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

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Title

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

Plain

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

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 $S$.

Settled by

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

Kind

well-posed

Answer type

classification

References

(empty)

Literature status

partially-resolved

Status 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.

Posed since

(empty)