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p(extinction) when number of prey species=10

The CEG model asserts that food web structure plays a role in extinction. The intricate patterns of relationships among species in a community distribute the effects of changes in one species to others in its community. Therefore, while the ultimate causes of increased extinction in an interval of time may be abiotic, and might affect only some species directly, the effects could be felt more broadly.

Topographic secondary extinction.–The narrow definition of secondary extinction, where a species becomes extinct because it has lost all its resources, is termed topological secondary extinction (Roopnarine, 2009). Topological refers to the dependence of extinction solely upon the topology (pattern) of the network. Note that topological secondary extinction affects the network only in a bottom-up fashion, that is, in the direction of energy flow from producers to consumers of increasing trophic level. Measuring or estimating topological secondary extinction in a food web, in response to a particular perturbation, can be approached in three ways, again depending on whether one assumes accuracy of a higher-level representation of the food web (e.g. a metanetwork) or precision of a species-level network. Here I will outline a probabilistic approach using metanetworks.

Let a perturbation of magnitude \omega be equal to the number of species removed randomly from the network. The probability that a species x_{i} will become secondarily extinct is the probability that all its links are to species that are a subset of the \omega set. This is determined from a hypergeometric distribution, where we can first ask: Given an in-degree of r_{i}^{x}, what is the probability that n_{i}^{x} of them will be lost?
p(n_{i}^{x}\vert \omega) = \binom{r_{i}^{x}}{n_{i}^{x}} \binom{S-1-r_{i}^{x}}{\omega - n_{i}^{x}} \binom{S-1}{\omega}^{-1}
where there are S-1 other species in the network and 0\leq\omega\leq S-1. The probability of x_{i} becoming extinct occurs when n_{i}^{x}=r_{i}^{x},
p(e_{i}^{x}\vert \omega) = \binom{r_{i}^{x}}{r_{i}^{x}} \binom{S-1-r_{i}^{x}}{\omega - n_{i}^{x}} \binom{S-1}{\omega}^{-1} = \frac{\omega ! (S-1-r_{i}^{x})!}{(S-1)!(\omega -r_{i}^{x})!}
The formula can be re-stated interestingly as
\boldmath{p(\text{extinction of } x_{i}) = \frac{(\text{perturbation magnitude)}!(\text{species richness - trophic breadth)}!}{(\text{species richness})!(\text{perturbation magnitude - trophic breadth})!} }
where trophic breadth is the number of species consumed by species x_{i}, out of a pool of potential prey (species richness). For example, in a network of 10 species, 1\leq r_{i}^{x}\leq 9 and 0\leq\omega\leq 9, and the probability of extinction increases as \omega\rightarrow \text{S-1}, and decreases as r_{i}^{x}\rightarrow\text{S-1} (see figure). Species with more resources are thus more resistant to topological secondary extinction. This is the same as saying that the more trophically generalized a species, the greater its resistance to extinction.

This explains in part the suggestion that food webs of greater connectance (C) are more resistant to topological secondary extinction (Dunne et al., 2002). Overall food web or network resistance to this type of extinction has been termed structural robustness (Dunne and Williams, 2009). Food web connectance does not increase, however, because of uniform increases in the in-degrees of all species in the network, but increases instead because of the presence of highly linked species. The skewed, long-tailed in-link distributions discussed earlier indicate the non-uniformity of in-degrees within real food webs. The above formula for extinction shows that p(e_{i}^{x})<p>r_{i}^{y}, that is, x_{i} is of greater in-degree than y_{i}. This will also be the case if the species on which x_{i} preys are more resistant to extinction, even if r_{i}^{x}=r_{i}^{y}. The presence of generalist consumers therefore enhances robustness both because of their own greater resistance, and the resistance which they confer upon their consumers.

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