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Figure 5.14. The basic worm model with a Poisson distribution (panel a) or negative binomial distribution of worms across hosts (panels b, c, and d). Common parameters are a = 2, b = 1, p = 1, H0 = 5, a = 0.5, and l = 4 and k = 100 (panel b), 10 (panel c) or 1 (panel d). In each case the parasite population is the one with the larger maximum size.

Figure 5.14. The basic worm model with a Poisson distribution (panel a) or negative binomial distribution of worms across hosts (panels b, c, and d). Common parameters are a = 2, b = 1, p = 1, H0 = 5, a = 0.5, and l = 4 and k = 100 (panel b), 10 (panel c) or 1 (panel d). In each case the parasite population is the one with the larger maximum size.

In Figures 5.14b-d, I show the resulting dynamics for three values of k (= 100, 10, and 1). Note that in each case the dynamics are stabilized. When one conducts an analysis of the linearized system (see Anderson and May 1978, Appendix A), a remarkable result emerges: the eigenvalues have negative real part as long as a/k> 0; but both of these parameters are positive so we reach the conclusion that any level of aggregation - meaning any deviation from a completely random distribution - is sufficient to stabilize the dynamics.

The papers of Anderson and May (1978) and May and Anderson (1978) are well worth examining, to study the other various cases, for which you are now set. We, however, move on.

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