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A mathematical analysis of Jones:s site model for spruce budworm infestations

By D. C. Hassell, D. J. Allwright and A. C. Fowler


Abstract. The budworm site model of Jones (1979) is a complicated representation of the interaction between the spruce budworm and the forests of Northeastern Canada, which describes the way in which the budworm population undergoes periodic outbreaks, which lead to severe defoliation and widespread tree mortality. We show how this model can be systematically simpli"ed without gainsaying the essential description of the budworm/forest interaction. The main simpli"cation is the collapse of the year to year age structure to allow for three classes of young, mature, and old trees; we then obtain a reduced set of six di!erence equations for the six variables: larval population, new and old foliage, and the three age classes. Further analysis and reduction of the model is then possible on the basis of formal asymptotic limits, and we analyse a reduced model consisting of three di!erence equations for larvae, (old) foliage, and the area fraction of mature trees. In practice, the latter variable is inessential to the mechanism of oscillation, which can be understood via the slow cycling of the foliage variable round a hysteresis loop of quasi-steady states, mediated by the varying larval population as control parameter. Key words: Spruce budworm} Di!erence equations} Site model} Hysteresis} Oscillations 1

Year: 1997
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