{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"ad2e273a7fdd8a7702347e50fcdfb413d7b5b3ef60265e01ce7698fa2fdb7b97","created":"2026-10-03T07:17:57Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"9fb3cd3e180f7719c7f6101e028668f586212c8f5ee8c78e66365076b9c66717","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"classification","assisted_by":[],"external_id":"bio.ecosystem-statphys.lv-phase-diagram","kind":"well-posed","literature_status":"partially-resolved","n":"1","parents":[],"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.","posed_since":"","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 $S$.","problem_ref":null,"references":"","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 $\\gamma$.","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.","title":"Phase diagram of random many-species Lotka-Volterra dynamics beyond 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