Singular perturbations of generalized Holling type III predator-prey models with two canard points

Abstract

We study the coexistence of limit cycles in a predator-prey model of Leslie type with generalized Holling type III functional response. When the prey reproduces much faster than the predator, we prove for this model that: (i) the existence of the configuration of one large stable limit cycle enclosing two small unstable limit cycles, (ii) the cyclicity of singular double-head canard cycles is three and reached, and (iii) the coexistence of two stable limit cycles surrounding three equilibria. The last result gives a positive answer to Coleman's problem on the coexistence of two ecologically stable limit cycles in predator-prey models.