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Mixtures of stably coexisting species overyield

Mario DesallaisiD, Michel LoreauiD, Jean‐François ArnoldiiD, Rudolf P. RohriD

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Abstract

Diverse plant communities usually produce more biomass than monocultures, an advantage that strengthens over time and is attributed mainly to the complementarity effect - the extra biomass produced, relative to monocultures, when species exploit different resources, share fewer enemies or facilitate one another. Yet whether this advantage is a mere tendency of diverse communities, or something that every persistent community must have, has remained unknown. Here we show that stable coexistence requires a positive complementarity effect, with vanishingly rare exceptions, and that underyielding communities are significantly more likely to lose a species within two years or sampling steps, across four long-term biodiversity experiments, with odds of losing a species nearly three times higher than in other communities. In the twenty-five years since the complementarity effect was defined, the link between biodiversity effects and species coexistence had only been established theoretically for two species in competition or observed as a correlation; our theoretical result makes it a general property of multispecies communities, in which the complementarity effect scales with two established measures of how securely species coexist - local and structural stability. Our analyses are, to our knowledge, the first to test a coexistence prediction - impending species loss - prospectively in biodiversity experiments, from a functioning measure alone, turning a measure of what diversity does into a diagnostic of whether diversity can persist. Stable coexistence and higher productivity are thus intrinsically linked: biodiversity does not merely benefit ecosystem production on average; biodiversity that endures cannot but benefit it through complementarity. We anticipate that this diagnostic will turn existing biodiversity experiments, sown grasslands and intercrops into early-warning systems for species loss, since it requires only the biomass of each species in mixture and in monoculture - data these systems already collect, and that agronomists already combine into the land equivalent ratio - and that it will extend to forests, through the growing networks of tree diversity experiments and mixed-species plantations where the same measurements are made.

In-depth summary

Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (27 Oct 2026) · CC BY 4.0

In generalized Lotka–Volterra communities, stable coexistence almost always requires overyielding; in four long-term biodiversity experiments, underyielding raised species-loss odds within two observations (pooled odds ratio 2.61).

Biodiversity–ecosystem functioning research and coexistence theory have developed largely apart, and a link between the complementarity effect and coexistence had been shown only for two competing species, particular community structures, or as correlations in grassland experiments. This work extends the link to multispecies Lotka–Volterra communities, ties the size of complementarity to stability and feasibility, and tests prospectively whether underyielding precedes species loss.

Key results

  1. ShownIn generalized Lotka–Volterra communities where every species persists alone and no pair's combined competition exceeds the bound aij + aji > −2, any feasible equilibrium overyields (RYT > 1, positive complementarity effect); with equal growth rates, any stable feasible equilibrium overyields regardless of the bound.

    Analytical proof of Theorem 1 (SI S3.2); with equal growth rates RYT is tied to the sum of Jacobian eigenvalues. In S = 10 simulations, communities meeting the bound never fall below RYT = 1, and violators are overwhelmingly unstable. In the Adler et al. compilation of measured plant–plant interactions, 88% of all pairs and 80% of strictly competitive pairs satisfy the bound. · Stable coexistence leads to overyielding in…; Community assembly, a filter that retains only…; Figure legends

  2. SuggestedHigher RYT goes with a more negative leading eigenvalue λmax (local stability) and a wider feasibility domain Ω. For two species, the λmax link is exact at equal growth rates and an average trend otherwise, and a first-order Ω curve bounds it from below; simulated 10-species communities show the same trend.

    Two species: λmax follows RYT along one line (exact at equal growth rates, average trend for ri ∈ [1/3,3]); Ω widens monotonically with RYT above 1, with an analytical lower bound. At S = 10: Spearman ρS = −0.83 (λmax vs RYT) and 0.94 (Ω vs RYT), correlations descriptive and not tested; robustness across S ∈ {3,5,10,15} and four interaction structures is in SI. · The complementarity effect measures how securely…; Stable coexistence leads to overyielding in…; Statistical analysis; Figure legends

  3. SuggestedAway from equilibrium, total yield settles much faster than composition shifts, so a transient RYT reports the current interactions; in simulations, a community that underyields almost always loses a species, with compositional recovery or stable underyielding as rare exceptions.

    At S = 4 (Fig. 3D), 4,410 communities (23%) underyielded; 4,398 (99.7%) lost a species and seven (0.16%) recovered with all species kept. None of the 2,601 communities whose RYT fell below 0.9 recovered intact. Time-scale separation held across growth-rate heterogeneity, heterogeneous interactions and multispecies communities (Fig. S7). · Away from equilibrium, underyielding forecasts…; Non-equilibrium analysis; Figure legends

  4. ShownIn four long-term experiments (protist microcosms, Cedar Creek E120, the Jena Dominance Experiment and the Agrodiversity Experiment), underyielding at two consecutive observations raised the odds that a species was persistently lost within the next two observations.

