Competition models with Allee effects

Abstract

In this paper, we study a generalized two-species contest-competition model with an Allee effect. We provide a complete analysis of the global dynamics of the system. In particular, we determine all the invariant manifolds, the extinction, the exclusion and the coexistence regions. We use tools from topology and dynamical systems to show that all orbits must converge to one of the equilibrium points of the system. The analysis shows that there are several potential scenarios including competition coexistence, exclusion and extinction.

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

“…An example is an invasive species x that strongly competes with a native species y. The terms e −m 1 /(1+s 1 x) and e −m 2 /(1+s 2 y) represent the positive density dependence of x and y, respectively, which is the hallmark of the Allee effect caused by predator saturation [1,2,9,12,13,16,19,21,24,25,27]. The parameter m i represents the predation intensity, and s i is a parameter proportional to the 'handling time', that is, the duration of the capture and eating phase, i = 1, 2.…”

Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect

“…See [10,11,[13][14][15][16][17][18] for related results and [19] for dynamics of competitive system with a singular point at the origin. The biological interpretation of a related system is given in [20,21] and similar system is treated in [22]. The specific feature of our results is that no equilibrium point in the interior of the first quadrant is computable and so our analysis is based on geometric analysis of the equilibrium curves.…”

Hierarchical competition models with Allee effects

“…(See Figure 2 and for comparison, we include Figure 1 for the non-Allee effect case.) Allee effects in two-species models were studied in [8,14,18,19]. Strong Allee effect in multispecies models is characterized by the presence of an extinction region where both species would go to extinction if their densities (sizes) fall into that region [18,19].…”

“…Allee effects in two-species models were studied in [8,14,18,19]. Strong Allee effect in multispecies models is characterized by the presence of an extinction region where both species would go to extinction if their densities (sizes) fall into that region [18,19]. For instance, a planar competition model is said to display the strong Allee effect property if its phase space exhibits an extinction region and a non-extinction region separated by an Allee threshold curve (a manifold of dimension n − 1 where n is the dimension of the phase space).…”