## Variation in attack rate

The classic (Anderson and May 1978) means of stabilizing the Nicholson-Bailey model is to recognize that not all hosts are equally susceptible to attack, for one reason or another. To account for this variability, we replace the attack rate a by a random variable A, with E{A} = a, so that the fraction of hosts escaping attack is exp(— AP). However, to maintain a deterministic model, we average over the distribution of A; formally Eq. (4.1) becomes

H (t + 1)= RH (t)EA{e-APW} P(t + 1) = H (t)(1 - Ea {e-APW})

where EA{ } denotes the average over the distribution of A. For the distribution of A, we choose a gamma density with parameters a and k. We then know from Chapter 3 that the resulting average of exp(-AP(t)) will be the zero term of a negative binomial distribution, so that

Since the mean of a gamma density with parameters a and k is k/a, it would be sensible for this to be the average value of the attack rate so that a = k/a; we choose a = k/a. We then multiply top and bottom of the right hand side of Eq. (4.14) by k/a to obtain

This modification of the Nicholson-Bailey model is sufficient to stabilize the population dynamics (Figure 4.6). To help understand the intuition that lies behind this stabilization, I note the following remarkable feature (Pacala et al. 1990): the stabilization occurs as long as the overdispersion parameter k < 1. I have illustrated this point in

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