Hierarchical competition models with the Allee effect II: the case of immigration

Abstract

This is part II of an earlier paper that dealt with hierarchical models with the Allee effect but with no immigration. In this paper, we greatly simplify the proofs in part I and provide a proof of the global dynamics of the non-hyperbolic cases that were previously conjectured. Then, we show how immigration to one of the species or to both would, drastically, change the dynamics of the system. It is shown that if the level of immigration to one or to both species is above a specified level, then there will be no extinction region where both species go to extinction.

Hierarchical competition models with the Allee effect III: multispecies

“…Finally, to avoid the extinction of all species, one may use seeding or immigration to the species, see for instance [4].…”

“…A complete study was done of the non-hyperbolic case under the assumption that the centre manifolds of the non-hyperbolic fixed points are stable. Unfortunately, this is not the case for most competition models [3,4] in which the centre manifold is semi-stable. Hence it is desirable to remove this condition.…”

“…Then we apply these results to hierarchical competition models with the Allee effect that are represented by triangular maps. By this we complete the stability analysis that was initiated in Assas et al [3,4]. By a hierarchical model, we mean a dynamical system model with a networked hierarchy of state variables rather than the random parameter models of statistics [6,7,12,21,23,24].…”

“…It is a phenomenon in biology that is characterized by a positive correlation between population density and its per capita growth rate. One can make a distinction between the strong Allee effect and the weak Allee effect: a strong Allee effect refers to a population that exhibits a 'critical size or density' below which the population declines to extinction, while a weak Allee effect refers to a population that lacks a 'critical density', but where, at lower densities, the population growth rate rises with increasing densities [4,11]. With few exceptions, the literature on the Allee effect is exclusively for single-species [8, 14, 15, 18-20, 28-30, 32].…”

On the global dynamics of periodic triangular maps

“…In the paper [2] Assas et al have been study a two-species hierarchical competition model with a strong Allee effect with no immigration (the Allee effect is assumed to be caused by predator saturation). Later on, in [1] the authors simplified the techniques employed in the previous paper and provided a proof in global dy-namics of the non-hyperbolic cases that were previously conjectured. They have been shown how immigration to one of the species or to both would, drastically, change the dynamics of the system.…”