Fixed-Point Abundances and the Onset of Extinctions in Multispecies Lotka-Volterra Ecosystems

Abstract: I investigate the average and variance of species abundance distributions for fixed-point solutions to generalized Lotka-Volterra ecosystems. Two different analytical methods are constructed. First, I build up on the perturbative Random Matrix Theory approach developed in my earlier work [Phys. Rev. E. 113, L062202 (2026)], and formalize this method diagrammatically. The abundance average is given by a sum of planar diagrams with non-crossing vertex pairings which can be encoded as Dyck paths. This allows me to express the combinatorial factor multiplying perturbative terms at each order as Catalan and Narayana numbers, following which the abundance average is given by the latter's generating functions. In this way, the infinite perturbation expansion is exactly resummed in a closed-form formula. I have not been able to find a similar resummation for the abundance variance, however diagrammatic Feynman rules are expressed, following which the perturbative series can be calculated to any desirable order. Second, I do the same calculations using the cavity method, which gives closed-form formulas for both the abundance average and variance. For the average, the cavity formula gives the exact same result as the perturbative approach when the average interspecies interaction vanishes. When this is not the case, the two results are functionally different, however they are hardly distinguishable in their parametric range of validity. Comparison with numerical data indicate that the cavity result is more accurate over a larger parameter range.