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would suggest. However, if species from different pairs differ in their responses to environmental fluctuations (p<1), then increasing the number of species n in the community decreases the CV of combined species densities (Figure 8.2b). The decrease in CV is due to a statistical averaging effect (Doak et al., 1998); when the environment is bad for one species, it is not necessarily bad for another (since p <1), and therefore the combined density of both species does not fluctuate as much as either species separately. There is still no effect on the magnitude of l* (Figure 8.2a), which is not surprising because species-species interactions remain identical among predator-prey pairs. Also, using the community matrix for a single predator-prey pair (eqn 8.3) as a linear approximation of eqns 8.1 gives fairly good predictions of the CV calculated by simulating eqns 8.1 (Figure 8.2b, dashed lines). This indicates that the role of species-species interactions in the stability of the model with n predator-prey pairs is predicted by the predator-prey interactions of just a single predator-prey pair4.

In case 2, predator-prey pairs differ in demographic parameters. To create these differences, suppose values of a and K for a given predator-prey pair are drawn from random variables with means a and K, and standard deviations a scaled relative to the mean. Values of a, K, and the other parameters are set so that when a — 0, case 2 gives exactly the same model as in case 1 under the assumption that p — 0 (Figure 8.2a and b). Differences in a and K among predator-prey pairs decrease the endogenous stability of the community; the dominant eigenvalue l* of the n-species community matrix increases with even small differences (a — 0.05) among predator-prey pairs (Figure 8.2c). The effect of this is apparent in the CV in combined prey densities, which decreases but then increases with increasing n when there are large differences in values of a and K among pairs (Figure 8.2d). The net CV of combined prey density is driven by two forces. First, increasing n

4 The deviation between the predicted and observed CV is due to nonlinearities in eqns 8.1, rather than the inability of the dynamics of the single predator-prey pair to predict the dynamics of the system with n pairs.

causes a decrease in CV due to differences among species in species-environment interactions. This causes the CV to decline in the absence of differences among pairs (a — 0). Counteracting the effect of differences in species-environment interaction, differences in species-species interactions (a>0) raise the CV. Thus, these two types of diversity act in opposition.

I included the final case, case 3, to address the issue of making comparisons. Making comparisons requires assumptions about what to keep equal. In both cases 1 and 2, predator-prey pairs were added while keeping the strength of coupling among pairs the same (a — 3). This has the effect of maintaining the overall strength of competition and the overall strength of predation in the system as n increases; formally, when a — 3, the sum of effects of competitors and predators on the per-capita prey population growth rates at equilibrium, ± and £ te!^, are independent of n and a — 3; the same is true for the per-capita population growth rate of predators. For the predation rate, this means that the same proportion of the total prey population is eaten regardless of either the number of predator-prey pairs or the strength of coupling between them. One could envision alternative scenarios, with for example increasing numbers of pairs leading to greater impacts of predation relative to competition. This might occur if prey occupy different habitats and therefore do not compete, whereas predators move among habitats and feed on all prey. In mathematical terms, this scenario would mean a — 0 and 3>0.

When the coupling between predator-prey pairs only occurs through predation (a — 0 and 3>0), increasing the number of pairs n is strongly destabilizing, causing increases in both the magnitude of l* and the CV in combined prey density (Figure 8.2e and f). There is a simple explanation for this. Consider eqn 8.3, which gives the dynamics of the combined prey and predator densities in the 2n-species community. The element in the top left-hand corner of matrix C, dx(t + 1)/dx(t), gives the effect of prey density on the prey population growth rate. When a — 0,

A = 1 + (n — 1)^; therefore, from eqn 8.3, increasing n decreases the strength of competition among prey. Similarly, increasing n increases the element in the top right corner of matrix C, thereby increasing the strength of predation on the prey population growth rate. Thus, increasing n has exactly the same effect as reducing competition and increasing predation in the community matrix for a single predator-prey pair (eqn 8.2). Not surprisingly, increasing predation relative to competition is destabilizing in the single predator-prey pair, so increasing predation relative to competition is similarly destabilizing when there are n predator-prey pairs. In fact, the stability of the n-pair system is well predicted using the community matrix for a single pair once this increase in predation relative to competition is incorporated (Figure 8.2f, dashed lines; in Figure 8.2e the lines coincide perfectly).

To summarize, adding species to a community only affects stability if the new species are somehow different. If species respond to the environment differently (case 1), this can confer stability, at least if the measure of stability depends on species-environment interactions (i.e. CV but not l*). If species differ in strengths of interactions (case 2), then adding species decreases stability, at least in this model. Therefore, the effect of diversity on stability depends on how species differ. These conclusions were made by comparing communities that contain different numbers of predator-prey pairs, but nonetheless have equal overall impacts of competition and predation on prey and predator per-capita population growth rates. If instead adding species changes the average strength of interactions among species, this can change stability. In case 3, increasing predator-prey pairs decreased stability, because this increased the overall impact of predation relative to competition on prey per-capita population growth rates. Is this an effect of diversity, or is it an effect of increasing predation? From a theoretician's point of view, I would prefer to call it an effect of predation, not diversity, because the increase in predation would have identical impacts on the stability of a single predator-prey pair. Therefore, it doesn't really have anything to do with the numbers of species and the differences among them. Nonetheless, stepping outside theory into ugly reality, the importance of predation relative to competition might in fact increase in more species-rich communities. This is an empirical question. For theory, it seems to make most sense to pick those comparisons that are conceptually most informative, thereby isolating and separating effects that do not have anything directly to do with diversity.

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