pith. sign in

arxiv: 2606.09494 · v1 · pith:63XJSQTZnew · submitted 2026-06-08 · 🧬 q-bio.PE · cond-mat.dis-nn

Percolation and clustering in ecological communities: A dynamical theory

Dario Sergo , C\'edric Koller , Vittorio Erba , Lenka Zdeborov\'a This is my paper

Pith reviewed 2026-06-27 14:11 UTC · model grok-4.3

classification 🧬 q-bio.PE cond-mat.dis-nn
keywords percolationclusteringecological communitiesLotka-Volterra modelrandom graphsdynamical accessibilitycompetitive interactionssurviving sites
0
0 comments X

The pith

Dynamical accessibility of equilibria in a discrete Lotka-Volterra model determines the onset of percolation and clustering on random interaction graphs.

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

The paper develops a discrete version of the generalized Lotka-Volterra model for competitive ecological systems defined on random interaction graphs. This formulation preserves key macroscopic features of continuous dynamics while permitting analytical treatment of collective behavior. It establishes that the equilibria reachable by the dynamics control whether percolating clusters form and how surviving sites organize in space. A sympathetic reader would care because the work supplies an analytical account of percolation and clustering phenomena previously seen mainly in experiments and simulations, adding a dynamical layer to classical equilibrium descriptions.

Core claim

Within the discrete generalized Lotka-Volterra framework on random graphs, the set of equilibria that the dynamics can actually reach governs the emergence of percolating clusters and the spatial organization of surviving sites, so that dynamical accessibility, rather than the mere existence of equilibria, sets the thresholds for clustering and percolation.

What carries the argument

The discrete generalized Lotka-Volterra model on random interaction graphs, whose reachable equilibria dictate the onset of percolation and clustering.

If this is right

  • Percolation occurs only when the dynamics can reach equilibria that support a giant connected component of surviving sites.
  • The spatial patterns of surviving species are fixed by the subset of equilibria the dynamics can attain.
  • Clustering thresholds are set by the boundary between reachable and unreachable equilibria rather than by static stability alone.
  • The discrete formulation yields explicit analytical conditions for the transition to percolating states.

Where Pith is reading between the lines

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

  • The same reachability constraint may limit collective organization in continuous-time versions or in graphs with different degree distributions.
  • Controlled perturbations that block specific equilibria could be used to test whether percolation disappears as predicted.
  • The approach suggests a general route for analyzing dynamical selection of phases in other network-based population models.

Load-bearing premise

The discrete version of the generalized Lotka-Volterra model preserves the key macroscopic features of continuous ecological dynamics.

What would settle it

A concrete counter-example would be a realized random graph where the dynamics reach only non-percolating equilibria yet percolation is nevertheless observed in the steady state, or where an accessible equilibrium predicts percolation but none appears.

Figures

Figures reproduced from arXiv: 2606.09494 by C\'edric Koller, Dario Sergo, Lenka Zdeborov\'a, Vittorio Erba.

