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.

Figure 1

Phase diagram showing the strength of the demographic noise $T$ as a function of the degree of heterogeneity $\sigma$ at fixed $\mu =10$ and selected value of the cutoff $N_c={10}^{-2}$. Upon decreasing the noise, three different phases can be detected: (i) a single equilibrium phase, (ii) a multiple equilibria regime between the light blue and the orange lines, and (iii) a Gardner phase, which turns out to be characterized by a hierarchical organization of the equilibria in the free-energy landscape.

Figure 2

Cartoon of the landscape in the 1RSB ansatz. The parameters $q_0$ and $q_1$, which can be exactly obtained by solving saddle-point equations, are respectively associated with the size of the largest basin and the innermost basin (large overlaps are associated with small basins).

Figure 3

Correlation $C(t,t^{\prime })$ as a function of $t-t^{\prime }$ in the one-equilibrium phase for different $t^{\prime }$: $t^{\prime }=1$ (blue), 10 (orange), 100 (green), 500 (red). The inset shows the decorrelation time versus $(T-T_{1\mathrm{RSB}})$ in log-log scale. The blue points correspond to numerical data, while the dashed red line is a fit.

Figure 4

Rescaled correlation as a function of $(t-t^{\prime })$, for different age $t^{\prime }$ of the system, showing aging dynamics. The dashed black and red lines correspond respectively to the theoretical predictions for $q_1$ and $q_0$, both rescaled by the analytical prediction $C(t^{\prime },t^{\prime })\equiv q_d$.