Expansion or extinction: deterministic and stochastic two-patch models with Allee effects

Abstract

We investigate the impact of Allee effect and dispersal on the long-term evolution of a population in a patchy environment. Our main focus is on whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction. Our study is based on the combination of analytical and numerical results for both a deterministic two-patch model and a stochastic counterpart. The deterministic model has either two, three or four attractors. The existence of a regime with exactly three attractors only appears when patches have distinct Allee thresholds. In the presence of weak dispersal, the analysis of the deterministic model shows that a high-density and a low-density populations can coexist at equilibrium in nearby patches, whereas the analysis of the stochastic model indicates that this equilibrium is metastable, thus leading after a large random time to either a global expansion or a global extinction. Up to some critical dispersal, increasing the intensity of the interactions leads to an increase of both the basin of attraction of the global extinction and the basin of attraction of the global expansion. Above this threshold, for both the deterministic and the stochastic models, the patches tend to synchronize as the intensity of the dispersal increases. This results in either a global expansion or a global extinction. For the deterministic model, there are only two attractors, while the stochastic model no longer exhibits a metastable behavior. In the presence of strong dispersal, the limiting behavior is entirely determined by the value of the Allee thresholds as the global population size in the deterministic and the stochastic models evolves as dictated by their single-patch counterparts. For all values of the dispersal parameter, Allee effects promote global extinction in terms of an expansion of the basin of attraction of the extinction equilibrium for the deterministic model and an increase of the probability of extinction for the stochastic model.

Population dispersal and Allee effect

“…Kang and Lanchier [14] extend model (2.1) by incorporating the symmetric density-independent dispersals between two patches and analyze attractors of the model for the case of the strong Allee effect. The first objective of this paper is to consider the general case of population dispersals among n patches:…”

“…For example, the baseline dynamics of population growth are governed by the Logistic equation. However, there have been abundant evidences that Allee effect plays an important role in diverse bio- [1,5,[7][8][9][10][14][15][16]18] and the references cited therein). Allee effect, defined as positive effects in the growth rate at low population density, may be strong or weak.…”

“…The dispersal rule in [1] is density-dependent and assumes that all dispersers enter a common pool and are then redistributed between the two patches. Kang and Lanchier [14] considered whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction by using a two-patch model with strong Allee effect. The model in [14] adopts a symmetric density-independent dispersal rate.…”

“…Kang and Lanchier [14] considered whether a population already established in one patch either successfully invades an adjacent empty patch or undergoes a global extinction by using a two-patch model with strong Allee effect. The model in [14] adopts a symmetric density-independent dispersal rate. The purpose of the present paper is twofold.…”

A seasonal SIR metapopulation model with an Allee effect with application to controlling plague in prairie dog colonies

“…If a disease incurs a higher death rate or lower individual fitness compared to a disease-free population, then the disease reduces patch population densities. The Allee effect and seasonal birth rates together may then drive such patches to extinction and population dispersal may govern the recolonization [23,29,30,47].…”