Bifurcations of a predator–prey system of Holling and Leslie types

A predator–prey model with simplified Holling type-IV functional response and Leslie type predator’s numerical response is considered. It is shown that the model has two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of co-dimension 2 and the other is a multiple focus of multiplicity one. When parameters vary in a small neighborhood of the values of parameters, the model undergoes the Bogdanov–Takens bifurcation and the subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. And it is further shown that by choosing different values of parameters the model can have a stable limit cycle enclosing two equilibria, or a unstable limit cycle enclosing a hyperbolic equilibrium, or two limit cycles enclosing a hyperbolic equilibrium. However, the model never has two limit cycles enclosing a hyperbolic equilibrium each for all values of parameters. Some computer simulation are presented to illustrate the conclusions.