Allee thresholds and basins of attraction in a predation model with double Allee effect

Abstract

We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle‐node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

Invariant manifolds and global bifurcations to find multiple wave solutions in a reaction-diffusion system

“…In such case, the Allee threshold is typically a one-dimensional object in the (x, y)-plane. Recent findings reveal that this threshold can be the basin boundary of a positive (coexistence) equilibrium, or a limit cycle, or a homoclinic orbit [4,18].…”

“…Since the Allee effect is related to population extinction, an understanding of this phenomenon is very relevant for ecological administration and conservation [56]. Many studies have been carried out in this regard, as can be seen in [4,18,21,56]. Different models have shown that the Allee effect increases the risk of population extinction as the Allee threshold rises [56].…”