Phase diagram of random many-species Lotka-Volterra dynamics beyond stability

Consider $dN_{i}/dt=N_{i}(1-N_{i}-\sum_{j}\alpha_{ij}N_{j})+\lambda$, $i=1..S$, 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 $S$.

A dynamical mean-field or replica analysis, confirmed by large-$S$ simulations, that gives the phase boundaries and characterizes each phase for all $\gamma$.