## The Nicholson Bailey model

Turning now to parasitoids, the basic model (Nicholson & Bailey, 1935) is again not so much realistic as a reasonable basis from which to start. Let Ht be the number of hosts, and Pc the number of parasitoids in generation t; r is the intrinsic rate of natural increase of the host. If Ha is the number of hosts attacked by parasitoids (in generation t), then, assuming no intraspecific competition amongst the hosts (exponential growth - see Section 4.7.1), and that each host can support only one parasitoid (commonly the case):

In other words, hosts that are not attacked reproduce, and those that are attacked yield not hosts but parasitoids.

To derive a simple formulation for Ha, let Ec be the number of host-parasitoid encounters in generation t. Then, if A is the parasitoid's searching efficiency:

and:

Note the similarity to the formulation in Equation 10.2. Remember, though, that we are dealing with parasitoids, and hence a single host can be encountered several times, although only one encounter leads to successful parasitization (i.e. only one parasitoid develops). Predators, by contrast, would remove their prey and prevent re-encounters. Thus, Equation 10.2 dealt with instantaneous rates, rather than numbers.

If encounters are assumed to occur more or less at random, then the proportions of hosts that are encountered zero, one, two or more times are given by the successive terms in the appropriate 'Poisson distribution' (see any basic statistics textbook). The proportion not encountered at all, p0, would be given by e~Et'Ht, and thus the proportion that is encountered (one or more times) is 1 — e~Et'Ht. The number encountered (or attacked) is then:

Using this and Equation 10.12 to substitute into Equations 10.9 and 10.10 gives us:

of the two populations is a possibility, but even the slightest disturbance from this equilibrium leads to divergent coupled oscillations.