Canard cycles for predator–prey systems with Holling types of functional response

By using the singular perturbation theory developed by Dumortier and Roussarie and recent work of De Maesschalck and Dumortier, we study the canard phenomenon for predator–prey systems with response functions of Holling types. We first develop a formula for computing the slow divergence integrals. By using the formula we prove that for the systems with the response function of Holling types III and IV the cyclicity of any limit periodic set is at most two, that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations. We also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two.