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REVIEW 3 major objections 4 minor 59 references

Emergent self-inhibition governs the landscape of stable states in complex ecosystems

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read In the random Generalized Lotka–Volterra model, the likelihood of reaching a stable ecosystem state rises sharply with that state's total biomass, and the entire landscape of state likelihoods can be predicted from macroscopic observables a

desk verdict A genuinely derived biomass–likelihood relation in random GLV, with an honest but uncontrolled mean-field step that should be flagged in the abstract and checked against the χ>0 condition. read the letter →

arxiv 2511.06697 v2 pith:RPTK6OIM submitted 2025-11-10 q-bio.PE cond-mat.dis-nncond-mat.stat-mech

Nitesh Kumar Patro , Washington Taylor , Akshit Goyal This is my paper · ORCID
classification q-bio.PEcond-mat.dis-nncond-mat.stat-mech MSC 92D2592D4060G70
keywords generalizedLotka-Volterramultistabilityattractorbasinsstatelikelihoodbiomassemergentself-inhibitioncoarse-grainingrandominteractionmatrices
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks why some stable states of a species-rich ecosystem are reached far more often than others. In the random Generalized Lotka–Volterra model, where the same species pool can settle into many alternative states, the authors show that a state's likelihood increases sharply with its total biomass, often by orders of magnitude. They explain this by coarse-graining each stable state into a single effective unit whose self-inhibition is the inverse of its biomass, and derive an analytic formula for state likelihoods that depends only on each state's biomass, the mean interaction strength, and average species diversity. The formula predicts simulated basin sizes for hundreds of states in disordered ecosystems without detailed knowledge of the interaction matrix. The result matters because biomass and diversity are measurable in real communities, so state probabilities could be estimated experimentally.

What carries the argument

The central device is the block coarse-graining of stable states. Each observed stable state is treated as an effective species with self-inhibition equal to the inverse of its equilibrium biomass, and with an initial abundance that is the sum over its member species, approximated as Gaussian by the central limit theorem. Dressed initial conditions then determine the winner, and Eq. (5) computes each state's likelihood as an extreme-value probability. This machinery converts a high-dimensional dynamical problem into a one-dimensional competition among a few effective units.

What would settle it

Run a large number of simulations from random initial conditions on a fixed GLV matrix in the multistable regime, and compare each observed stable state's frequency with Eq. (5) using only its biomass and mean diversity; if two states with equal biomass systematically differ in likelihood, or if the rank order is reversed for a substantial fraction of states, the claim is falsified. The paper's own data already show the formula overpredicts the rarest states, so the falsifier should target mid- and high-likelihood states where the match is claimed to be good.

Watch

Extended reading notes

Core claim

The central claim is that basin size—the fraction of random initial conditions leading to a given stable state—is controlled by an emergent, state-level self-inhibition that equals the inverse of the state's equilibrium biomass. Because high-biomass states have low self-inhibition, they grow faster and outcompete other states, making them sharply more likely outcomes. This is made quantitative in a block model: each state behaves as a Gaussian-dressed initial abundance, and the probability that a state wins is the probability that its dressed initial abundance is the largest, giving Eq. (5). The same formula predicts the observed likelihoods of hundreds of stable states in random GLV matrice

Load-bearing premise

The disordered interaction matrix is replaced by an effective block model with a single mean inter-state inhibition and a common average block size, while fluctuations within each state are assumed to dissipate quickly; the paper itself calls this an 'uncontrolled' mean-field approximation.

