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Ever since Lord Robert May challenged Robert MacArthur’s assertion that there is a positive relationship between diversity and stability, the argument has raged as to whether there really is a relationship between the proxies, species richness and connectance. May demonstrated that, at least within randomly connected food webs (more properly graphs), diversity does not beget stability, and that there is a critical connectance above which the system becomes unstable. I say that richness and connectance are proxies because diversity is more than richness, and stability is more than a critical point of connectance. Many workers, stimulated by May’s contention, have since shown that the non-random connection topologies of food webs matter; that is, functional diversity and hierarchical arrangements of species interactions allow real food webs, apparently, to be far more complex than allowed in May’s framework. Is there then no limit, or indeed no relationship between species richness and food web connectance?

I showed in an earlier post that there is a positive relationship between node richness and the number of links spanning a broad array of food web types. The same has been demonstrated before, most recently by Ings et al. Indeed, workers such as Jennifer Dunne and others have hypothesized that increased connectance confers greater robustness on food webs, hence allowing increases in richness as long as complexity also increases. I, on the other hand, doubt that this relationship actually exists for several reasons. First, the data upon which these hypotheses are based are extremely heterogeneous, and it is unclear whether connectance as measured across the array of food webs is actually the same thing from one web to the next. Second, measures of robustness typically are incapable of assessing robustness against anything other than the bottom-up perturbation of unparameterized systems; that is, no link strengths, population sizes, etc. Additionally, there should hence be no expectation of similarity of connectance values among any food webs.

In continuing our work on Greater Antillean coral reef food webs, I wanted to examine this relationship for our three food webs, namely those of the Cayman Islands, Cuba and Jamaica. The food web models differ only in vertebrate richness, and are ordered as Caymans>Jamaica>Cuba. This ordination corresponds nicely with sampling events and efforts. Yet, Jamaica has by far the greatest connectance of the three. Is this unusual or unexpected? We assessed this by stochastically drawing food webs of varying vertebrate richness, ranging from 80 to 160 species, from the regional species pool, and calculating their connectances. We did this for about 9000 food webs, and discovered this very nice, linear relationship. Connectance clearly increases, linearly in this case, with increasing richness. Why? The explanation is rather simple. Recall that the in-degree distributions of the island food webs, and hence the regional pool, is modal, yet with a significant right long tail. As one increases the number of species in a randomly drawn food web, the probability of drawing species from the long tail, those of high in-degree, also increases. Think of those species as being more “link dense”. Connectance will therefore increase, and will not be a constant value.

A wonderful surprise, however, is that the real food webs do not necessarily conform to this. The Cayman and Cuban food webs are indeed indistinguishable from the random food webs, but look at Jamaica! It’s connectance is well above random expectation. We know why, but I won’t tell you yet.

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