Phase diagram of random many-species Lotka-Volterra dynamics beyond stability
In plain words
In the standard model of many interacting species with random interaction strengths, weak interactions lead every community to the same single steady state. Beyond a threshold strength the behavior changes, and it is not settled whether the community then has many possible steady states, keeps fluctuating chaotically, or both.
Precise statement
Consider $dN_{i}/dt=N_{i}(1-N_{i}-\operatorname{sum}_{j}\alpha_{ij}N_{j})+\lambda$, $i=1..S$, with $\alpha_{ij}$ of mean $\mu/S$, standard deviation $\sigma/\sqrt{S}$, correlation $\gamma$ between $\alpha_{ij}$ and $\alpha_{ji}$, and immigration $\lambda$. Dynamical mean-field theory gives a unique globally attracting fixed point for $\sigma<\sigma_{c}=\sqrt{2}/(1+\gamma)$ as $S\to\infty$. Classify the phases for $\sigma>\sigma_{c}$ as functions of $\gamma$ in $[-1,1]$ and $\mu$, with $\lambda\to 0$: number and stability of fixed points, presence of chaos or aging, and whether the transitions are sharp. An answer is a phase diagram with order parameters, derived or established numerically at large $S$.
What would settle it
A dynamical mean-field or replica analysis, confirmed by large-$S$ simulations, that gives the phase boundaries and characterizes each phase for all $\gamma$.
Status in the literature
Unverified note
The typical number of equilibria was computed in 2023 and found exponential in $S$; a 2026 preprint distinguishes two kinds of multiple-equilibria phase, and the chaotic region for $\gamma < 1$ is not fully mapped.