The Dynamical Behavior of a Predator-Prey System with Holling Type II Functional Response and Allee Effect

Abstract

In this paper, we mainly considered the dynamical behavior of a predator-prey system with Holling type II functional response and Allee-like effect on predator, including stability analysis of equilibria and Hopf bifurcation. Firstly, we gave some sufficient conditions to guarantee the existence, the local and global stability of equilibria as well as non-existence of limit cycles. By using the cobweb model, some cases about the existence of interior equilibrium are also illustrated with numerical outcomes. These existence and stability conclusions of interior equilibrium are also suitable in corresponding homogeneous reaction-diffusion system subject to the Neumann boundary conditions. Secondly, we theoretically deduced that our system has saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation under certain conditions. Finally, for the Hopf bifurcation, we choose d as the bifurcation parameter and presented some numerical simulations to verify feasibility and effectiveness of the theoretical derivation corresponding to the existence of k y , respectively. The Hopf bifurcations are supercritical and limit cycles generated by the critical points are stable.

Complexity Analysis of a Modified Predator-Prey System with Beddington–DeAngelis Functional Response and Allee-Like Effect on Predator

“…is eorem 1 holds obviously after introducing an auxiliary function z � e 1 x + y [1,16]. e following theorem with the help of comparison principle in ODEs is about permanence of system (2a) and system (2b).…”

“…In this section, we will discuss the existence conditions of trivial equilibria of system (2a) and system (2b). Firstly, from [1], it is obvious that system (2a) and system (2b) have trivial equilibria:…”

“…e point E 1 exists when r 1 > m 1 , while the points E (k) 2 all exist if r 2 > m 2 and Δ > 0. If r 2 > m 2 but Δ � 0, then the two equilibria E (1) 2 and E (2) 2 collide with each other and we denote this equilibrium as E 2 � (0, y 2 ).…”

“…For the trivial equilibrium E 0 with r 1 ≠ m 1 and the axial equilibrium E 1 with (q 2 x 1 /(bx 1 + a)) ≠ m 2 , we omit their stability [1]. When r 1 � m 1 , the transformation τ � − m 2 t and equations (2a) and (2b) yield _…”

“…In reference [1], the authors simply considered a predatorprey system with Holling type II functional response and Allee-like effect on predator, which is described by the following nonlinear ordinary differential equations (ODEs):…”