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Simple trophic network

I’ve run into a curious dilemma regarding fractional trophic level. Fractional trophic level, or ftl, measures the trophic level of a species or food web node based on the ftl of its prey. The concept dates back to a paper by Odum and Heald (1975). You begin by assigning a ftl of 1 to all primary producers. Subsequent ftl is then 1 plus the mean ftl of all your prey.
\mathrm{ftl} = 1 + \frac{1}{n}\sum_{i=1}^{n}\mathrm{ftl}_{i}
Therefore, a primary consumer will have a ftl of 2. That’s very nice and intuitive. Beyond that, ftl measures can be continuous numbers, which really captures a lot of the complexity, e.g. omnivory or feeding at multiple trophic levels that are characteristic of real food webs. A slightly different way of calculating ftl, but one that remains true to the concept, is given by Pauly as
\mathrm{ftl} = 1 + \sum_{i=1}^{n}\frac{\mathrm{ftl}_{i}}{i^{*}}
where i^{*} is the fraction of prey i in the consumer’s diet. Whereas the earlier formula is simply the mean ftl of prey, this is a weighted mean.

Okay, my question. How does one handle recursive situations? By this, I refer specifically to instances of inter-node predation? Take a look at the simple food web in the figure. ftl can be calculated easily for nodes 1 and 2, and either 3 or 4. But the calculations for 3 and 4 include the ftls of each other. This is a classic recursive or coupled equation situation. But trophic level really should be a fixed, stationary number in a food web network, changing only if topology (or other parameters if included), change. I think that I have a solution suitable for my purposes of a trophic network model. But any thoughts, ideas, solutions, or pointers to solutions that already exist and which I have missed, are most welcome! Leave a comment.

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