Editorial extensions

If this is right

  • In the multistable region of the random GLV model, replicate communities started from random initial conditions should show a strongly skewed distribution of outcomes, with high-biomass states orders of magnitude more frequent than low-biomass ones.
  • State probabilities can be estimated from macroscopic measurements—total biomass and species richness per state—without knowing the species-level interaction matrix, which is usually unmeasurable.
  • The biomass–likelihood ordering survives moderate interaction asymmetry, variation in carrying capacities, changes in system size, and different initial-condition sampling schemes.
  • The analytic formula for monodominant and block-structured models is exact, giving a benchmark for studying more complex or overlapping state structures.
  • Because high biomass corresponds to low self-inhibition and faster growth, transient growth rates rather than detailed equilibration dynamics may be enough to rank outcomes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the biomass–likelihood relation generalizes beyond random matrices to the structured microbial communities grown from parallel enrichments, then simple cell-count and 16S data could rank alternative stable states without long time-series.
  • The paper's surprising finding that using each state's own diversity worsens predictions suggests that overlapping species renormalize effective block size; a testable extension would be to predict the optimal effective block size from overlap statistics between states.
  • The hyperbolic form of the biomass–likelihood relationship is shown to be only one possible shape; the curvature depends on the distribution of self-inhibition, so deviations in real data could be used to infer the underlying inhibition structure.
  • Outside the symmetric, strong-interaction setting, nonreciprocal interactions can lead to chaos, where the very notion of a stable state and its basin needs replacement; whether a biomass-biased occupation measure survives in chaotic regimes is an open question the paper does not resolve.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the multi-stable phase of the random Generalized Lotka-Volterra (GLV) model with strong interactions, asking what determines the relative likelihood (basin volume fraction) of each stable state. Numerically, for a fixed random interaction matrix, state likelihoods span orders of magnitude and increase sharply with the state's equilibrium biomass. To explain this, the authors first solve exactly a monodominant model in which each stable state contains a single species: the winner is the species with the largest dressed initial condition χ_i N_i(0), and the likelihood is given by Eq. (4). They then treat each stable state as an effective block with effective self-inhibition equal to the inverse of the block's equilibrium biomass, obtaining the block-likelihood formula Eq. (5) as the probability that one Gaussian-dressed block initial condition is the maximum. Finally, they apply Eq. (5) to random disordered matrices by replacing the inter-state inhibition D with the global mean interaction μ and using a common block size equal to the mean state diversity. This predicts simulated likelihoods for most states but systematically overpredicts the rarest, low-biomass states.

Significance. If the random-matrix extension were fully justified, the paper would establish a striking and practically useful result: ecological basin sizes are controlled by macroscopic state properties (biomass, diversity, and global mean interaction) rather than by the detailed interaction matrix. The monodominant solution Eq. (4) and the block-model integral Eq. (5) are genuine analytic contributions, and the block-model simulations validate them in their intended setting. The paper also contains a clear numerical demonstration that the biomass-likelihood relationship is robust across system sizes, asymmetry, carrying-capacity heterogeneity, and initial-condition distributions. The weakness is that the step from perfect block models to disordered random matrices relies on approximations the authors themselves describe as 'uncontrolled' (Appendix G); since this step carries the paper's central claim about random ecosystems, the strength of the final conclusion is currently disproportionate to the evidence.

