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Columbia River

Understanding the dual role of in-degree on resistance (previous post) makes it possible to examine the structural robustness of an entire metanetwork in response to a specific perturbation. First, the network is perturbed by the removal (extinction) of several species from one or more guilds. If enough species, or the “right” ones are removed, this could in turn cause topological secondary extinctions of species that consume the extinct ones. Whether that actually happens or not depends on the probabilities of extinction. Therefore, to estimate topological secondary extinction in our network, after perturbing it, we have to follow the paths of propagation and estimate the levels of resulting secondary extinction from the probabilities of extinction.

Let guild G_{j} comprise species of different in-degrees, y_{j}, and hence probabilities of secondary extinction. If x_{i} (x_{i}\in G_{i}) is a species that potentially preys upon species in G_{j}, then its probability of extinction, given a measured level of extinction of y_{j}, denoted \psi_{j}^{y}, is
p(e_{i}^{x}\vert \psi_{j}^{y}) = \frac{\psi_{j}^{y}!(\vert y_{j}\vert - r_{ij}^{x})!}{\vert y_{j}\vert! (\psi_{j}^{y}-r_{ij}^{x})!}
Note that this is a simple re-statement of the earlier formula given for the probability of topological secondary extinction, with two differences. First, r_{ij}^{x} is the expected number of x_{i}‘s links that come from y_{j}-type species. It is estimated simply as the proportion of y_{j}-type species out of the total number of species available as prey to x_{i}:
E(r_{ij}^{x}) = a_{ij}\frac{\vert y_{j}\vert}{b_{i}} r_{i}^{x}
\vert y_{j}\vert itself can be estimated from the trophic in-link distribution of guild G_{j} as
\vert y_{j}\vert \approx \lfloor \frac{\int_{r_{j}^{y}-1}^{r_{j}^{y}}P(r_{j})}{\int_{0}^{\vert G_{j}\vert}P(r_{j})} \rfloor
Second, \psi_{j}^{y} is substituted for \omega as a generalization. Whereas \omega referred specifically to the primary perturbation of the network, \psi_{j}^{y} refers to any species removal, including secondary extinction in guild G_{j} caused by perturbation elsewhere in the network.

The total expected level of secondary extinction of species of in-degree x_{i} can now be estimated by applying the above formula to all types of prey species in all guilds with in-links leading to G_{i}.
E(\psi_{i}^{x}) = \vert x_{i}\vert \prod_{j=1}^{\vert U\vert} \prod_{r_{j}^{y}=1}^{b_{j}}p(e_{i}^{x}\vert \psi_{j}^{y}\vert)
Incorporating the estimates of secondary extinction, \psi for each class (in-degree) of species in each guild, we can see how an iterative estimate of secondary extinction can be made for the entire network. Say that the perturbation was a disruption of primary productivity and that guild G_{j} is a guild of primary consumers. Then \psi_{j}^{y} is an estimate of the level of topological secondary extinction of y_{j} species. If guild G_{i} is a guild of secondary consumers, carnivores, with species that prey on those in G_{j}, then we see why topological secondary extinction of species in G_{i}, x_{i}, is a function of \psi_{j}^{y}. The actual order in which the calculations is made is important, because the perturbation will propagate along paths in an order defined and constrained by the metanetwork and food web topologies.

The formulas for p(e_{i}^{x}\vert \psi_{j}^{y}) and E(\psi_{i}^{x}) are significant, because they enable the estimation of the magnitude of extinction in a paleocommunity in response to the extinction of any single species or guild.

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