Geometric aspects of bifurcations for a classical predator-prey model

Abstract

The bifurcation and dynamics of the classical predator-prey model with the generalized Holling type III functional response have been studied from different aspects. When the denominator of the response function has at least one zero, its global dynamics has been classified. When the denominator does not vanish, its local bifurcation was classified in 2008 from analytic point of view. Here we first characterize the local bifurcation via geometry of the critical curve. Then utilizing these geometric aspects of the bifurcations, we further classify all global topological dynamics of this model in the slow-fast setting, where we can also exhibit not only the birth and disappearance but also the locations and shapes of the limit cycles.