Multiple bifurcations in a predator-prey system of modified Holling and Leslie type with double Allee effect and nonlinear harvesting

In this paper, a modified Leslie-type predator-prey model with simplified Holling type IV functional response is established, in which double Allee effect on prey and nonlinear prey harvesting are considered. The analysis of the model shows that there exists a Bogdanov-Takens singularity (focus case) of codimension 4, and also multiple other nonhyperbolic and degenerate equilibria. Bifurcations are explored and it is found that transcritical bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension 2, degenerate cusp type Bogdanov-Takens bifurcation of codimension 3, and degenerate focus type Bogdanov-Takens bifurcation of codimension 4 occur as parameters vary. The bifurcations result in complex dynamic behaviors, such as double limit cycle, triple limit cycle, quadruple limit cycle, cuspidal loop, (multiple) homoclinic loop, saddle-node loop, and limit cycle(s) simultaneously with homoclinic loop. We run numerical simulations to verify the theoretical results, and it is found that the system admits bistability, tristability, or even tetrastability.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

In this paper, a modified Leslie-type predator-prey model with simplified Holling type IV functional response is established, considering double Allee effect on prey and nonlinear prey harvesting. The analysis of the model reveals the existence of a Bogdanov-Takens singularity (focus case) and multiple nonhyperbolic and degenerate equilibria. Various bifurcations are explored, including transcritical bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension 2, degenerate cusp type Bogdanov-Takens bifurcation of codimension 3, and degenerate focus type Bogdanov-Takens bifurcation of codimension 4. These bifurcations result in complex dynamic behaviors, such as double limit cycle, triple limit cycle, quadruple limit cycle, cuspidal loop, (multiple) homoclinic loop, saddle-node loop, and simultaneous existence of limit cycle(s) with homoclinic loop. Numerical simulations confirm the theoretical results, showing bistability, tristability, or even tetrastability in the system.