Generating functional analysis of Lotka-Volterra equations with Hebbian couplings

We study a benchmark model in theoretical ecology for population dynamics, the general- ized Lotka-Volterra equations, for the case of random Hebbian couplings. A set of binary traits describes each species in the ecosystem and we assume the interaction between any two species to be stronger the more traits they share. We use the generating functional method to derive an effective process with the same statistical properties as the Lotka-Volterra dynamics. This effec- tive process is then used to study the resulting dynamically evolved communities, the different phases of the system, and the transitions between them. We check the predictions of the the- ory against numerical simulations. We find that increasing the number of traits leads to reduced community sizes and increases instability.