Complex Dynamics of a Diffusive Holling-Tanner Predator-Prey Model with the Allee Effect

Abstract

We investigate the complex dynamics of a diffusive Holling-Tanner predation model with the Allee effect on prey analytically and numerically. We examine the existence of the positive equilibria and the related dynamical behaviors of the model and find that when the model is with weak Allee effect, the solutions are local and global stability for some conditions around the positive equilibrium. In contrast, when the model is with strong Allee effect, this may lead to the phenomenon of bistability; that is to say, there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spots-stripes mixtures, and stripes patterns. The results show that the impact of the Allee effect essentially increases the models spatiotemporal complexity.

A Holling–Tanner Predator–Prey Model with Strong Allee Effect

“…The growth function (x) = rx(1 − x/K)(x − m) has an enhanced growth rate as the population increases above the threshold population value m. If (0) = 0 and (0) ≥ 0 -as it is the case with m ≤ 0 -then (x) represents a proliferation exhibiting a weak Allee effect, whereas if (0) = 0 and (0) < 0 -as it is the case with m > 0 -then (x) represents a proliferation exhibiting a strong Allee effect [37]. The aim of this manuscript is to study the dynamics of the Holling-Tanner predator-prey model with strong Allee effect on prey and functional response Holling Type II, that is (3) with m > 0.…”

A modified May-Holling-Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

“…This manuscript also extends the properties of the May-Holling-Tanner model with multiple Allee effects studied in [36] that is (7) with c = 0 by showing the impact of the inclusion of alternative food sources for predators. In addition, it complements the results of the May-Holling-Tanner model considering only alternative food for the predator studied in [7,20] and the model considering only a single Allee effect on the prey and no alternative food for the predator studied in [49,52]. Model (1) with functional response Holling type II, i.e.…”

Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems

“…As mentioned in the introduction, the bistable structure can occur in many biological models, such as the insect population model [5], genetic control model [1], interactive model between two populations, including mating between the sexes [20], obligate mutualism between two species [21], predator-prey relationship [22,23], etc., epidemic model [4,24,25], and so on. Here, we briefly introduce four biological models, all of them exhibit a bistable structure.…”