Source
Blumenthal, Rocks and Mehta (2024), LIT-1236, read in NOTE-1366; the stability test is the one Bunin (2017), LIT-1234, read in NOTE-1397, uses for random Lotka–Volterra communities.
What was actually shown
The model: S species and M logistic resources, dNᵢ/dt = Nᵢ(Σ_α cᵢα R_α − mᵢ), dR_α/dt = R_α(K_α − R_α) − Σᵢ Nᵢ eᵢα R_α, with random cᵢα and eᵢα of mean µ/M and fluctuations σ/√M correlated by ρ. When e = c (ρ = 1) the effective species interactions Aᵢⱼ = Σ_α cᵢα eⱼα Θ(R_α) are symmetric and the steady state minimizes an objective.
In the limit S, M → ∞ with M/S fixed, the cavity method under a replica-symmetric, self-averaging ansatz gives self-consistent equations for the surviving fractions φ_N, φ_R and the susceptibilities χ (resources to carrying capacity) and ν (species to mortality). Perturbing every survivor by independent noise, the second moments x of the species response and y of the resource response satisfy x = a(y + 1), y = b(x + 1), with a = φ_R/(σρχ)² and b = σ²γ⁻¹φ_N/D², D = 1 − ρσ²γ⁻¹ν. Both diverge when ab = 1, i.e. (ρχD)² = γ⁻¹φ_Nφ_R. The cavity equation for a resource gives χ = φ_R/D, so the condition is ρ²φ_R² = γ⁻¹φ_Nφ_R, ρ*² = γ⁻¹φ_N/φ_R = (S φ_N)/(M φ_R): the number of surviving species over the number of non-depleted resources. Beyond it the variances are negative, so the replica-symmetric fixed point does not exist as a stable state.
What could have come out otherwise: the boundary could have depended on σ, µ or the carrying capacities separately; it depends on them only through the packing ratio. And simulations could have kept reaching steady states below ρ*; in the paper’s runs (S = M up to 1024) they do not, with positive largest Lyapunov exponents, and a finite-size fit at two parameter points puts the probability of reaching steady state towards zero with size.
What this does not say
- Not that non-reciprocity alone ends the optimization picture. Any ρ < 1 removes the symmetric objective, but the fixed point stays stable down to ρ*. A sparsely packed community (S*/M* small) tolerates strong asymmetry; one near the exclusion bound tolerates little.
- Not a derivation of chaos. The cavity method locates the loss of stability; that the other side is chaotic, rather than multistable or periodic, is numerical, at moderate sizes and few parameter points.
- Not beyond the ansatz. Replica symmetry is assumed and checked against simulations, not proved; no symmetry-broken solution is given.
- Not for structured interactions. The interactions are random and unstructured, the resources self-replenishing, one consumer level.
- Not that symmetry guarantees a unique outcome in general. In random Lotka–Volterra communities (LIT-1234) a multiple-attractor phase exists even for symmetric interactions; what keeps the symmetric consumer–resource model unique is that its interaction matrix is a product, as in CLAIM-369.
Bearing
It sharpens the reading the synthesis gave LIT-1236 for QUESTION-083: the limit of reading a community of competing processes as one optimization is a boundary in reciprocity that depends on how many processes each shared resource must support, not a switch that any asymmetry throws.