pith. sign in

arxiv: 2509.23977 · v2 · submitted 2025-09-28 · 🧬 q-bio.PE · cond-mat.dis-nn· cond-mat.stat-mech

Emergent frequency-dependent selection predicts mutation outcomes in complex ecological communities

Shing Yan Li , Zhijie Feng , Akshit Goyal , Pankaj Mehta This is my paper

Pith reviewed 2026-05-18 12:39 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.dis-nncond-mat.stat-mech
keywords frequency-dependent selectionfixation probabilityecological interactionsmutation outcomescommunity ecologyeco-evolutionary dynamicspopulation genetics
0
0 comments X

The pith

Ecological interactions create frequency-dependent selection that suppresses fixation of moderately beneficial mutations

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

This paper shows that interactions within a species-rich community generate an effective frequency-dependent selection between a parent strain and a new mutant. The selection is summarized by one parameter that quantifies the strength of ecological feedbacks. The resulting analytic formula for fixation probability extends the classic Kimura expression and shows that moderately beneficial mutations become harder to fix because the frequency dependence prolongs coexistence between the two lineages. The suppression grows stronger as effective population size increases and as the number of open niches in the ecosystem rises. The framework therefore allows mutation outcomes to be predicted from a simple model even when the surrounding community is complex.

Core claim

Ecological interactions result in emergent frequency-dependent selection between parents and mutants, characterized by a single parameter measuring the strength of ecological feedbacks. This result generalizes classical population genetics models to highly diverse communities and enables predictions of mutation outcomes in these eco-evolutionary settings. We derive an analytic expression for fixation probability that extends Kimura's formula and reveals that ecological interactions strongly suppress the fixation of moderately beneficial mutations. This suppression arises because frequency-dependent selection leads to prolonged coexistence between parent and mutant lineages, which acts as a a

What carries the argument

Emergent frequency-dependent selection characterized by a single parameter for the strength of ecological feedbacks, which alters the fixation probability between parent and mutant lineages.

If this is right

  • Fixation probabilities of mutations in complex communities can be calculated with a single-parameter extension of the Kimura formula.
  • Moderately beneficial mutations experience a barrier to fixation due to extended coexistence with the parent lineage.
  • The magnitude of suppression increases with larger effective population size and greater numbers of open niches.
  • Evolutionary outcomes remain predictable from simple models once the strength of community feedbacks is known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Natural ecosystems with high diversity may evolve more slowly than isolated-population models predict.
  • Community structure could buffer against rapid sweeps by single mutations, favoring gradual or collective adaptation.
  • The approach could be tested in controlled microbiome systems to predict which mutations persist under different resource regimes.

Load-bearing premise

The community is large and diverse enough that average ecological feedbacks can be captured by a single strength parameter without tracking every specific pairwise interaction or small-population fluctuations.

What would settle it

Direct measurement of fixation probability for a moderately beneficial mutation in a microbial community experiment where diversity and niche availability are varied, compared against the extended fixation formula.

Figures

Figures reproduced from arXiv: 2509.23977 by Akshit Goyal, Pankaj Mehta, Shing Yan Li, Zhijie Feng.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Ecological interactions can dramatically alter evolutionary outcomes in complex communities. Yet, the framework of population genetics largely neglects interactions from a species-rich community. Here, we bridge this gap by using dynamical mean-field theory to integrate community ecology into classical population genetics models. We show that ecological interactions result in emergent frequency-dependent selection between parents and mutants, characterized by a single parameter measuring the strength of ecological feedbacks. This result generalizes classical population genetics models to highly diverse communities and enables predictions of mutation outcomes in these eco-evolutionary settings. We derive an analytic expression for fixation probability that extends Kimura's formula and reveals that ecological interactions strongly suppress the fixation of moderately beneficial mutations. This suppression arises because frequency-dependent selection leads to prolonged coexistence between parent and mutant lineages, which acts as a barrier to fixation. The strength of these effects increases with effective population size and the number of open niches in the ecosystem. Our study establishes a framework for integrating ecological interactions into population genetics, showing that evolutionary outcomes can be predicted using simple models even in the presence of complex community feedbacks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper applies dynamical mean-field theory (DMFT) to species-rich ecological communities to show that interactions produce an emergent frequency-dependent selection between parent and mutant lineages, fully characterized by a single ecological feedback strength parameter. This yields an analytic fixation probability extending Kimura's formula, with the key result that ecological feedbacks strongly suppress fixation of moderately beneficial mutations via prolonged coexistence; the suppression intensifies with larger effective population size and more open niches.

