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Figure 4.7, showing that if k = 0.99 the dynamics are stable (the oscillations have decreasing amplitude), but if k > 1 they are not (the oscillations have increasing amplitude).

Recall that the coefficient of variation of the gamma density with parameters a and k is 1/Vk, so that k < 1 is equivalent to the rule that the coefficient of variation is greater than 1. Pacala et al. (1990) call this the CV2 > 1 rule (but also see Taylor (1993) who notices that the specific properties of the dynamics will depend not only upon k but also upon R). Also recall that when k < 1, the probability density for the attack rate is large when the attack rate is small 0. This means that arbitrarily small values of the attack rate have substantial probability associated with

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Figure 4.7. The dynamics determined by Eq. (4.14) when k = 0.99 (panel a), 1.01 (panel b), or 1.02 (panel c) showing that the dynamics are unstable when k > 1. All other parameters as in Figure 4.6. In each case, the hosts are the upper curve, the parasitoids the lower curve.

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Figure 4.7. The dynamics determined by Eq. (4.14) when k = 0.99 (panel a), 1.01 (panel b), or 1.02 (panel c) showing that the dynamics are unstable when k > 1. All other parameters as in Figure 4.6. In each case, the hosts are the upper curve, the parasitoids the lower curve.

them, even though the mean attack rate is held constant. But very small attack rates mean that some hosts are essentially invulnerable to attack or that a refuge from attack exists. A host refuge is clearly one way to stabilize the dynamics. For example, the stable dynamics shown in Figure 4.3d involve a 30% refuge for the host.

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