    Pooled random-effects odds ratio 2.61 (95% CI [1.40, 4.87]) at the two-observation horizon; 2.29 one and 2.31 three observations ahead. Largest in the protist microcosms (odds close to six-fold), clear in the Agrodiversity Experiment, same direction in both grasslands; cluster-robust GEE estimates give the same pooled conclusions. · Underyielding precedes species loss in long-term…; Statistical analysis; Figure legends

  5. SuggestedHigher RYT lowered the odds of loss in every experiment, but underyielding predicted a given species' loss beyond its own rarity only in the recently assembled protist and agricultural systems, not in the two mature grasslands; the authors attribute this to assembly filtering.

    Slope of log-odds per unit ln RYT negative in all four (Agrodiversity β = −0.73, P < 0.001); pooled β = −0.41 [−0.84, +0.02], not significant. Against a rarity-only null (10,000 simulated datasets), the species-level odds ratio exceeded the null 95th percentile in the protist microcosms and the Agrodiversity Experiment only. · Underyielding precedes species loss in long-term…; Community assembly, a filter that retains only…; Statistical analysis

Methods, data and reproducibility

Generalized Lotka–Volterra theory and simulations relate relative yield total (RYT > 1 means a positive complementarity effect) to feasibility, feasibility-domain size Ω and leading Jacobian eigenvalue λmax. Underyielding was tested as a predictor of species loss in four long-term experiments via Firth logistic regression, generalized estimating equations, a rarity-only null and random-effects meta-analysis.

  • Pennekamp et al. protist microcosms: six bacterivorous species, six temperatures from 15 to 25 °C, 19 observations over 40 days (Zenodo, CC0) · protists · n = 580 mixture microcosms
  • Cedar Creek E120: 16-species perennial-prairie pool, annual species-sorted sampling 2001–2021 except 2009 (Environmental Data Initiative) · plants (perennial prairie species) · n = 56 mixture plots
  • Jena Experiment, Dominance Experiment: species-resolved grassland biomass (PANGAEA) · plants (grassland) · n = 188 plots
  • Agrodiversity Experiment: grass–legume mixtures at 28 of 33 sites (27 in Europe, one in Canada), multiple harvests per year (Ecological Archives) · plants (grass and legume species) · n = 796 plots
  • Adler et al. compilation of pairwise plant competition coefficients (Dryad, CC0), used to check the interaction bound · plants
  • Simulated generalized Lotka–Volterra communities (Fig. 2 at S = 10; robustness over S ∈ {3,5,10,15} and four interaction structures; Fig. 3D at S = 3, 4, 6, 8)
Methods & tools
generalized Lotka–Volterra model, Firth penalized logistic regression, generalized estimating equations (GEE), random-effects meta-analysis (DerSimonian–Laird, Hartung–Knapp–Sidik–Jonkman), Python 3.13, numpy, pandas, scipy, statsmodels, deSolve (R), mvtnorm (R), CVXR (R)
Software & environment
Python 3.13 with numpy 2.4.4, pandas 3.0.2, scipy 1.18.1, statsmodels 0.14.6; Firth regression and meta-analysis implemented by the authors; Fig. 2 simulations in R (deSolve, mvtnorm, CVXR; versions not stated). Code under MIT licence.
Reproducibility gaps
Code on Zenodo becomes public only on publication (printed link is reviewer access). Simulations for Fig. 2 and Figs. S4–S5 had no fixed seed; outputs are archived as data. Source datasets are named by repository only, without accession numbers.
Limitations (6)
  • Inherits generalized Lotka–Volterra assumptions such as linear per capita interactions; in two species, non-linear functional responses are already known to break the overyielding–coexistence link. · stated by the authors
  • Relative yields are defined for a single trophic level, with no equivalent for multitrophic food webs; obligate predator–prey interactions fall outside the model. · stated by the authors
  • Requires contemporaneous monoculture yields, so the diagnostic cannot be applied to the many communities observed without monocultures. · stated by the authors
  • The theorem runs one way: overyielding does not guarantee stability (coalitions of mutually positive species can destabilize), and stable underyielding or compositional recovery can give false positives, though simulations find them rare. · stated by the authors
  • Pooling rests on four experiments (t distribution with 3 degrees of freedom) with no multiple-comparison correction, and the pooled continuous slope's interval includes zero. · noted in this summary
  • In the two mature grasslands the species-level estimate did not exceed the 95th percentile of the rarity-only null, so underyielding was not shown to add information beyond the lost species' own rarity; the assembly-filter explanation offered for this weaker signal is not tested independently. · noted in this summary

Read this if you work on biodiversity–ecosystem functioning or coexistence theory and want a Lotka–Volterra argument, tested prospectively in protist, grassland and agricultural experiments, that underyielding mixtures are headed for species loss.

Citations
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