Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Harvesting

¹ School of Mathematics and Statistics, Central China Normal University Wuhan, Hubei 430079, P. R. China
² Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA
³ School of Mathematics and Statistics, Northeast Normal University Changchun, Jilin 130024, P. R. China

^⋆ Corresponding author. E-mail: hjc@mail.ccnu.edu.cn

Abstract

In this paper we study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant-yield for prey, it is shown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established by numerical simulations. When the harvesting is seasonal for both species, sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and transition from invariant tori to periodic solutions, respectively, as the amplitude of seasonal harvesting increases.