, ,

If you’re just jumping into this blog, it will be very important that you read the previous posts in order to understand this one

There are elements or SLNs (species-level food web networks) in S that cannot exist because they possess topologies inconsistent with observed guild species richnesses. We define those SLNs as having probabilities equal to zero, and are therefore interested in the subsets of F, which are those comprising SLNs of probability greater than zero. The arguments on ensemble composition in the earlier post conclude that the members of F cannot be easily enumerated, but the probability of any specific SLN can be determined and if exceeding zero, could therefore have been an actual food web of the paleocommunity. Let species x_{i} (x_{i}\in G_{i}) have r_{i}^{x} in-links, or prey, and topology t_{i}^{x}. The probability of t_{i}^{x}, constrained by the metanetwork topology and guild species-richnesses, is given by the multinomial probability
p(t_{i}^{x}) = \frac{r_{i}^{x}!} {a_{i1}k_{1}!a_{i2}k_{2}!\ldots a_{i\vert U\vert}k_{\vert U\vert}!} p_{1}^{a_{i1}k_{1}}p_{2}^{a_{i2}k_{2}}p_{\vert U\vert}^{a_{i\vert U\vert}k_{\vert U\vert}}
where k_{n} is the number of links from species in guild G_{n} to x_{i}, \sum_{n=1}^{\vert U\vert}k_{n}=r_{i}^{x} , and p_{n} is the probability that a link, drawn randomly between x_{i} and any other species, will connect to a species in G_{n}. Factorial zero is defined conventionally as equal to one. p_{n} is estimated empirically from the data as
p_{n} = \frac{a_{in}\vert G_{n}\vert}{\sum_{j=1}^{\vert U\vert}a_{ij}\vert G_{j}\vert}
The multinomial formula calculates the number of ways in which r links can be arranged among the guilds with k_{1} assigned to G_{1}, k_{2} to G_{2}, and so on, and then multiplies this number by the product of the probabilities, which is equal to the probability of drawing k_{1} links between x_{i} and G_{1}, and k_{2} links between x_{i} and G_{2}, and so on. In order for p(t_{i}^{x} to be greater than zero, no $k_{n} can exceed the number of species in G_{n}. The probability of any SLN may now be defined formally as
p(t_{i}^{x}) = \frac{r_{i}^{x}!} {a_{i1}k_{1}!a_{i2}k_{2}!\ldots a_{i\vert U\vert}k_{\vert U\vert}!} p_{1}^{a_{i1}k_{1}}p_{2}^{a_{i2}k_{2}}p_{\vert U\vert}^{a_{i\vert U\vert}k_{\vert U\vert}} \qquad \mathrm{if}\quad 0\leq k_{n}\leq G_{n} \forall n

It now becomes immediately obvious that among SLNs of probability greater than zero, some are of greater probability than others. Examine the SLNs in the figure, arranged in order of descending probability. p1 is 0.36, p2 is 0.16 and p3 is zero. The most probable food web is therefore one where the predatory species, with two in-links, preys exclusively on species in G2. The third SLN is of probability zero since the metanetwork (earlier post) forbids links between G3 and G4. Most food webs are of course significantly more complicated than the example used here, but the probability of a SLN is simply the product of the topological probabilities of all species in the food web given the metanetwork,
p(\mathrm{SLN}) = \prod_{i=1}^{\vert U\vert}\prod_{n=1}^{\vert G_{n}\vert} p(t_{i}^{n})
Finally, although the set of possible SLNs cannot be delimited analytically, the preceding formula may be used as the basis of a likelihood model, facilitating Markov Chain Monte Carlo exploration of (S,F,P).