Ecological stationary currents

### PDF page 26 19. Conclusion Relative abundance data are bounded compositions, and closure-preserving change becomes additive in log-ratio coordinates, where neutral processes are diffusion and niche processes are drift. Niche drift has two parts. The gradient part — stabilising coexistence and habitat filtering — reshapes the abundance distribution and is already detectable in it. The circulating part — intransitive and cyclic dynamics — moves probability in closed loops, is mathematically invisible to any abundance distribution, and reveals itself only in time-ordered observables, as an irreversible current with a positive entropy-production rate. The niche–neutral split is therefore partly a non- identifiability: a stationary distribution fixes a density but not a current. The framework turns the debate into a two-axis regime map and supplies a covariate-free diagnostic — the irreversibility of log-ratio trajectories — for finding the niche structure the histogram cannot see. Seen this way, the niche signal is the symmetry-irreducible, cyclic residual of the community’s probability current: the part that no change of frame, no waiting, no environmental conditioning, and no relabeling of species can remove. Detecting it therefore rests on a discrete invariant rather than a bias-prone rate — which is what makes the programme defensible. In one line: abundance distributions are projections of community dynamics that preserve density and erase current; gradient niche structure is visible in the density, cyclic niche structure lives in the erased current, and the missing coordinate is the arrow of time. ### PDF page 27 Appendix A. Proofs A.1 Theorem 1 Positivity of T_a(p)ᵢ follows from positivity of pᵢ and exp(aᵢ). Closure follows because the denominator is the sum of all positive weighted components, so Σᵢ T_a(p)ᵢ = 1. For composition, applying exp(aᵢ) then exp(bᵢ) multiplies pᵢ by exp(aᵢ + bᵢ); the renormalisations collapse into one, and any constant added to every aᵢ cancels in the ratio, giving T_b T_a = T_(a+b) modulo a∘ common additive constant. For the additive update, write zᵢ′ = log(pᵢ′ / p_S′); the common denominator cancels, leaving zᵢ′ = zᵢ + (aᵢ − a_S). ∎ A.2 Theorem 2 (non-identifiability) Let z follow dz = b dt + Σ^(1/2) dW with constant diffusion D = ½ Σ and smooth, positive stationary density ρ. The stationary Fokker–Planck equation is ·J = 0 with current J = b ρ − D ρ, and the∇ ∇ stationary velocity is v = J/ρ. Define b_DB = D log ρ, the unique gradient drift whose current∇ vanishes identically (J = 0), giving detailed balance. Any admissible drift may be written b = b_DB + u, where u = v is the residual; stationarity ·J = 0 is then equivalent to the divergence-free∇ condition ·(u ρ) = 0. Conversely, adding to b any field u with ·(u ρ) = 0 leaves the stationary∇ ∇ equation — and hence ρ — unchanged. Therefore the set of drifts consistent with a fixed (ρ, D) is the affine family b_DB + { u : ·(u ρ) = 0 }, and every functional of the unordered marginal of p,∇ which depends on ρ and D only, is constant across this family. Two processes sharing (ρ, D) thus share an identical abundance distribution for arbitrary admissible u. Detailed balance holds iff u ≡ 0; otherwise the steady-state entropy-production rate is strictly positive, σ = ∫ v(z)ᵀ D ¹ v(z) ρ(z) dz > 0,⁻ so the circulating component u — the niche structure invisible to the abundance distribution — is exactly the component that makes the log-ratio trajectory irreversible. (For state-dependent D the same decomposition holds with the current J = bρ − ·(Dρ); the gradient/divergence-free split is∇ then taken in the metric set by D.) ∎ A.3 Theorem 3 (current decomposition) By the Helmholtz decomposition any sufficiently regular field on the open, simply connected log- ratio domain writes uniquely as J = ψ + J_circ with ·J_circ = 0; at stationarity ·J = 0 makes ψ∇ ∇ ∇ harmonic, and with the decay (finite-energy/normalizability) condition that ρ and J vanish at infinity the harmonic part vanishes, so ψ = 0 and a steady-state current is purely circulatory and any∇ divergent part signals non-stationarity (transient relaxation). The continuity condition ·J = 0 is∇ here the metric-free statement of stationarity; the diffusion metric enters the Section 10 split only through the detailed-balance drift b_DB = D log ρ and the rate σ, not through this divergence. For∇ a neutral process the generator commutes with the species-permutation group G, so averaging J over G (the Reynolds projection) returns J in distribution; a neutral current therefore lies in the G- invariant (exchangeable) subspace J_circ^sym and its species-specific complement J_circ^asym vanishes. Equivalently, a species-label permutation is a measure-preserving symmetry of the neutral law, so the label-permutation surrogate has the same expected entropy production as the data; any significant positive excess rejects exchangeability and, restricted to stationary windows and homogeneous strata, isolates J_circ^asym. Finally, J = bρ − ·(Dρ) is the unique current∇ whose vanishing is equivalent to detailed balance for state-dependent D, so assessing reversibility against it removes the multiplicative-noise (Itô) artifact. ∎ ---

