Chapter 11

Spruce budworm infestations

Spruce budworm infestations occur sporadically in the forests of North America. The chapter mainly analyses a model proposed by Ludwig, Jones and Holling in 1978, which can explain how periodic outbreaks can occur (this occurred historically in New Brunswick, at intervals of 35 years). The model consists of three ordinary differential equations for the variables B, the budworm larval density, S, a measure of pine branch surface area density, and E, a measure of the foliage density. It is possible to explain quantitatively how oscillations occur, using conceptually simple ideas (but complicated in detail) akin to those of chapter 10.

The budworm model is a powerful pedagogical tool, but is much simpler than the simulation model described by Jones in 1979, termed the budworm site model. Apart from the inclusion of spatial variation, the site model includes a detailed description of age structure, and also evolves the system forward in discrete time steps of one year. This makes the simulation model much more complicated and less easy to analyse. Exercise 5 in the present chapter suggests a possible age-structured model which might be used to analyse the age-dependent disease susceptibility, but still in the framework of the continuous Ludwig et al. model.

More recently (1996), David Hassell wrote an M. Sc. dissertation on the Jones site model, and this work has been written up as a paper, submitted to the Journal of Mathematical Biology. In this paper we show that the full Jones site model can be effectively asymptotically reduced to a much simpler third order difference equation, which exhibits hysteresis similar to that in the Ludwig et al. model, though for different reasons.

Actually, the whole issue of continuous versus discrete models is somewhat contentious, for example in the analysis of epidemics, see D. Mollison (ed.), Epidemic Models, C.U.P. 1995. Although the use of continuous models is attractive to those with a differential equations background, there are grounds for supposing that continuous models, particularly for populations with seasonally related reproductive dynamics, are fundamentally inappropriate.