Can high-diversity chaos persist in a closed community without immigration?
In plain words
Models show that many species can coexist if their populations keep fluctuating chaotically, but each crash brings some species close to zero, where they can die out. It is unknown whether a closed community with no new arrivals keeps its diversity or slowly loses species until it settles into a simple steady state.
Precise statement
In random generalized Lotka-Volterra or consumer-resource models with $\lambda=0$ (no immigration) and discrete individuals or an extinction threshold $N_c$, chaotic phases drive rare species to abundances that decrease with $S$. Determine whether a fluctuating state with extensive diversity (surviving fraction of order 1 as $S\to\infty$) persists for times growing exponentially in $S$, or whether extinctions accumulate until the community reaches a stable fixed point. An answer is a yes or no with the scaling of lifetime and survivor number with $S$ and $N_c$.
What would settle it
An analytic treatment of the extinction-limited chaotic dynamics, checked by simulations over a range of S large enough to fix the scaling.
Status in the literature
Unverified note
Theory in 2020 and 2024 found that spatial structure or immigration from a reservoir sustains chaotic diversity; the closed well-mixed case is unsettled.