Generalized Lotka-Volterra Equations with Random, Nonreciprocal Interactions: The Typical Number of Equilibria

We compute the typical number of equilibria of the generalized Lotka-Volterra equations describing species-rich ecosystems with random, nonreciprocal interactions using the replicated Kac-Rice method. We characterize the multiple-equilibria phase by determining the average abundance and similarity between equilibria as a function of their diversity (i.e., of the number of coexisting species) and of the variability of the interactions. We show that linearly unstable equilibria are dominant, and that the typical number of equilibria differs with respect to the average number.

Figure 1

Quenched complexity $\mathrm{\Sigma }(\varphi ,\sigma )$ of uninvadable equilibria for uncorrelated interactions ($\gamma =0$). Black lines correspond to vanishing complexity; the green dotted line to the diversity ${\varphi }_{\text{May}}(\sigma )$ above which equilibria are linearly unstable (red area); the orange dotted line to the transition between the unique ($\sigma <{\sigma }_c$) and the multiple ($\sigma >{\sigma }_c$) equilibria phases.

Figure 2

Complexity of equilibria as a function of their diversity, for $\gamma =0$. Main panel: Complexity in the multiple equilibria phase (at $\sigma =4$). A difference between quenched (magenta) and annealed (blue) is apparent. All the equilibria are unstable ($\varphi >{\varphi }_{\text{May}}$). Inset: Annealed complexity in the unique equilibrium phase (at $\sigma =1$), negative except at the diversity predicted by the cavity formalism consistent with the existence of a unique equilibrium.

Figure 3

Typical averaged population size as a function of diversity $\varphi$ for $\sigma =4$ and $\mu =30$, in the annealed (blue) and quenched (magenta) calculation. More diverse equilibria have a smaller averaged population size $m$, which for $\varphi >{\varphi }_{\mathrm{cav}}$ is underestimated by the annealed approximation. The inset is a enlarged plot.

Figure 4

Diversity vs variability diagram. The range of possible diversities is indicated by the gray region. Curves of maximal complexity are shown in magenta (quenched) and blue (annealed). The black squares give ${\varphi }_{\mathrm{cav}}$. The orange dashed line corresponds to ${\varphi }_{\text{May}}$ above which all equilibria are linearly unstable. Inset: Enlargement in the vicinity of ${\sigma }_c=\sqrt{2}$.