The authors study the following predator-prey model with Holling type-IV functional response and Leslie type numerical response for the predator
where and represent the densities of the prey and the predator population at time , respectively. and are positive parameters. It is shown that system (1) admits two non-hyperbolic positive equilibria for some values of parameters, one is a cusp of codimension 2 and the other is a multiple focus of multiplicity one. When the parameters vary in a small neighborhood of the values of parameters, system (1) undergoes a Bogdanov-Takens bifurcation and a subcritical Hopf bifurcation in two small neighborhoods of these two equilibria, respectively. And it is further derived that by choosing different values of parameters, system (1) can have a stable limit cycle enclosing two equilibria, or an unstable limit cycle enclosing a hyperbolic equilibrium, or two limit cycles enclosing a hyperbolic equilibrium. It is also proved that system (1) never has two limit cycles enclosing a hyperbolic equilibrium each for all values of parameters.