Fig 1.
Conceptual map of covariance-testing.
(A) The dynamics of a 15-species neutral community of 10,000 individuals and migration probability m = 0.0002 (shown here) can be approximated by a WFP with λ = 20 (see eq 1). If the community is neutral, then the CVTs should yield homoskedastic plots of νt versus ft. We test neutrality by randomly drawing from the 2n possible CVTs, performing homoskedasticity tests on νt versus ft, and then testing the uniformity of the resulting P-value distribution using a modified KS-test (see details in S1 Text, part 3). (B) The relative abundances of 15 independent, mean-reverting geometric brownian motions, d log Xt = μ(b − log Xt)dt + σdWt with μ = 15, σ = 30, b = 10. Neutrality is rejected by the highly non-uniform distribution of P-values. The left-skewed P-value distribution indicates many CVTs had volatilities that depended on the state variable, ft.
Fig 2.
Testing a volatility-stabilized market model (VSM).
Pal (2011) proved that the relative abundances / market shares in the VSM follow a WFP, i.e. are neutral. (A) Community dynamics under a volatility-stabilized market model in eq (7) over the time window t ∈ [0, 1], with n = 50 species and δ = 0.02. (B) The scaling of the variance of population sizes, Var[Nt], against their mean, E[Nt], over the time window. A common assumption is that neutrality can only yield a linear relationship bewteen the mean and variance, yet the VSM provides a clear counter-example. (C) The scaling of DΔt vs. Δt for different initial population sizes in the VSM. A common assumption is that the dependence of the slope of Var[DΔt] vs Δt on the initial population size rejects neutrality, yet the VSM provides a clear counter-example. (D) While the existing methods suggest the VSM is not neutral, despite proven otherwise, applying our test to the VSM correctly reveals that the VSM simulated here is very close to neutrality.
Fig 3.
Applying our test to time-series datasets reveal non-neutral competitive dynamics in microbial and financial systems.
P-values displayed are from a modified KS-test which accounts for dependence among the observations and these P-values serve as a quantification of the incompatibility of the data with neutral fluctuations. Despite decent fits of neutral species-abundance distributions, our time-series test reveals that competitive asymmetries are important drivers in all systems except the female palm bacteria, and the extent of non-neutrality varies from system to system.
Fig 4.
Investigating the competitive asymmetry (A-D).
Analyzing the non-neutrality of competitive systems (A) The negative relationship between νt vs. ft indicates mean reversion. Overlaying νt vs. ft scatter plot from a particular CVT from the male tongue data onto the results from 4,000 WFP trajectories with long sampling intervals, Δt, shows that mean reversion can be accounted for by sparse time-sampling of the data. (B) However, even when correcting for sparse time-sampling, the left-skewed P-value distribution in the male tongue indicates stronger signal of non-neutral volatility than 16,000 surrogate WFPs. (C) The parameter β2 from significantly (P < 0.001 for male tongue, P < 0.01 for surrogate data) heteroskedastic auxiliary regressions in eq 8 reveals significantly more β2 > 0 than β2 < 0 in the data. The different P-value cutoffs are for visualization—the same bias for β2 > 0 holds for a standard cutoff of P < 0.01 (D) Overlaying scatterplots of the residuals, , from all heteroskedastic cases (P < 0.01) of the male tongue data, reveals the empirical pattern of heteroskedasticity. Compared to surrogate neutral data, the male tongue is more volatile when the groupings are uneven, suggesting that either rare or abundant groups are more volatile—or equal groupings are relatively less volatile—than neutrality would predict. (E) All datasets have the same over-abundance of β2 > 0 for heteroskedastic (P < 0.05) CVTs.