coordinates already require (Section 3). The calibrated null. Entropy-production estimators over-report irreversibility in finite, high- dimensional samples, and the bias points toward the hypothesis. Neutrality, however, is by definition species-exchangeable: its generator commutes with permutations of species labels, so a label-permutation surrogate is a valid reversible-or-exchangeable reference carrying the identical dimensionality and sample size. The bias therefore enters the observed statistic and the surrogate equally and cancels in their contrast; significance is read from the permutation

mapped individuals, taxonomic resolution, and spatial structure (Condit et al. 2019), and lets the same system be analysed both as an abundance distribution and as a temporal–spatial trajectory. The decisive result the framework predicts is a split at the same site: neutral-looking abundance distributions accompanied by significant irreversibility in log-ratio trajectories. Such a split would be the predicted geometric signature, not a contradiction. A natural objection is that eight censuses over three decades are too sparse to estimate a current in a high-dimensional log-ratio space. But estimating a current does not require one long trajectory. The stationary current is a local property of the increment distribution conditioned on state, so it can be estimated from an ensemble of many short transitions. A mapped forest with

either way, but the probability-current reading is a modelling choice here, and the recovered 2- cycle is trophic predator–prey coupling, not within-guild intransitive competition. 14. Predictions and how to falsify the framework The framework should not be judged by whether a neutral model fits an abundance distribution — that is precisely the projection in which circulating niche structure hides. It should be judged by whether structured, irreversible drift appears when the same community is analysed as a bounded trajectory. It is falsified if log-ratio trajectories are statistically reversible and increments exchangeable wherever abundance distributions look neutral, across scales and groupings — in

particular, if communities with independently documented intransitive interactions show no irreversibility (Table 4). Table 4 Predictions of the framework and their operational tests. Prediction Operational test Expected result P1. A neutral histogram can hide a niche current. Fit neutral models to abundance distributions, then test log-ratio increments for irreversibility. the covariant current exceeds the label-permutation surrogate. P2. Neutrality strengthens within guilds. Repeat within guilds, trait clusters, or habitat classes. Within-guild Péclet and entropy production fall relative to cross- guild contrasts. P3. Gradient niche tracks gradients. Estimate drift across topography, soil, moisture, light. Reversible gradient drift aligns with environment and shows in the distribution. P4. Scale controls the answer. Vary quadrat size and census interval. Detectability of niche drift rises with interval like √Δt. P5. Projection determines interpretation. Compare unordered summaries with ordered trajectory models. The same community supports neutral- and niche-looking claims by projection. P6. Intransitivity implies irreversibility. Compare communities with documented intransitive vs hierarchical interactions. Intransitive communities show structure in the histogram with low irreversibility. P7. The niche current is cyclic. Estimate loop circulation (curl) of the current within habitat strata. Intransitive communities show

and with how much irreversibility? The strongest version of the argument is empirical. The decisive test is to take a canonical dataset such as Barro Colorado Island — using its spatial replication to estimate currents — and demonstrate the predicted pattern: neutral-looking abundance distributions together with significant, structured irreversibility in temporal–spatial log- ratio trajectories, ideally corroborated in a denser time series. If that pattern appears, the contradiction between neutral-looking histograms and non-neutral structure stops being a paradox and becomes a measurement.