Significance. If the central reduction holds, the work is significant for bridging community ecology and population genetics with a tractable, low-parameter model that makes falsifiable predictions for mutation outcomes in diverse systems. The emergence of a single feedback parameter directly from DMFT averaging, rather than fitting, and the analytic extension of a classical result are clear strengths that could guide microbial evolution experiments.

major comments (2)
  1. [DMFT setup and reduction to single parameter] DMFT setup and reduction to single parameter (likely §3 and associated equations): the claim that the effective frequency-dependent selection is fully captured by one emergent parameter without residual pairwise interactions or finite-size fluctuations is load-bearing for the fixation formula; the abstract notes that effects strengthen with population size and open niches, implying an asymptotic regime, yet no explicit error bounds or finite-N corrections are provided to assess robustness in moderately large communities.
  2. [Extended fixation probability derivation] Extended fixation probability derivation (the generalized Kimura expression): the suppression result for moderate selection coefficients depends on the coexistence time being accurately predicted by the mean-field frequency dependence; without direct comparison to individual-based simulations that retain explicit interactions, it remains unclear whether the single-parameter model over- or under-estimates fixation probabilities when demographic stochasticity is non-negligible.
minor comments (2)
  1. [Abstract] Abstract: the range of selection coefficients for which suppression is strongest could be stated more quantitatively to guide readers.
  2. [Notation] Notation: ensure the ecological feedback parameter is introduced with a clear symbol and used consistently in all equations and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The comments identify important aspects of the DMFT reduction and the validation of the extended fixation probability that warrant clarification and strengthening. Below we respond point by point to the major comments. We will incorporate the suggested additions in a revised manuscript.

read point-by-point responses

  1. Referee: DMFT setup and reduction to single parameter (likely §3 and associated equations): the claim that the effective frequency-dependent selection is fully captured by one emergent parameter without residual pairwise interactions or finite-size fluctuations is load-bearing for the fixation formula; the abstract notes that effects strengthen with population size and open niches, implying an asymptotic regime, yet no explicit error bounds or finite-N corrections are provided to assess robustness in moderately large communities.

    Authors: We agree that the reduction to a single emergent feedback parameter is central and that the derivation is performed in the thermodynamic limit of DMFT. The single-parameter characterization arises because the random interaction matrix is averaged self-consistently, eliminating residual pairwise terms in the large-N limit. We acknowledge that explicit error bounds or finite-N corrections are not derived in the current manuscript. In the revision we will add a dedicated subsection on the regime of validity, including a scaling argument for the leading finite-N corrections to the effective selection coefficient and a brief numerical illustration for community sizes N ≈ 100–500 that are still large enough for the mean-field approximation to be useful. revision: yes

  2. Referee: Extended fixation probability derivation (the generalized Kimura expression): the suppression result for moderate selection coefficients depends on the coexistence time being accurately predicted by the mean-field frequency dependence; without direct comparison to individual-based simulations that retain explicit interactions, it remains unclear whether the single-parameter model over- or under-estimates fixation probabilities when demographic stochasticity is non-negligible.

    Authors: We recognize that direct validation against individual-based simulations retaining the full interaction matrix would strengthen in the quantitative accuracy of the fixation formula, especially for moderate selection strengths where coexistence times are long. The present work emphasizes the analytic DMFT derivation and its consistency with the generalized Kimura expression. To address the concern we will include, in the revised manuscript, a new figure and accompanying text that compares the analytic fixation probabilities to stochastic simulations of finite communities (N = 200–1000) with explicit random interactions drawn from the same ensemble used in the DMFT. These simulations will quantify the deviation attributable to demographic stochasticity and finite-size effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity: DMFT yields independent effective selection parameter

full rationale

The derivation applies dynamical mean-field theory to a community ecology model with many species and open niches, producing an effective frequency-dependent selection between parent and mutant that is characterized by a single emergent feedback parameter. This effective model is then inserted into the standard branching-process or diffusion approximation to obtain an analytic extension of Kimura's fixation probability formula. The feedback parameter is obtained from the DMFT averaging procedure applied to the ecological dynamics and is not defined in terms of, or fitted to, the fixation probabilities themselves. No self-citation chain, self-definitional step, or renaming of a known result is required for the central claim. The result remains falsifiable against explicit simulations of finite communities and is therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central result rests on the validity of the dynamical mean-field approximation for a large, diverse community and on the existence of a well-defined effective feedback parameter that captures all higher-order interactions.

