The Role of Dispersal in Interacting Patches Subject to an Allee Effect

Abstract

This article is concerned with a stochastic multi-patch model in which each local population is subject to a strong Allee effect. The model is obtained by using the framework of interacting particle systems to extend a stochastic two-patch model that has been recently introduced by Kang and the author. The main objective is to understand the effect of the geometry of the network of interactions, which represents potential migrations between patches, on the long-term behavior of the metapopulation. In the limit as the number of patches tends to infinity, there is a critical value for the Allee threshold below which the metapopulation expands and above which the metapopulation goes extinct. Spatial simulations on large regular graphs suggest that this critical value strongly depends on the initial distribution when the degree of the network is large whereas the critical value does not depend on the initial distribution when the degree is small. Looking at the system starting with a single occupied patch on the complete graph and on the ring, we prove analytical results that support this conjecture. From an ecological perspective, these results indicate that, upon arrival of an alien species subject to a strong Allee effect to a new area, though dispersal is necessary for its expansion, strong long range dispersal drives the population toward extinction.

Survival and extinction results for a patch model with sexual reproduction

“…On the other hand, when b = 0, the projection of the influence set on Z reduces to a singleton therefore S t s (c) = 0 and (9) is trivial. Finally, we use that γ(θ) → ∞ as c → ∞ together with (9) to conclude that, for all γ > 0,…”

“…This birth mechanism (sexual reproduction) reflects the difficulty to locate mates in patches at low density, which has been identified by ecologists as one of the most common causes of Allee effect. In particular, while the Allee effect is forced into the model in the form of a threshold parameter in [3,8,9], it is on the contrary naturally induced by the birth mechanism in the model considered in this paper. A useful consequence of this fact is both our model and its mean-field equations have a dual process, moreover the stochastic model's dual process is a straightforward truncation of the mean-field's dual process.…”

“…The model is used to show that, when only small flocks of individuals can migrate from patch to patch, the metapopulation goes extinct whereas survival is possible if large enough flocks of individuals can migrate. The framework of interacting particle systems has also been used in [8,9] to study the consequence of an Allee effect. There, the model is a modification of the averaging process that also includes a threshold: local populations below this threshold go extinct whereas local populations above this threshold expand to their carrying capacity, each at rate one.…”

Management of invasive Allee species

“…If this threshold is high then a species may have difficulty persisting and expanding into new territory (Keitt et al, 2001;Lanchier, 2013), particularly if its movement is restricted (Ackleh, Allen, & Carter, 2007).…”

The timing of life history events in the presence of soft disturbances

“…For the simplest among this models, the branching random walk, much has been done: for instance in [4,5,7,32,45,47] one finds characterization of the persistence/disappearance of genotypes (seen as locations for the model), on general space structures; the same can be found, for some random graphs, in [8,35]. Stochastic modelling and interacting particle systems have been successfully applied to biology and ecology (see [1,2,3,12,13,24,25,26,28,19,48] just to mention a few). Although stochastic modelling is very interesting and complex, here we will assume that over many generations, our populations have been sufficiently large to justify the use of a model where stochasticity appears only in the random time at which the disturbance strikes.…”