major comments (3)
  1. [Appendix G, Eq. (5)] Equation (5) is derived for a block-structured matrix with a single inter-state inhibition D, and the dressed prefactor is χ_α = D − A^eff_αα. In the random-matrix application this becomes χ_α = μ − 1/B_α. The derivation requires χ_α > 0 for every block; otherwise the Gaussian 'dressed initial condition' has negative mean and the maximum-of-independent-Gaussians picture in Eq. (5) is no longer the correct basin-volume integral. The manuscript does not check whether B_α > 1/μ holds for the states to which Eq. (5) is applied, nor does it state the condition as a validity criterion. This is load-bearing because Eq. (5) is the quantitative prediction underlying the 'entire landscape' claim.
  2. [Fig. 3b and Appendix G] The mean-field approximation fails precisely in the regime the paper claims to predict: rare, low-likelihood states are overpredicted by orders of magnitude (the gray region in Fig. 3b, and the text notes these states can be 10^5 times less probable than the most likely state). The authors acknowledge this, but the abstract and main text nevertheless state that the model 'accurately predicts the entire landscape.' Since the overprediction is systematic in the tail of the likelihood distribution, the central claim needs to be qualified, and the manuscript should provide an error analysis or an a priori criterion for when the block/mean-field description applies.
  3. [Appendix D, Eq. (D8)] The reduction from the exact block dynamics Eq. (D6) to the effective single-unit form Eq. (D8) drops the intra-block fluctuation term (A_αα − 1) Σ(N_i − N̄_α)^2 / B_α, asserting it 'rapidly dissipates.' No timescale separation is demonstrated, and for the block sizes used (L ≈ 4–6) the fluctuations are not obviously negligible at early times. In the random-matrix setting, where states overlap and the block structure is only approximate, this uncontrolled truncation is a second independent approximation. The good agreement in Fig. S12 shows the truncation is harmless in the clean block model, but it does not establish that it is harmless for the disordered matrices that the main text targets.
minor comments (4)
  1. [Abstract and main text] The phrase 'accurately predict the entire landscape' is stronger than what Fig. 3b and Appendix G support. Suggest revising to 'the likelihoods of the vast majority of states' or adding a quantified statement about the low-likelihood tail.
  2. [Fig. 1c caption] The hyperbolic fit log(p) ∝ (B* − B)^−1 is introduced in the main text, but B* is not defined in the caption or in the surrounding text. Since B* is a fitted parameter, its role should be clarified so readers do not mistake the fit for a parameter-free prediction.
  3. [Appendix B] Typos: 'biomass-likelihood relationsip' should be 'relationship'; 'Radaumethod' should be 'RADA method' or similar; 'uninvasible' should be 'uninvadable.'
  4. [Appendix D] The text refers to 'Fig. S12b–c' when discussing the block-model comparison; the figure has panels (a) and (b). Please correct the cross-reference.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the biomass–likelihood relation is derived from the monodominant and block-model dynamics rather than fitted; the random-matrix application rests on an admitted uncontrolled mean-field approximation, which is a correctness risk, not a circular step.

full rationale

The paper's central derivation is self-contained. In the monodominant model, the likelihood p_i is solved exactly from the dressed initial condition χ_i N_i(0), with χ_i = D − A_ii (Appendix C, Eq. C15/C16), and biomass is separately B_i = 1/A_ii; the biomass–likelihood relation therefore follows from the dynamics, not from a fit. In the block model, the effective self-inhibition A_eff_αα is derived, not assumed, as the inverse of the equilibrium biomass (Appendix D, Eqs. D8–D9), and Eq. (D18) [main-text Eq. (5)] is obtained by integrating over the Gaussian dressed initial conditions; this formula is validated against independent block-model simulations with no likelihood data used to set parameters. When applied to the disordered random matrix, the paper replaces the inter-state inhibition D by the global mean μ, uses the observed biomass B_α, and uses the mean diversity L across states; Appendix G explicitly calls this 'an uncontrolled approximation'. That is an admitted approximation whose failure for rare states with a dominant species is a correctness/accuracy concern, not a circular reduction of the prediction to its inputs. The choice of average L over state-specific L is made by comparing predictive performance (Fig. S16) but L is a macroscopic summary statistic, not a parameter fitted to the target likelihoods. Self-citations, including [44] (Taylor & O'Dwyer) and [41,52,56], are used for context, generalization statements, and discussion, not as the load-bearing justification of the main derivation; the exactness of the cone decomposition is proven here in Appendix C rather than imported. The multistability boundary in Appendix H is obtained by rescaling a prior weak-interaction result [19], and the authors note the approximation involved; again this is a modeling assumption, not circularity. Overall, no equation or claim reduces by construction to its own inputs.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central relationship is derived from the model, but the ledger is not empty. Free parameters: B* in the headline hyperbolic fit, the mean-diversity block size L chosen for random-matrix predictions (state-specific diversity is rejected because it predicts worse), and the numerical threshold ladder that defines 'surviving' species. Axioms: several domain assumptions (GLV dynamics, symmetric strong-interaction random matrices, uniform random initial conditions) plus two explicitly ad hoc steps — the quasi-equilibration truncation in Eq. (D8) and the re-scaling of the weak-interaction multistability boundary in Appendix H. No invented entities: 'emergent self-inhibition' is a derived quantity (A_eff = 1/B*, Eq. D9), not a new postulated object with independent evidence.

