Mixed-Mode Oscillations Due to a Singular Hopf Bifurcation in a Forest Pest Model

In a forest pest model, young trees are distinguished from old trees. The pest feeds on old trees. The pest grows on a fast scale, the young trees on an intermediate scale, and the old trees on a slow scale. A combination of a singular Hopf bifurcation and a “weak return” mechanism, characterized by a small change in one of the variables, determines the features of the mixed-mode oscillations.

Period-doubling and saddle-node bifurcations lead to closed families (called isolas) of periodic solutions in a bifurcation corresponding to a singular Hopf bifurcation.