k = 2.

The point D = 0 is excluded from the diagram since it refers to isolated populations with threshold α_c = λ/4 = 0.25. In all the cases $n_0=\frac{{10}^4}{n_s}$, where n₀ is the initial subpopulation size. The lines are obtained from Eqs (3) and (4). At the bottom of each plot we show the optimal dispersal D* and the corresponding maximum allowed mortality rate α_max(D*) below which the population still stay in the survival phase.

Fig 4. Survival area in the phase diagram α × D versus the number of sources n_s for 0 < D < 1.

The case D = 1 is excluded because it implies no reproduction/death. The case D = 0 is excluded because it implies no migration between the patches. In all cases we keep the initial subpopulation size fixed $n_0=\frac{{10}^4}{n_s}$ and we use N = 10⁴ L.

Fig 5. Phase diagram α vs D for networks with increasing number of neighbors k = 2, 4, 6, 8 (decreasing spatial constraints).

The theoretical lines (red) comes from numerical integration of Eqs (3) and (4). The optimal dispersal and the corresponding maximum α are: (a) D* = 0.09 and α_max = 0.122; (b) D* = 0.092 and α_max = 0.089.

Fig 6. Regime diagram of the dependence between threshold mortality α_c vs dispersal rate D for L = 50.

The vertical line that separates the two regimes is k_threshold = 30. For k < k_threshold: α_max > α_{D = 0.5} then α_c × D displays a nonmonotonic dependence. For k ≥ k_threshold: α_max = α_{D = 0.5} then α_c × D exhibits a monotonic dependence.