free parameters (3)
  • B* (upper biomass reference in hyperbolic fit) = not stated numerically; fitted per dataset (Fig. 1c)
    The curve log(p) ∝ (B*−B)^−1 in Fig. 1c and Appendix I requires B* to be fitted to the simulated likelihood–biomass data. It is presented as an approximation to the relationship rather than as the derived prediction.
  • Common block size L (mean diversity over observed states) = mean observed diversity; roughly 4–8 for S=100 (Fig. S13)
    For random matrices the prediction uses one global L instead of the per-state diversity, because per-state diversity gives worse predictions (Fig. S16b, Appendix G). This is a modeling choice evaluated on the same data it predicts.
  • Extinction threshold ladder and integration time = thresholds 10^-5 … 10^-1; t = 300, extended to 50300
    The set of 'surviving' species — and hence the catalog of stable states and their biomasses — depends on these hand-set numerical thresholds (Appendix A). Not a theory parameter, but it shapes the state catalog the predictions are tested on.
assumptions (7)
  • domain assumption GLV dynamics (Eq. 1) with K_i = 1 and symmetric interactions capture the relevant ecosystem behavior
    The entire analysis lives inside this model; natural communities are not tested here. Symmetry and K_i = 1 are relaxed only in Appendix B.
  • domain assumption Quenched random interaction matrix with strong-interaction scaling (μ, σ finite as S grows)
    Adopted from the statistical-physics tradition (Refs. [19,20,23,25,33,36,53]); the paper explicitly contrasts this with weak-scaling treatments.
  • domain assumption Uniform [0,1] independent initial abundances define the likelihood measure
    Likelihood is defined relative to this sampling scheme; Appendix B shows robustness to truncated-Gaussian sampling, changing likelihood values but not the qualitative relationship.
  • standard math Block initial abundances are Gaussian with mean L/2 and variance L/12
    Central-limit/Irwin–Hall approximation used in Eqs. (D12)–(D13); validated for L ≥ 5 (Fig. S9).
  • ad hoc to paper Within-block quasi-equilibration N_i ≈ B_α/L_α and neglect of the fluctuation term in Eq. (D8)
    This is what lets the self-inhibition block model map onto the monodominant solution. The paper says the fluctuation term 'should not significantly impact the relative probabilities... over most ranges of the parameters' — asserted, not proven.
  • ad hoc to paper A block model with a single cross-state inhibition D applies to disordered matrices, with D replaced by μ and a common L
    Explicitly called 'an uncontrolled approximation' (Appendix G). States in the random matrix overlap (Appendix E), violating the block assumption; the approximation's failure is visible in the rare-state overprediction.
  • ad hoc to paper The weak-interaction multistability boundary rescales to strong interactions
    Appendix H substitutes μ̃→μS and σ̃→σ√S in Eq. (H1) while admitting that linear response — central to the original derivation — is no longer valid; numerical agreement is the only support.

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Pith. "Pith review of Emergent self-inhibition governs the landscape of stable states in complex ecosystems." pith.science (2026). https://pith.science/paper/RPTK6OIM

@misc{pith2026251106697,
  author       = {Pith},
  title        = {Pith review of: Emergent self-inhibition governs the landscape of stable states in complex ecosystems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPTK6OIM}},
  note         = {Machine review of arXiv:2511.06697}
}
read the original abstract

