Forest defoliation scenarios

We consider the mathematical model originally created by Ludwig, Jones, and Holling to model the infestation of spruce forests in New Brunswick by the spruce budworm. With biologically plausible parameter values, the dimensionless version of the model contains small parameters derived from the time scales of the state variables and smaller parameters derived from the relative importance of different population change mechanisms. The small time-scale parameters introduce a singular perturbation structure to solutions, with one variable changing on a slow time scale and two changing on a fast time scale. The smaller process-scale parameters allow for the existence of equilibria at vastly different orders of magnitude. These changes in scale of the state variables result in fast dynamics not associated with the time scales. For any given set of parameters, the observed dynamics is a mixture of time-scale effects with process-scale effects. We identify and analyze the different scenarios that can occur and indicate the relevant regions in the parameter space corresponding to each.