On the role of Allee effect and mass migration in survival and extinction of a species

Abstract

We use interacting particle systems to investigate survival and extinction of a species with colonies located on each site of Z d . In each of the four models studied, an individual in a local population can reproduce, die or migrate to neighboring sites.We prove that an increase of the death rate when the local population density is small (the Allee effect) may be critical for survival, and that the migration of large flocks of individuals is a possible solution to avoid extinction when the Allee effect is strong. We use attractiveness and comparison with oriented percolation, either to prove the extinction of the species, or to construct nontrivial invariant measures for each model.

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

“…The stochastic two-patch model we introduced and studied in [6] is simply the process (1) when the network of interactions consists of two vertices connected by a single edge. This result suggests that 1 2 is a critical value for the Allee threshold. both patches in state 0 eventually, or to a global expansion, i.e.…”

“…We refer the reader to [2] for a general overview of the literature on the Allee effect and also mention more particularly the recent work of Borrello [1], who, similarly to this paper, focused on a stochastic process based on the framework of interacting particle systems. If, in addition, the growth rate becomes negative at low density, this monotone relationship results in the existence of a so-called Allee threshold below which populations are at high risk of being driven toward extinction, a phenomenon which is referred to as the strong Allee effect and may be due to various ecological factors: failure to locate mates, inbreeding depression, failure to satiate predators, lack of cooperative feeding, etc.…”

“…In particular, our model includes the same two components as the model introduced in [1]: an Allee threshold and migrations between patches. Each vertex becomes active at rate 1, which results in the population density at that vertex to jump to state 0 if it is below the Allee threshold and to jump to 1 if it is above the Allee threshold.…”

Survival and extinction results for a patch model with sexual reproduction

“…In particular, letting v(s) be the solution to (3) with v x (s) = v(s−(t−t 0 )) for all x, using Lemma 12 and monotonicity, we conclude that…”

“…• Given u(0), the restriction K u x (t) is defined to be the solution to (3) where only the values u x (t) with |x| ≤ k are determined by (3), and using the given boundary values u y (t) if y ∼ x and |y| > K.…”