The study of the interactions between populations of predators and their prey forms the foundations on which the theory and practice of biological control rests. Mathematical models play a central role in the study of predator-prey interactions because they allow us to study the dynamic consequences of the interaction between populations of predators and their prey, and to ask questions about the underlying basis of natural control. With models we can study the consequences that we might not be able to observe directly in the field. This course, therefore, is mainly concerned with mathematical models and their use to study the interaction between insects and their natural enemies. We will concentrate, in particular, on models of insect parasitoids and their insect hosts. This is because much of the theoretical work by entomologists has centered on host-parasitoid interactions. With some caution, however, host parasitoid models can be extended to include interactions with other kinds of natural enemies.

Royama (1971), in his detailed and thorough analysis of models for predation and parasitism, recognized two basic kinds of attack functions which he called "instantaneous" and "overall" hunting equations. Dynamic, or time varying, instantaneous equations are normally written as continuous-time differential equations and hold only over an instant in time. Overall equations, on the other hand, are integrated over a unit of time, usually a day, year, or generation, and hence the dynamic models are normally written as discrete-time difference equations, which may have different forms for parasitoids and predators. The parasitoid form acknowledges the fact that a host may be parasitized more than once by the same or different parasitoids whereas the predator form recognizes that prey are removed so that prey density is reduced during an interval of time. Difference equations are widely used to study the dynamics of insect populations in north temperate regions because many have discrete generations, but discrete models can also apply to populations with overlapping generations provided age structure is included in the model and when the period of observation is shorter than a generation.

Reference

Royama, T. 1971. A comparative study of models for predation and parasitism. Researches on Population Ecology, Supplement No. 1.