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One possible food web for Late Miocene shallow marine community, Dominican Republic.

The metanetwork is a higher guild-level representation of the community food web or trophic network, incorporating uncertainty and variability of the species-level links. The number of species-level networks that can be derived from the network is finite, but large. This species-level network space encompasses all the possible configurations and behaviours that can be exhibited by the community. An interesting excercise is to calculate the size of the space under particular parameterizations of the metanetwork. Let’s do that for the Late Miocene community whose metanetwork is described here. An example of one of the species-level food webs is illustrated in this figure. Okay, here we go…

A metanetwork U is a set of guilds, $\{ G_{i},G_{2}\ldots ,G_{\vert U \vert}\}$. $\vert U\vert$ represents the power of U, or the number of guilds, and $\vert G_{i}\vert$ is the species-richness or number of species in guild $G_{i}$. The number of potential prey species for each species in $G_{i}$ is the sum of the species-richnesses of all guilds designated by the topology of U as prey to $G_{i}$. This number is calculated as

$\vert R_{i}\vert = \sum_{j=1}^{\vert U\vert} a_{ij}\vert G_{j}\vert$

where $\vert R_{i}\vert$ is the number of potential prey species and $a_{ij}$ is the $ij^{th}$ element of U’s binary adjacency or connectivity matrix. The order ij designates i as a predator of j, and $a_{ij}=$1 if this is true, and zero otherwise.

If for simplicity we assume that all species in the community are extreme specialists, that is, they consume specifically a single resource, then the number of different species link configurations possible in $G_{i}$ is $\vert R_{i}\vert ^{\vert G_{i}\vert}$. Therefore in U there are exactly

$\prod_{i=1}^{\vert U\vert} \vert R_{i}\vert ^{\vert G_{i}\vert}$

possible species-level networks. Given that links within each guild are described, however, by a trophic link distribution, and that species within a guild are expected to differ in their dietary breadths, the expected (average) number of species-level networks is found by first calculating the number of possible ways in which the links of a single species may be arranged among potential prey nodes, and the applying that to the above product.

$\prod_{i=1}^{\vert U\vert} \prod_{r_{x}=1}^{\vert R_{i}\vert} \left( \begin{array}{c} \vert R_{i}\vert \\ r_{x} \end{array} \right) ^{\vert x_{i}\vert}$

where $\vert x_{i}\vert$ is the number of species in $G_{i}$ of in-degree (with number of prey) $r_{x}$. (The formula for calculating $\vert x_{i}\vert$ was given in a previous post.)