Species-rich ecosystems often exhibit multiple stable states with distinct species compositions. Yet, the factors determining the likelihood of each state's occurrence remain poorly understood. Here, we characterize and explain the landscape of stable states in the random Generalized Lotka-Volterra (GLV) model, in which multistability is widespread. We find that the same pool of species with random initial abundances can result in different stable states, whose likelihoods typically differ by orders of magnitude. A state's likelihood increases sharply with its total biomass, or inverse self-inhibition. We develop a simplified model to predict and explain this behavior, by coarse-graining ecological interactions so that each stable state behaves as a unit. In this setting, we can accurately predict the entire landscape of stable states using only two macroscopic properties: the biomass of each state and species diversity. Our analyses also provide insight into the biomass-likelihood relationship: High-biomass states have low self-inhibition and thus grow faster, outcompete others, and become much more likely. These results reveal emergent self-inhibition as a fundamental organizing principle for the attractor landscape of complex ecosystems---and provide a path to predict ecosystem outcomes without knowing microscopic interactions.

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Reference graph

Works this paper leans on

59 extracted references · 1 canonical work pages

  1. [44]

    Taylor and J

    W. Taylor and J. O’Dwyer, Theoretical Ecology18, 1 (2025)

    2025

  2. [1]

    Petraitis,Multiple Stable States in Natural Ecosystems (Oxford University Press, 2013)

    P. Petraitis,Multiple Stable States in Natural Ecosystems (Oxford University Press, 2013)

    2013

  3. [2]

    A. C. Staver, S. Archibald, and S. A. Levin, Science334, 230 (2011)

    2011

  4. [3]

    Stritih, R

    A. Stritih, R. Seidl, and C. Senf, Landscape Ecology38, 933 (2023)

    2023

  5. [4]

    J. C. Aleman, A. Fayolle, C. Favier, A. C. Staver, K. G. Dexter, C. M. Ryan, A. F. Azihou, D. Bauman, M. te Beest, E. N. Chidumayo,et al., Proceedings of the National Academy of Sciences117, 28183 (2020)

    2020

  6. [5]

    C. A. Lozupone, J. I. Stombaugh, J. I. Gordon, J. K. Jansson, and R. Knight, Nature489, 220 (2012)

    2012

  7. [6]

    Van de Guchte, S

    M. Van de Guchte, S. D. Burz, J. Cadiou, J. Wu, S. Mon- dot, H. M. Blotti` ere, and J. Dor´ e, Microbiome8, 153 (2020)

    2020

  8. [7]

    R. J. Schmitt, S. J. Holbrook, S. L. Davis, A. J. Brooks, and T. C. Adam, Proceedings of the National Academy of Sciences116, 4372 (2019)

    2019

Show all 59 references
  1. [8]

    P. S. Petraitis, E. T. Methratta, E. C. Rhile, N. A. Vi- dargas, and S. R. Dudgeon, Oecologia161, 139 (2009)

    2009

  2. [9]

    C. I. Abreu, V. L. Andersen Woltz, J. Friedman, and J. Gore, PLoS computational biology16, e1007934 (2020)

    2020

  3. [10]

    Lopes, D

    W. Lopes, D. R. Amor, and J. Gore, Nature Communi- cations15, 4709 (2024)

    2024

  4. [11]

    Estrela, J

    S. Estrela, J. C. Vila, N. Lu, D. Baji´ c, M. Rebolleda- G´ omez, C.-Y. Chang, J. E. Goldford, A. Sanchez- Gorostiaga, and ´A. S´ anchez, Cell Systems13, 29 (2022)

    2022

  5. [12]

    Tkacz, J

    A. Tkacz, J. Cheema, G. Chandra, A. Grant, and P. S. Poole, The ISME journal9, 2349 (2015)

    2015

  6. [13]

    S. M. Kearney, E. Thomas, A. Coe, and S. W. Chisholm, Environmental Microbiome16, 1 (2021)

    2021

  7. [14]

    Scheffer and E

    M. Scheffer and E. Jeppesen, Ecosystems10, 1 (2007)

    2007

  8. [15]

    Y. Zou, C. M. Zohner, C. Averill, H. Ma, J. Merder, M. Berdugo, L. Bialic-Murphy, L. Mo, P. Brun, N. E. Zimmermann,et al., Nature Communications15, 4658 (2024)

