Hopf Bifurcation Analysis in the Forest Pest System

Abstract

In this paper, we analyze dynamical behaviors of the non-even-aged forests affected by insect pest system. This system is described by a cubic system of three ordinary nonlinear differential equations with five real parameters. We confirm that the forest pest system displays local Hopf bifurcations under certain conditions. Moreover, we show that a Hopf bifurcation occurs at four equilibrium points for the system. Also, we obtain sufficient conditions for supercritical and subcritical bifurcations via the normal form theory.
More precisely, we show that the forest pest system admits limit cycles. Numerical examples are given to validate the theoretical analysis.

Keywords: Stability equilibrium point; Hopf bifurcation; limit cycles; Forest pest system.

2010 Mathematics Subject Classification. 34A34; 34C23; 34C25; 34D20