free parameters (1)
  • ecological feedback strength parameter
    Single parameter introduced to quantify the strength of community-level feedbacks that generate the frequency dependence; its value is not derived from first principles but emerges from the DMFT treatment.
axioms (1)
  • domain assumption Dynamical mean-field theory accurately averages over species interactions in the limit of many species and open niches.
    Invoked to reduce the many-species community dynamics to an effective single-parameter frequency-dependent selection term.

pith-pipeline@v0.9.0 · 5729 in / 1451 out tokens · 51508 ms · 2026-05-18T12:39:35.718581+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    effectively neutral

    To model the relatedness of parent and mutant, we assume that the mutant’s interaction parameters,A mj, with other species in the community are correlated with those of the parent,A pj. We denote the strength of this correlation asρ. 3 Fixation Extinction Fixation Extinction Classic population genetics parent mutant Population genetics in complex communit...

    work page

  2. [2]

    Kimura, On the probability of fixation of mutant genes in a population, Genetics47, 713 (1962)

    M. Kimura, On the probability of fixation of mutant genes in a population, Genetics47, 713 (1962)

    work page 1962

  3. [3]

    Kimura, Theoretical foundation of population genet- ics at the molecular level, Theoretical population biology 2, 174 (1971)

    M. Kimura, Theoretical foundation of population genet- ics at the molecular level, Theoretical population biology 2, 174 (1971)

    work page 1971

  4. [4]

    M. Kimura, Stochastic processes and distribution of gene frequencies under natural selection (1955), Population Genetics, Molecular Evolution, and the Neutral The- ory: Selected Papers. The University of Chicago Press, Chicago144(1994)

    work page 1955

  5. [5]

    W. J. Ewens and W. Ewens,Mathematical population ge- netics: theoretical introduction, Vol. 27 (Springer, 2004)

    work page 2004

  6. [6]

    B. H. Good and O. Hallatschek, Effective models and the search for quantitative principles in microbial evolution, Current opinion in microbiology45, 203 (2018)

    work page 2018

  7. [7]

    M¨ ohle and S

    M. M¨ ohle and S. Sagitov, A classification of coalescent processes for haploid exchangeable population models, Annals of Probability , 1547 (2001)

    work page 2001

  8. [8]

    B. H. Good, M. J. McDonald, J. E. Barrick, R. E. Lenski, and M. M. Desai, The dynamics of molecular evolution over 60,000 generations, Nature551, 45 (2017)

    work page 2017

  9. [9]

    L. M. Wahl, P. J. Gerrish, and I. Saika-Voivod, Evaluat- ing the impact of population bottlenecks in experimental evolution, Genetics162, 961 (2002)

    work page 2002

  10. [10]

    J. F. Crow,An introduction to population genetics theory (Scientific Publishers, 2017)

    work page 2017

  11. [11]

    D. E. Dykhuizen and D. L. Hartl, Selection in chemostats, Microbiological reviews47, 150 (1983)

    work page 1983

  12. [12]

    Cvijovi´ c, B

    I. Cvijovi´ c, B. H. Good, E. R. Jerison, and M. M. Desai, Fate of a mutation in a fluctuating environment, Pro- ceedings of the National Academy of Sciences112, E5021 (2015)

    work page 2015

  13. [13]

    M. C. Whitlock, Fixation probability and time in subdi- vided populations, Genetics164, 767 (2003)

    work page 2003

  14. [14]

    Hallatschek and D

    O. Hallatschek and D. R. Nelson, Gene surfing in expand- ing populations, Theoretical population biology73, 158 (2008)

    work page 2008

  15. [15]

    G. F. Fussmann, M. Loreau, and P. A. Abrams, Eco- evolutionary dynamics of communities and ecosystems, Functional ecology , 465 (2007)

    work page 2007

  16. [16]

    A. P. Hendry,Eco-evolutionary dynamics(Princeton uni- versity press, 2017)

    work page 2017

  17. [17]

    Govaert, E

    L. Govaert, E. A. Fronhofer, S. Lion, C. Eizaguirre, D. Bonte, M. Egas, A. P. Hendry, A. De Brito Martins, C. J. Meli´ an, J. A. Raeymaekers,et al., Eco-evolutionary feedbacks—theoretical models and perspectives, Func- tional Ecology33, 13 (2019)

    work page 2019

  18. [18]

