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Food Web Assembly Rules for Generalized Lotka-Volterra Equations

Fig 1

Food web interaction matrix and application of perfect matching.

a, White (gray) boxes indicate nonzero (zero) matrix elements, orange boxes are unity matrix elements for the primary producers; dark and light blue squares indicate a possible path chosen, allowing to be nonzero. Here, No = n1 + n3 + n5 = 13 and Ne = n2 + n4 + n6 = 12, and Eq 6 is fulfilled with Δ = 1. Inset: Schematic of a possible pairing for the chosen path. Note that the invariance property of was used, yielding only n1 non-vanishing matrix elements in the lower right block (Details: SI). b, Perfect matching [25] applied to simple food webs where competitive exclusion rules out coexistence due to lack of niches (i) and where enough niches are available for coexistence (ii). (iii) and (iv) are two additional examples, where coexistence is ruled out by the assembly rules. In (iii), n1 = 2, n2 = n3 = 1. In (iv), n1 = n3 = n4 = 1, n2 = 2. In both, Δ ≡ NoNe ∉ {0, 1}, see Eq 5.

Fig 1