Date Published: June 20, 2019
Publisher: Public Library of Science
Author(s): Marcelo A. Pires, Sílvio M. Duarte Queirós, Sergio Gómez.
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.
The Allee effect is an influential finding named after the ecologist Warder Clyde Allee  concerning a phenomenon typically manifested by the departure from the standard logistic growth that enhances the susceptibility to extinction of an already vulnerable sparse population. Curiously, W. C. Allee did not provide a definition of the effect , but in general terms it can be defined as “the positive correlation between the absolute average individual fitness in a population and its size over some finite interval”. . The strong Allee effect, which is the focus of this work, corresponds to the case when the deviation from the logistic growth includes an initial population threshold below which the population goes extinct . On the other hand, there exists a weak version of the Allee effect which treats positive relations between the overall individual fitness in the population density and does not present population size nor density thresholds.
Consider a metapopulation [27, 28] with L subpopulations composed of agents that are able to move, die or reproduce. As usual in metapopulation dynamics , we assume a well-mixed subpopulation, i.e., inside each subpopulation all individuals have the possibility to interact one another. In Statistical Physics parlance that is to say that our local dynamics exhibits a mean-field character. The mobility is implemented as a random walk between neighbor subpopulations such that migration occurs at each time step with probability D. At a given time step, if migration does not take place (with probability 1 − D) then one of the two events is chosen : death of an agent with probability α or reproduction with probability λ when two agents meet. Mating limitation is an important source of the Allee effect [25, 29] which in our model is incorporated in the reproduction event.
In this section, we present the results for metapopulations of sizes 10 ≤ L ≤ 50 and increasing k. By means of Monte Carlo simulation and bearing in mind Eqs (1) and (2) we confirmed that the results are still valid for larger networks that correspond to the limit of a macroscopic system (the so-called thermodynamic limit). For the sake of simplicity and without loss of generality the results we show are for λ = 1.
In this work, we have investigated the spectrum of scenarios arising from a metapopulation dynamics under the Allee Effect using a minimal agent-based model which points at describing fundamental mechanisms thereof. Employing numerical and analytical tools we have shown that the survival-extinction boundary undergoes a monotonicity transition: it has a nonmonotonic behavior marked by an optimal dispersal for severe spatial constraints, but a monotonic behavior for loose spatial constraints. The verification of this qualitative change in the dependence of the mortality threshold as a function of the dispersal highlights the importance of the triangular interplay between the Allee Effect, dispersal and geometric constraints for the persistence of populations.