## The general model

The basis of the theoretical approach to parasitoid-prey interactions has largely revolved around the generalized model:

where:

Nt and Pt are the sizes of the initial prey and parasitoid populations and Nt+1 and Pt+1 are the number in the next generation; Fg(Nt) is the per capita net rate of increase of the host population and g(Nt) is a density dependent function ranging from zero to one;

c is the average number of progeny produced per prey attacked and s is the proportion of progeny that are female. The function (Nt, Pt) defines prey survival and includes all assumptions about para-sitoid searching efficacy (Hassell and Waage, 1984; May and Hassell, 1988). (Some of these functions are given in Table 6.2.)

The progress in the development of these models has led to a step-wise identification and exploration of the important factors influencing parasitoid-prey dynamics, always with the intention of explaining their role in the regulation of the two populations. The factors have been evaluated on the basis of the magnitude of their effects on the prey equilibrium levels and the

Table 6.2. A number of examples of specific expressions that have been used for f(Nf PI) in equations 6.1 and 6.2. a, a' and Q are searching efficiencies, Th is the handling time as a fraction of total time, m is an interference constant, a and p are the fraction of hosts and parasitoids respectively in the ith of n patches, k is the clumping parameter from the negative binomial distribution (after Hassell and Waage, 1984).

Form of f

Nicholson and Bailey (1 935) Rogers (1972)

Hassell and Varley (1969) Hassell and May (1 973)

May (1978)

k stability of this equilibrium population. Both these criteria have obvious importance for classical biological control because a successful introduction should reduce the prey population to a new low level and maintain it there indefinitely. The effects of different components of the models (6.1 and 6.2) on equilibrium levels and their stability are considered separately in the following sections. Although treated in this way, it should be realized that all factors affect the potential equilibrium levels to some degree.

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