LIMIT-CYCLES IN SLOW FAST FOREST PEST MODELS

A simple age-structured forest model is considered in this paper to prove that, in contrast with some recent findings, a forest can exhibit periodic behavior even in the case where the insect pest is adapted only to mature trees. The insect pest is assumed to have a very fast dynamics with respect to trees and the analysis is carried out through singular perturbation arguments. The method is based only upon simple geometric characteristics of the equilibrium manifolds of the fast, intermediate, and slow variables of the system and allows one to derive explicit conditions on the parameters that guarantee the existence of a limit cycle in the extreme case of very fast-very slow dynamics. Nevertheless, simulation shows that the limit cycle disappears through a Hopf bifurcation only when the dynamics of the different components of the system become almost comparable. © 1992.