Bifurcation analysis of a discrete-time ratio-dependent predator–prey model with Allee Effect

Abstract

A discrete-time predator–prey model with Allee effect is investigated in this paper. We consider the strong and the weak Allee effect (the population growth rate is negative and positive at low population density, respectively). From the stability analysis and the bifurcation diagrams, we get that the model with Allee effect (strong or weak) growth function and the model with logistic growth function have somewhat similar bifurcation structures. If the predator growth rate is smaller than its death rate, two species cannot coexist due to having no interior fixed points. When the predator growth rate is greater than its death rate and other parameters are fixed, the model can have two interior fixed points. One is always unstable, and the stability of the other is determined by the integral step size, which decides the species coexistence or not in some extent. If we increase the value of the integral step size, then the bifurcated period doubled orbits or invariant circle orbits may arise. So the numbers of the prey and the predator deviate from one stable state and then circulate along the period orbits or quasi-period orbits. When the integral step size is increased to a critical value, chaotic orbits may appear with many uncertain period-windows, which means that the numbers of prey and predator will be chaotic. In terms of bifurcation diagrams and phase portraits, we know that the complexity degree of the model with strong Allee effect decreases, which is related to the fact that the persistence of species can be determined by the initial species densities.