    2024

  9. [16]

    S´ anchez-Pinillos, V

    M. S´ anchez-Pinillos, V. Dakos, and S. K´ efi, Biological Conservation289, 110409 (2024)

    2024

  10. [17]

    Dubinkina, Y

    V. Dubinkina, Y. Fridman, P. P. Pandey, and S. Maslov, eLife8, e49720 (2019)

    2019

  11. [18]

    Aguad´ e-Gorgori´ o and S

    G. Aguad´ e-Gorgori´ o and S. K´ efi, Journal of Physics: Complexity5, 025022 (2024)

    2024

  12. [19]

    Bunin, Physical Review E95, 042414 (2017)

    G. Bunin, Physical Review E95, 042414 (2017)

    2017

  13. [20]

    J. G. Mart ´ ınez, S. de Monte, and M. Barbier, arXiv:2411.14969 [q-bio.PE] 10.48550/arXiv.2411.14969 (2024), v3, last revised 18 Mar 2025

    2024 doi

  14. [21]

    Tikhonov and R

    M. Tikhonov and R. Monasson, Physical Review Letters 118, 048103 (2017)

    2017

  15. [22]

    Altieri, F

    A. Altieri, F. Roy, C. Cammarota, and G. Biroli, Physi- cal Review Letters126, 258301 (2021)

    2021

  16. [23]

    Garc’ia Lorenzana, A

    G. Garc’ia Lorenzana, A. Altieri, C. Cammarota, and G. Biroli, Physical Review E105, 024307 (2022)

    2022

  17. [24]

    W. Cui, R. Marsland, and P. Mehta, Physical Review E 104, 034416 (2021)

    2021

  18. [25]

    W. Cui, R. Marsland, and P. Mehta, Les houches lectures on community ecology: From niche theory to statisti- cal mechanics (2024), les Houches Theoretical Biophysics Summer School 2023, arXiv:2403.05497 [q-bio.PE]

    2024 arXiv

  19. [26]

    Marsland, Robert and P

    I. Marsland, Robert and P. Mehta, Physical Review Let- ters125, 048101 (2020)

    2020

  20. [27]

    Altieri and S

    A. Altieri and S. De Monte, EPL (Europhysics Letters) 150, 51002 (2025)

    2025

  21. [28]

    P.-J. Ke, P. Mehta, and R. M. III, Ecology Letters27, 1011 (2024)

    2024

  22. [29]

    Advani, G

    M. Advani, G. Bunin, and P. Mehta, arXiv1707.03957 (2017)

    2017 arXiv

  23. [30]

    R. M. III, W. Cui, J. Goldford, A. Sanchez, K. Ko- rolev, and P. Mehta, PLOS Computational Biology15, e1006793 (2019)

    2019

  24. [31]

    Allesina, J

    S. Allesina, J. Grilli, G. Barab´ as, S. Tang, J. Aljadeff, and A. Maritan, Nature communications6, 7842 (2015). 6

    2015

  25. [32]

    Allesina and S

    S. Allesina and S. Tang, Nature483, 205 (2012)

    2012

  26. [33]

    T. A. de Pirey and G. Bunin, Physical Review X14, 011037 (2024)

    2024

  27. [34]

    M. T. Pearce, A. Agarwala, and D. S. Fisher, Proceed- ings of the National Academy of Sciences117, 14572 (2020)

    2020

  28. [35]

    Blumenthal, J

    E. Blumenthal, J. W. Rocks, and P. Mehta, Physical Review Letters132, 127401 (2024)

    2024

  29. [36]

    Mallmin, A

    E. Mallmin, A. Traulsen, and S. D. Monte, Proceedings of the National Academy of Sciences121, e2312822121 (2024)

    2024

  30. [37]

    Y. Liu, J. Hu, H. Lee, and J. Gore, Physical Review X 15, 011003 (2025)

    2025

  31. [38]