Figure 1
Figure 1. Figure 1: Spatial phase diagram of the discrete gLV model, on a d-regular graph with uniform competitive interactions. Critical values of α at which some biomasses become zero for some sites (αext), at which the percolation transition occurs (αperc), and at which the clusters of occupied sites become linear (αlin). A graphical representation of the different phases is also provided. This phase diagram is produced fo… view at source ↗
Figure 2
Figure 2. Figure 2: BDCM accurately captures typical dynamics and structure, illustrated for K = 2 on d = 3–random regular graphs. (Top) Extinction fraction ρ0 and largest cluster fraction ϕLC as a function of α, computed from BDCM on the dominant (p/c)–attractors (p = 3, c = 2). Black dots denote numerical simulations (S = 104 , averaging 5 simulations, error bars are negligible), lines represent theoretical predictions, sho… view at source ↗
Figure 3
Figure 3. Figure 3: Full occupied entropy and atypical full-survival threshold. (Left) The entropy ΦFull.Occupied (p/c) , which counts the log-number of atypical initial conditions leading to a fully occupied attractor, for multiple values of K, d, and p. For K = 2, d = 3, the entropies coincide for all p. For visual clarity all the negative entropies correspond to the marker < 0. (Right) Threshold values αatyp and αext for f… view at source ↗
Figure 4
Figure 4. Figure 4: Simulation of discrete gLV model on a grid, at convergence. (First two rows) Coupling of neighboring sites sampled from a truncated Gaussian, with standard deviation σ and mean µ. The white spots indicate extinct sites (at convergence), dark green occupied sites. The fragmentation of the community is visible in the first row. (Third row) Uniform coupling case, where the emergence of ring-like structures ca… view at source ↗
Figure 5
Figure 5. Figure 5: Empirical comparison of discrete generalized Lotka-Volterra with its continuous counterpart, uni￾form interactions. Simulations on a random 3−regular graph with S = 105 show that discrete gLV closely reproduces the behavior of continuous gLV. (Top row) For small α both models reach a uniform state, whereas for larger α extinction and heterogeneous biomass levels emerge. This is consistent with what is repo… view at source ↗
Figure 6
Figure 6. Figure 6: Numerical results for heterogeneous interactions. Numerical measurements of the occupied fraction ρ0 and of the largest-cluster fraction ϕLC for the discrete model on random 3-regular graphs with S = 104 sites and carrying capacity K = 100, for symmetric, nonreciprocal, and directed truncated Gaussian couplings αij . For comparison, we also show the corresponding results for the continuous gLV model. As th… view at source ↗
Figure 7
Figure 7. Figure 7: A graphical representation of the edge dual factor graph, with the factor nodes as squares and the variable nodes [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: BDCM captures typical dynamics and structure for K = 3, with slower entropy convergence near percolation. (Top) Extinction fraction ρ0 and largest cluster fraction ϕLC as functions of α, computed from BDCM for the dominant (p/c)–attractor (c = 2) and multiple values of p (ϕLC has been computed with the exact expression, but for p = 4 the computation was carried out only slightly beyond the threshold of per… view at source ↗
Figure 9
Figure 9. Figure 9: The high degree limit of BDCM equation, K = 2. (Left) Numerical computation of the extinction threshold at high d (over 5 simulations, S = 10000, S = 20000). Notice the drift from αext = 1/2 (we report simulations for two sizes to illustrate that this drift is not a finite size effect). (Right) Theoretical prediction of ρ0 (in red, from solving numerically Eq. (D26)) in the fully connected limit, alongside… view at source ↗
Figure 10
Figure 10. Figure 10: The high degree limit of BDCM equation, K = 3. See caption of [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Computation of the Extinction-Forbidden entropy at K = 2, 3, 4 for d → ∞ We show the extinction forbidden entropy at d → ∞, to determine the α d→∞ atyp threshold. In the K = 4 plot note that the p = 0 entropy jumps from 0 to ≈ 0.5 once at α ≈ 1.2. 3. Is the dominant attractor a cycle or a fixed point? We show in [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Entropy Φ(p/c) for c = 1 and c = 2 at K = 2, 3, for 3–random-regular graphs. The entropy of c = 2 attractors is always larger or equal than the entropy of c = 1 fixed points (and is strictly larger at “small” α). At very large α, independent set fixed points become dominant and the two entropies are equal [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of the numerical results with theory for [PITH_FULL_IMAGE:figures/full_fig_p035_13.png] view at source ↗
read the original abstract

Ecological communities with structured interactions exhibit collective phenomena such as percolation and clustering of occupied sites. While these effects have been documented in experiments and simulations, systematic analytical understanding has remained limited. In this paper, we develop a dynamical theory of these phenomena for competitive ecological systems defined on random interaction graphs. We introduce a discrete version of the generalized Lotka-Volterra model that preserves key macroscopic features of continuous ecological dynamics while enabling analytical treatment. Within this framework, we characterize the emergence of percolating clusters and describe the spatial organization of surviving sites. Our analysis uncovers which equilibria can be reached by the dynamics and shows how this dynamical accessibility governs the onset of clustering and percolation. In doing so, our framework complements classical Lotka-Volterra theory by providing a dynamical perspective on the collective organization of structured communities.

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

0 major / 2 minor

Summary. The paper develops a dynamical theory of percolation and clustering for competitive ecological communities on random interaction graphs. It introduces a discrete generalized Lotka-Volterra model asserted to preserve key macroscopic features of the continuous dynamics, enabling analytical characterization of reachable equilibria; the central claim is that dynamical accessibility of these equilibria governs the onset and spatial organization of clustering and percolation, thereby complementing classical Lotka-Volterra theory.

Significance. If the modeling choice and equilibrium analysis hold, the work supplies an analytical link between dynamical reachability and collective network phenomena that have previously been studied mainly via simulation or experiment. This could furnish falsifiable predictions for structured communities and a reproducible discrete framework for further theoretical exploration.

minor comments (2)
  1. [Abstract] The abstract states that the discrete model 'preserves key macroscopic features' but does not indicate where in the manuscript a direct comparison (e.g., steady-state distributions or invasion criteria) with the continuous GLV is provided; adding a short explicit statement or reference to the relevant section would strengthen the claim.
  2. [Methods] Notation for the random interaction graph and the discrete update rule should be introduced with a single consolidated table or figure early in the methods to avoid repeated re-definition later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the work, recognition of its potential significance, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract presents the discrete GLV variant as an explicit modeling choice chosen to preserve macroscopic features while enabling analysis, with subsequent claims about reachable equilibria, clustering, and percolation framed as results of that analysis. No equations, fitted parameters, self-citations, or ansatzes are quoted that reduce any prediction or uniqueness claim back to the inputs by construction. The load-bearing step (preservation of macroscopic features) is stated as an assumption rather than derived, leaving the framework independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the modeling assumption that the discrete model preserves macroscopic features is noted under weakest_assumption instead.

