Qualitative Analysis Of Several Kinds Of Predator-prey Systems Incorporating Prey Refuge

This paper consists of three parts:Firstly, we considered a predator-prey model with Holling type II functionalresponse incorporating a constant prey refuge and a constant-rate prey harvesting.Depending on the constant prey refuge m, which provides a condition for protecting mof prey from predation, some sufficient conditions for the instability and globalstability of the equilibria and the existence and uniqueness of limit cycles of themodel are obtained. We also show the influence of prey refuge and harvesting effortson equilibrium density values. Numerical simulations are carried out to illustrate thefeasibilityof the obtained results and the dependence of the dynamic behaviour on theharvesting efforts or prey refuge.Secondly, we consider a predator-prey model with simplified Holling-type IVresponse function incorporating a prey refuge. Through qualitative analysis of themodel, at least two limit cycles around the positive equilibrium point with the result offocus value, the Hopf bifurcation and Heteroclinic bifurcation under a prey refuge areobtained. We also show the influence of prey refuge.Finally, We consider a Holling-Tanner Model with Modified Leslie-Grower. Theexistence and stability of the equilibria of the system are analyzed. It is proved thatthere is no limit cycle around the positive equilibra by constructing Dulac function,thus global asymptotic stability of the positive equilibria is proved in the firstquadrant.