BIFURCATIONS OF CODIMENSION THREE IN A LESLIE-GOWER TYPE PREDATOR-PREY SYSTEM WITH HERD BEHAVIOR AND PREDATOR HARVESTING

A Leslie-Gower type predator-prey system with herd behavior in prey and constant harvesting in predators is considered in this paper. It is shown that there are two non-hyperbolic equilibria, one is a nilpotent cusp of codimension at most three and the other one is a weak focus of multiplicity also at most three. A complete analysis on bifurcations with codimension three is given as the bifurcation parameters vary, which includes a Bogdanov-Takens bifurcation of codimension three and a degenerate Hopf bifurcation of codimension three. The results indicate that the Leslie-Gower type system exhibits richer bifurcations than the classic Leslie-Gower model and also reveal the complexity of the interaction between the prey, predators and humans.