pith-pipeline@v0.9.1-grok · 5679 in / 1135 out tokens · 17414 ms · 2026-06-27T14:11:33.451512+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

80 extracted references · 4 canonical work pages

  1. [1]

    Rietkerk and J

    M. Rietkerk and J. van de Koppel, Regular pattern for- mation in real ecosystems, Trends Ecol. Evol.23, 169 (2008)

    2008

  2. [2]

    Borgogno, P

    F. Borgogno, P. D’Odorico, F. Laio, and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Reviews of Geophysics47(2009)

    2009

  3. [3]

    Manor and N

    A. Manor and N. M. Shnerb, Facilitation, competition, and vegetation patchiness: from scale free distribution to patterns, J. Theor. Biol.253, 838 (2008)

    2008

  4. [4]

    Dakos, S

    V. Dakos, S. K´ efi, M. Rietkerk, E. H. van Nes, and M. Scheffer, Slowing down in spatially patterned ecosys- tems at the brink of collapse., The American Naturalist 177, E153 (2011), pMID: 21597246

    2011

  5. [5]

    K´ efi, M

    S. K´ efi, M. Rietkerk, C. L. Alados, Y. Pueyo, V. P. Pa- panastasis, A. Elaich, and P. C. de Ruiter, Spatial vege- tation patterns and imminent desertification in mediter- ranean arid ecosystems, Nature449, 213 (2007)

    2007

  6. [6]

    von Hardenberg, E

    J. von Hardenberg, E. Meron, M. Shachak, and Y. Zarmi, Diversity of vegetation patterns and deser- tification, Phys. Rev. Lett.87, 198101 (2001)

    2001

  7. [7]

    Rietkerk, M

    M. Rietkerk, M. C. Boerlijst, F. van Langevelde, R. Hil- lerislambers, J. van de Koppel, L. Kumar, H. H. T. Prins, and A. M. de Roos, Self-organization of vegeta- tion in arid ecosystems, Am. Nat.160, 524 (2002)

    2002

  8. [8]

    Hogeweg, Cellular automata as a paradigm for eco- logical modeling, Appl

    P. Hogeweg, Cellular automata as a paradigm for eco- logical modeling, Appl. Math. Comput.27, 81 (1988)

    1988

  9. [9]

    G. B. Ermentrout and L. Edelstein-Keshet, Cellular au- tomata approaches to biological modeling, J. Theor. Biol.160, 97 (1993)

    1993

  10. [10]

    Deutsch and S

    A. Deutsch and S. Dormann,Cellular automaton modeling of biological pattern formation, Modeling and Simulation in Science, Engineering & Technology (Birkh¨ auser Boston, Boston, MA, 2017)

    2017

  11. [11]

    A. M. Turing, The chemical basis of morphogenesis. 1953, Bull. Math. Biol.52, 153 (1990)

    1953

  12. [12]

    D’Odorico, F

    P. D’Odorico, F. Laio, and L. Ridolfi, Patterns as indi- cators of productivity enhancement by facilitation and competition in dryland vegetation, J. Geophys. Res. 111(2006)

    2006

  13. [13]

    Lejeune, P

    O. Lejeune, P. Couteron, and R. Lefever, Short range co-operativity competing with long range inhibition ex- plains vegetation patterns, Acta Oecol. (Montrouge) 20, 171 (1999)

    1999

  14. [14]

    K´ efi, V

    S. K´ efi, V. Guttal, W. A. Brock, S. R. Carpenter, A. M. Ellison, V. N. Livina, D. A. Seekell, M. Scheffer, E. H. van Nes, and V. Dakos, Early warning signals of eco- logical transitions: methods for spatial patterns, PLoS One9, e92097 (2014)

    2014

  15. [15]

    Vega and C

    E. Vega and C. Monta˜ na, Effects of overgrazing and rainfall variability on the dynamics of semiarid banded vegetation patterns: A simulation study with cellular automata, Journal of Arid Environments75, 70 (2011)

    2011

  16. [16]

    Ge, The hidden order of turing patterns in arid and semi-arid vegetation ecosystems, Proceedings of the Na- tional Academy of Sciences120, e2306514120 (2023)

    Z. Ge, The hidden order of turing patterns in arid and semi-arid vegetation ecosystems, Proceedings of the Na- tional Academy of Sciences120, e2306514120 (2023)

    2023

  17. [17]

    Volterra, Fluctuations in the abundance of a species considered mathematically, Nature118, 558–560 (1926)

