P

Equations (5.35b) and (5.37) need to be analyzed. By now, we know the drill: determine the steady states, examine the isoclines, and compute the eigenvalues of the system linearized around the steady states.

Show that the steady states are

We will not do the linearization around the steady state here, nor will I assign it as an exercise (although you might want to do it). Suffice to say: the eigenvalues are pure complex numbers, so that the steady state is neutrally stable and the dynamics are purely oscillatory. This point is illustrated in Figure 5.14a and should you remind you of our experience with the Nicholson-Bailey model when the fraction of hosts escaping parasitism was given by the zero term of the Poisson distribution.

We now turn to the case in which worms have a negative binomial distribution across hosts. Based on our experience with parasitoids, we expect that aggregation will stabilize the dynamics. We need to compute the second moment in Eq. (5.36), remembering that for the negative binomial distribution, the variance is m + (m2/k).

Exercise 5.10 (M)

Show that when worms have a negative binomial distribution across hosts, Eq. (5.37) is replaced by dP d — P

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