Limit Cycles in Slow-Fast Forest-Pest Models

Some of the most exciting current work in the environmental sciences involves simplified but analytically tractable versions of a few basic equations. IIASA's Environment Program has developed such an approach in its analysis of forest systems. A number of previous papers (WP-87-70 and WP-87-92) have demonstrated the progress that has been made. In this new work some of the ideas contained in those papers have been further developed.
In particular a simple aged-structure forest model is considered to prove that a forest can exhibit periodic behavior even in the case the insect pest is adapted only to mature trees. The insect pest 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.

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