    Z. Feng, E. Blumenthal, P. Mehta, and A. Goyal, A the- ory of ecological invasions and its implications for eco- evolutionary dynamics, ArXiv , arXiv (2025)

    work page 2025

  19. [19]

    McEnany and B

    J. McEnany and B. H. Good, Predicting the first steps of evolution in randomly assembled communities, Nature Communications15, 8495 (2024)

    work page 2024

  20. [20]

    B. H. Good and L. B. Rosenfeld, Eco-evolutionary feed- backs in the human gut microbiome, Nature Communi- cations14, 7146 (2023)

    work page 2023

  21. [21]

    R. D. Holt,On the integration of community ecology and evolutionary biology: historical perspectives and current prospects(Elsevier, 2005)

    work page 2005

  22. [22]

    G. G. Mittelbach and D. W. Schemske, Ecological and evolutionary perspectives on community assembly, Trends in ecology & evolution30, 241 (2015)

    work page 2015

  23. [23]

    Govaert, F

    L. Govaert, F. Altermatt, L. De Meester, M. A. Lei- bold, M. A. McPeek, J. H. Pantel, and M. C. Urban, Integrating fundamental processes to understand eco- evolutionary community dynamics and patterns, Func- tional Ecology35, 2138 (2021)

    work page 2021

  24. [24]

    P. M. Altrock and A. Traulsen, Fixation times in evo- lutionary games under weak selection, New Journal of Physics11, 013012 (2009)

    work page 2009

  25. [25]

    P. M. Altrock, C. S. Gokhale, and A. Traulsen, Stochas- tic slowdown in evolutionary processes, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics82, 011925 (2010)

    work page 2010

  26. [26]

    P. M. Altrock, A. Traulsen, and T. Galla, The mechanics of stochastic slowdown in evolutionary games, Journal of Theoretical Biology311, 94 (2012)

    work page 2012

  27. [27]

    McAvoy and B

    A. McAvoy and B. Allen, Fixation probabilities in evolu- tionary dynamics under weak selection, Journal of Math- ematical Biology82, 14 (2021)

    work page 2021

  28. [28]

    P. A. Marquet, G. Espinoza, S. R. Abades, A. Ganz, and R. Rebolledo, On the proportional abundance of species: Integrating population genetics and community ecology, Scientific reports7, 16815 (2017)

    work page 2017

  29. [29]

    Overcast, B

    I. Overcast, B. C. Emerson, and M. J. Hickerson, An integrated model of population genetics and community ecology, Journal of biogeography46, 816 (2019)

    work page 2019

  30. [30]

    K. R. Foster, J. Schluter, K. Z. Coyte, and S. Rakoff- Nahoum, The evolution of the host microbiome as an ecosystem on a leash, Nature548, 43 (2017)

    work page 2017

  31. [31]

    Ferreiro, N

    A. Ferreiro, N. Crook, A. J. Gasparrini, and G. Dantas, Multiscale evolutionary dynamics of host-associated mi- 10 crobiomes, Cell172, 1216 (2018)

    work page 2018

  32. [32]

    Lawrence, F

    D. Lawrence, F. Fiegna, V. Behrends, J. G. Bundy, A. B. Phillimore, T. Bell, and T. G. Barraclough, Species in- teractions alter evolutionary responses to a novel envi- ronment, PLoS biology10, e1001330 (2012)

    work page 2012

  33. [33]

    Calcagno, P

    V. Calcagno, P. Jarne, M. Loreau, N. Mouquet, and P. David, Diversity spurs diversification in ecological communities, Nature communications8, 15810 (2017)

    work page 2017

  34. [34]

    Venkataram, H.-Y

    S. Venkataram, H.-Y. Kuo, E. F. Hom, and S. Kryazhim- skiy, Mutualism-enhancing mutations dominate early adaptation in a two-species microbial community, Nature Ecology & Evolution7, 143 (2023)

    work page 2023

  35. [35]

    Schluter and M

    D. Schluter and M. W. Pennell, Speciation gradients and the distribution of biodiversity, Nature546, 48 (2017)

    work page 2017

  36. [36]

    J. P. Hall, E. Harrison, and M. A. Brockhurst, Competi- tive species interactions constrain abiotic adaptation in a bacterial soil community, Evolution Letters2, 580 (2018)

    work page 2018

  37. [37]

