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 yk, respectively. The Hopf bifurcations are supercritical and limit cycles generated by the critical points are stable.
Highlights
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
In [20], by using the Mawhins continuation theorem of coincidence degree and analysis techniques, the authors obtained some sufficient conditions of the existence of periodic solutions in a nonautonomous predator-prey system with Holling type II functional response, strong Allee effect and impulsive perturbation
In Subsection 3.1, some cases about the existence of interior equilibria E* are derived with the help of the cobweb model but we neglect the monotonicity (a) of functions sometimes, and we extend the zero-point theorem to the plane R+2 ; for instance, when yk do not exist, a point from the Equation (6a) should belongs to the domain Σ which is surrounded by the isocline in the Equation
Summary
We consider a predator-prey system with Holling type II functional. S. In [20], by using the Mawhins continuation theorem of coincidence degree and analysis techniques, the authors obtained some sufficient conditions of the existence of periodic solutions in a nonautonomous predator-prey system with Holling type II functional response, strong Allee effect and impulsive perturbation. They proved that their system has at least one positive ω -periodic solution.
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