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Let’s apply some network thinking to this problem now. First, the network, or food web, is perturbed by the removal (extinction) of several nodes from one or more guild. If enough nodes, or the “right” ones are removed, this could in turn cause 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 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. The formula for those probabilities was given in an earlier post. Here we restate it in the following framework.

Guild G_{j} comprises species of different in-degrees, y_{j} and hence probabilities of secondary extinction. If x_{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_{yj}, is

\mathrm{pr}(e, x_{i}|\psi_{yj}) = \psi_{yj}!\left( |y_{j}|-r_{xy}\right) ! \left[ |y_{j}|!\left( \psi_{yj}-r_{xy}\right) !\right]^{-1}

where

\mathrm{E}(r_{xy}) = \frac{|y_{j}|}{b_{i}}r_{x}a_{ij}

is the expected number of x_{i}‘s links which come from y_{j}-type species. The total expected level of secondary extinction of species of in-degree x_{i} is therefore

\mathrm{E}(\psi_{xi}) = |x_{i}|\prod_{j=1}^{|U|}\prod_{r_{y}=1}^{b_{j}} \mathrm{pr}(e, x_{i}|\psi_{yj})

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_{yj} 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_{yj}.

The actual order in which the calculations is made is important, because the perturbation will propagate along paths in an order defined by the metanetwork and food web topologies. I’ll cover the determination of that ordering in the next post.

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