The Hopf Bifurcation for a Predator-Prey System with θ -Logistic Growth and Prey Refuge

Abstract

The Hopf bifurcation for a predator-prey system with $\theta$-logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index-$\theta$ passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.