    Scheuerl, M

    T. Scheuerl, M. Hopkins, R. W. Nowell, D. W. Rivett, T. G. Barraclough, and T. Bell, Bacterial adaptation is constrained in complex communities, Nature communi- cations11, 754 (2020)

    work page 2020

  38. [38]

    B. H. Good, S. Martis, and O. Hallatschek, Adaptation limits ecological diversification and promotes ecological tinkering during the competition for substitutable re- sources, Proceedings of the National Academy of Sciences 115, E10407 (2018)

    work page 2018

  39. [39]

    Yamamichi, T

    M. Yamamichi, T. Gibbs, and J. M. Levine, Integrating eco-evolutionary dynamics and modern coexistence the- ory, Ecology Letters25, 2091 (2022)

    work page 2091

  40. [40]

    Sireci and M

    M. Sireci and M. A. Mu˜ noz, Statistical mechanics of phe- notypic eco-evolution: From adaptive dynamics to com- plex diversification, Physical Review Research6, 023070 (2024)

    work page 2024

  41. [41]

    L. Fant, I. Macocco, and J. Grilli, Eco-evolutionary dy- namics lead to functionally robust and redundant com- munities, bioRxiv , 2021 (2021)

    work page 2021

  42. [42]

    F. Roy, G. Biroli, G. Bunin, and C. Cammarota, Nu- merical implementation of dynamical mean field theory for disordered systems: Application to the lotka–volterra model of ecosystems, Journal of Physics A: Mathematical and Theoretical52, 484001 (2019)

    work page 2019

  43. [43]

    M. T. Pearce, A. Agarwala, and D. S. Fisher, Stabi- lization of extensive fine-scale diversity by ecologically driven spatiotemporal chaos, Proceedings of the National Academy of Sciences117, 14572 (2020)

    work page 2020

  44. [44]

    Bunin, Ecological communities with lotka-volterra dy- namics, Physical Review E95, 042414 (2017)

    G. Bunin, Ecological communities with lotka-volterra dy- namics, Physical Review E95, 042414 (2017)

    work page 2017

  45. [45]

    R. M. May, Will a large complex system be stable?, Na- ture238, 413 (1972)

    work page 1972

  46. [46]

    Mahadevan, M

    A. Mahadevan, M. T. Pearce, and D. S. Fisher, Spa- tiotemporal ecological chaos enables gradual evolution- ary diversification without niches or tradeoffs, Elife12, e82734 (2023)

    work page 2023

  47. [47]

    Advani, G

    M. Advani, G. Bunin, and P. Mehta, Statistical physics of community ecology: a cavity solution to macarthur’s consumer resource model, Journal of Statistical Mechan- ics: Theory and Experiment2018, 033406 (2018)

    work page 2018

  48. [48]

    Altieri, F

    A. Altieri, F. Roy, C. Cammarota, and G. Biroli, Proper- ties of equilibria and glassy phases of the random lotka- volterra model with demographic noise, Physical Review Letters126, 258301 (2021)

    work page 2021

  49. [49]

    Mahadevan and D

    A. Mahadevan and D. S. Fisher, Continual evolution in nonreciprocal ecological models, PRX Life3, 033008 (2025)

    work page 2025

  50. [50]

    Goyal, J

    A. Goyal, J. W. Rocks, and P. Mehta, Universal niche geometry governs the response of ecosystems to environ- mental perturbations, PRX Life3, 013010 (2025)

    work page 2025

  51. [51]

    Kimura and T

    M. Kimura and T. Ohta,Theoretical aspects of population genetics, 4 (Princeton University Press, 1971)

    work page 1971

  52. [52]

    C. K. Fisher and P. Mehta, The transition between the niche and neutral regimes in ecology, Proceedings of the National Academy of Sciences111, 13111 (2014)

    work page 2014

  53. [53]

    J. H. Gillespie, Some properties of finite populations experiencing strong selection and weak mutation, The American Naturalist121, 691 (1983)

    work page 1983

  54. [54]

    Sompolinsky, Statistical mechanics of neural networks, Physics Today41, 70 (1988)

    H. Sompolinsky, Statistical mechanics of neural networks, Physics Today41, 70 (1988)

    work page 1988

  55. [55]

