## 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.