Multiple bifurcations in a predator–prey system with monotonic functional response

摘要

In this paper, we study the dynamics of a predator–prey system with Holling type III functional response of the formx′=rx1-xK-x2x2+ay,y′=y-d+ωx2x2+a-h.When h = 0, Chen and Zhang (J.P. Chen, H.D. Zhang, The qualitative analysis of two species predator–prey model with Holling’s type III functional response, Appl. Math. Mech. 7 (1) (1986) 73–80) and Kazarinoff and Driessche (N.D. Kazarinoff, P. Van Den Driessche, A model predator–prey system with functional response, Math. Biosci. 39 (1–2) (1978) 125–134) gave qualitative analysis of the above system and obtained some conditions for the existence and stability of positive equilibria and limit cycles. The dynamical phenomena they concluded are normal, and there is no degenerate singularities. But when the constant harvesting is present, we will show that the system may have more complex and more rich dynamics. We carry out bifurcation analysis by choosing the death rate and the harvesting rate of the predator as the bifurcation parameters and show that the system can undergo the Bogdanov–Takens bifurcation. Also the figures of all degenerate structures are given.