    Levins, Theory of fitness in a heterogeneous environ- ment

    R. Levins, Theory of fitness in a heterogeneous environ- ment. i. the fitness set and adaptive function, The Amer- ican Naturalist96, 361 (1962)

    work page 1962

  56. [56]

    Macarthur and R

    R. Macarthur and R. Levins, The limiting similarity, convergence, and divergence of coexisting species, The American Naturalist101, 377 (1967)

    work page 1967

  57. [57]

    Marsland III, W

    R. Marsland III, W. Cui, J. Goldford, A. Sanchez, K. Ko- rolev, and P. Mehta, Available energy fluxes drive a tran- sition in the diversity, stability, and functional structure of microbial communities, PLoS computational biology 15, e1006793 (2019)

    work page 2019

  58. [58]

    W. Cui, R. Marsland III, and P. Mehta, Effect of resource dynamics on species packing in diverse ecosystems, Phys- ical review letters125, 048101 (2020)

    work page 2020

  59. [59]

    Kimura and T

    M. Kimura and T. Ohta, The average number of gener- ations until fixation of a mutant gene in a finite popula- tion, Genetics61, 763 (1969)

    work page 1969

  60. [60]

    Goyal, L

    A. Goyal, L. S. Bittleston, G. E. Leventhal, L. Lu, and O. X. Cordero, Interactions between strains govern the eco-evolutionary dynamics of microbial communities, Elife11, e74987 (2022)

    work page 2022

  61. [61]

    Roodgar, B

    M. Roodgar, B. H. Good, N. R. Garud, S. Martis, M. Avula, W. Zhou, S. M. Lancaster, H. Lee, A. Bab- veyh, S. Nesamoney,et al., Longitudinal linked-read se- quencing reveals ecological and evolutionary responses of a human gut microbiome during antibiotic treatment, Genome research31, 1433 (2021)

    work page 2021

  62. [62]

    Goyal, V

    A. Goyal, V. Dubinkina, and S. Maslov, Multiple stable states in microbial communities explained by the stable marriage problem, The ISME journal12, 2823 (2018)

    work page 2018

  63. [63]

    Lopes, D

    W. Lopes, D. R. Amor, and J. Gore, Cooperative growth in microbial communities is a driver of multistability, Na- ture Communications15, 4709 (2024)

    work page 2024

  64. [64]

    Blumenthal, J

    E. Blumenthal, J. W. Rocks, and P. Mehta, Phase tran- sition to chaos in complex ecosystems with nonreciprocal species-resource interactions, Physical review letters132, 127401 (2024)

    work page 2024

  65. [65]

    Arnoulx de Pirey and G

    T. Arnoulx de Pirey and G. Bunin, Many-species eco- logical fluctuations as a jump process from the brink of extinction, Physical Review X14, 011037 (2024)

    work page 2024

  66. [66]

    J. Hu, D. R. Amor, M. Barbier, G. Bunin, and J. Gore, Emergent phases of ecological diversity and dynamics mapped in microcosms, Science378, 85 (2022)

    work page 2022

  67. [67]

    M. M. Desai and D. S. Fisher, Beneficial mutation– selection balance and the effect of linkage on positive selection, Genetics176, 1759 (2007)

    work page 2007

  68. [68]

    Yoshida, S

    T. Yoshida, S. P. Ellner, L. E. Jones, B. J. M. Bohan- nan, R. E. Lenski, and N. G. Hairston Jr, Cryptic popu- lation dynamics: rapid evolution masks trophic interac- tions, PLoS biology5, e235 (2007)

    work page 2007

  69. [69]

    Dornic, H

    I. Dornic, H. Chat´ e, and M. A. Munoz, Integration of 11 langevin equations with multiplicative noise and the vi- ability of field theories for absorbing phase transitions, Physical review letters94, 100601 (2005)

    work page 2005

  70. [70]

    Garcia Lorenzana, A

    G. Garcia Lorenzana, A. Altieri, and G. Biroli, Interac- tions and migration rescuing ecological diversity, PRX Life2, 013014 (2024)

    work page 2024

  71. [71]

    strong-selection-weak-mutation

    W. Cui, J. W. Rocks, and P. Mehta, An elementary mean-field approach to the spectral densities of random matrix ensembles, Physica A: Statistical Mechanics and its Applications637, 129608 (2024). 12 SUPPLEMENTARY INFORMATION Appendix A: Lotka-Volterra model We begin with the details of our eco-evolution models. Consider an ecological community ofScompetin...

    work page 2024