    Mahadevan, M

    A. Mahadevan, M. T. Pearce, and D. S. Fisher, Elife12, e82734 (2023)

    2023

  32. [39]

    V. Ros, F. Roy, G. Biroli, G. Bunin, and A. M. Turner, Physical Review Letters130, 257401 (2023)

    2023

  33. [40]

    Dubinkina, Y

    V. Dubinkina, Y. Fridman, P. P. Pandey, and S. Maslov, Elife8, e49720 (2019)

    2019

  34. [41]

    Goyal, V

    A. Goyal, V. Dubinkina, and S. Maslov, The ISME jour- nal12, 2823 (2018)

    2018

  35. [42]

    Fried, N

    Y. Fried, N. M. Shnerb, and D. A. Kessler, Physical Re- view E96, 012412 (2017)

    2017

  36. [43]

    D. A. Kessler and N. M. Shnerb, Physical Review E111, 034408 (2025)

    2025

  37. [45]

    M. E. Gilpin and T. J. Case, Nature261, 40 (1976)

    1976

  38. [46]

    Srinivasan, A

    S. Srinivasan, A. Jnana, and T. S. Murali, Microbial ecol- ogy87, 56 (2024)

    2024

  39. [47]

    O˜ na, S

    L. O˜ na, S. K. Shreekar, and C. Kost, Trends in Micro- biology (2025)

    2025

  40. [48]

    Marrec, G

    L. Marrec, G. Bravo-Ruiseco, X. Zhou, A. G. Daodu, and K. Faust, Current Opinion in Biotechnology96, 103352 (2025)

    2025

  41. [49]

    Meroz, T

    N. Meroz, T. Livny, and J. Friedman, Current Opinion in Microbiology80, 102511 (2024)

    2024

  42. [50]

    P. Mira, P. Yeh, and B. G. Hall, Plos one17, e0276040 (2022)

    2022

  43. [51]

    J. E. Goldford, N. Lu, D. Baji´ c, S. Estrela, M. Tikhonov, A. Sanchez-Gorostiaga, D. Segr` e, P. Mehta, and A. Sanchez, Science361, 469 (2018)

    2018

  44. [52]

    Dal Bello, H

    M. Dal Bello, H. Lee, A. Goyal, and J. Gore, Nature ecology & evolution5, 1424 (2021)

    2021

  45. [53]

    R. M. May, Nature238, 413 (1972)

    1972

  46. [54]

    J. Hu, D. R. Amor, M. Barbier, G. Bunin, and J. Gore, Science378, 85 (2022)

    2022

  47. [55]

    We then extend to block ecosystems with disjoint groups of coexisting species

    (also studied in this context in [44]), where indi- vidual species compete so strongly that only one species survives in each stable state. We then extend to block ecosystems with disjoint groups of coexisting species. Fi- nally, we apply these results to random matrices, wher...

  48. [56]

    A. J. Lotka,Elements of mathematical biology(Dover Publications, 1956)

    1956

  49. [57]

    Goyal, L

    A. Goyal, L. S. Bittleston, G. E. Leventhal, L. Lu, and O. X. Cordero, Elife11, e74987 (2022). 7 SUPPLEMENT AR Y INFORMA TION Appendix A: Computing the likelihoods and biomass of stable states We first generate a random interaction matrixAwith meanµand standard deviationσ. Bel...

    2022

  50. [58]

    We then solve the GL V equations using theRadaumethod which has adaptive time steps. After running the simulation for sufficient time untilt= 300, we check whether we have reached a stable, uninvasible steady state in the following ways: Identifying surviving species.—To deter...

  51. [59]

    what is the likelihood of species 2 winning?

    Here we show that the biomass–likelihood relationship is robust to another reasonable sampling scheme: that of sampling the initial conditions from a Gaussian centred at 0 with standard deviation 0.3, truncated to the positive orthant to ensure nonnegative abundances. We still...

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Reviewed August 3, 2026 · model on record in the stance chip above.