Properties of Equilibria and Glassy Phases of the Random Lotka-Volterra Model with Demographic Noise

We study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, valid for symmetric interactions, shows that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of locally stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we unveil the presence of a second transition like the so-called "Gardner"transition to a marginally stable phase similar to that observed in the jamming of amorphous materials. We confirm and complement our analytical results by numerical simulations. Furthermore, we extend their relevance by showing that they hold for other interacting random dynamical systems such as the random replicant model. Finally, we discuss their extension to the case of asymmetric couplings.