    V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature118, 558–560 (1926)

    1926

  18. [18]

    Volterra, Fluctuations dans la lutte pour la vie, leurs lois fondamentales et de r´ eciprocit´ e, Bull

    V. Volterra, Fluctuations dans la lutte pour la vie, leurs lois fondamentales et de r´ eciprocit´ e, Bull. Soc. Math. France (1939)

    1939

  19. [19]

    Grilli, T

    J. Grilli, T. Rogers, and S. Allesina, Modularity and stability in ecological communities, Nat. Commun.7, 12031 (2016)

    2016

  20. [20]

    K. Z. Coyte, J. Schluter, and K. R. Foster, The ecology of the microbiome: Networks, competition, and stabil- ity, Science350, 663 (2015)

    2015

  21. [21]

    Faust and J

    K. Faust and J. Raes, Microbial interactions: from net- works to models, Nat. Rev. Microbiol.10, 538 (2012)

    2012

  22. [22]

    MacArthur, Species packing and competitive equilib- rium for many species, Theor

    R. MacArthur, Species packing and competitive equilib- rium for many species, Theor. Popul. Biol.1, 1 (1970)

    1970

  23. [23]

    Biroli, G

    G. Biroli, G. Bunin, and C. Cammarota, Marginally stable equilibria in critical ecosystems, New Journal of Physics20, 083051 (2018)

    2018

  24. [24]

    Altieri, F

    A. Altieri, F. Roy, C. Cammarota, and G. Biroli, Prop- erties of equilibria and glassy phases of the random lotka-volterra model with demographic noise, Physical Review Letters126, 10.1103/physrevlett.126.258301 (2021)

    work page doi:10.1103/physrevlett.126.258301 2021

  25. [25]

    Bunin, Ecological communities with lotka-volterra dynamics, Phys

    G. Bunin, Ecological communities with lotka-volterra dynamics, Phys. Rev. E95, 042414 (2017)

    2017

  26. [26]

    Marcus, A

    S. Marcus, A. M. Turner, and G. Bunin, Local and collective transitions in sparsely-interacting ecological communities, PLoS Comput. Biol.18, e1010274 (2022)

    2022

  27. [27]

    Tonolo, M

    T. Tonolo, M. C. Angelini, S. Azaele, A. Maritan, and G. Gradenigo, Generalized Lotka-Volterra model with sparse interactions: Non-Gaussian effects and topolog- ical multiple-equilibria phase, PRX Life4(2026)

    2026

  28. [28]

    V. Ros, F. Roy, G. Biroli, G. Bunin, and A. M. Turner, Generalized lotka-volterra equations with random, non- reciprocal interactions: The typical number of equilib- ria, Phys. Rev. Lett.130, 257401 (2023)

    2023

  29. [29]

    Chesson, MacArthur’s consumer-resource model, Theor

    P. Chesson, MacArthur’s consumer-resource model, Theor. Popul. Biol.37, 26 (1990)

    1990

  30. [30]

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

    2021

  31. [31]

    J. Gore, J. Hu, Y. He, M. Barbier, J. Song, and G. Bunin, Transition from global stability to multiple attractors in microcosms (2025)

    2025

  32. [32]

    Aguirre-L´ opez, Heterogeneous mean-field analysis of the generalized Lotka–Volterra model on a network, J

    F. Aguirre-L´ opez, Heterogeneous mean-field analysis of the generalized Lotka–Volterra model on a network, J. Phys. A Math. Theor.57, 345002 (2024)

    2024

  33. [33]

    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 Me- chanics: Theory and Experiment2018, 033406 (2018)

    2018

  34. [34]

    W. Cui, R. Marsland, and P. Mehta, Effect of re- source dynamics on species packing in diverse ecosys- tems, Physical Review Letters125, 10.1103/phys- revlett.125.048101 (2020)

    work page doi:10.1103/phys- 2020

  35. [35]

    B. R. Silliman, M. J. Hensel, J. P. Gibert, P. Daleo, C. S. Smith, D. J. Wieczynski, C. Angelini, A. B. Pax- ton, A. M. Adler, Y. S. Zhang, A. H. Altieri, T. M. Palmer, H. P. Jones, R. K. Gittman, J. N. Griffin, M. I. O’Connor, J. van de Koppel, J. R. Poulsen, M. Rietk- erk, Q. He, M. D. Bertness, T. van der Heide, and S. R. Valdez, Harnessing ecological ...

    2024

  36. [36]

    R. J. Orth, J. S. Lefcheck, K. S. McGlathery, L. Aoki, M. W. Luckenbach, K. A. Moore, M. P. J. Oreska, R. Snyder, D. J. Wilcox, and B. Lusk, Restoration 11 of seagrass habitat leads to rapid recovery of coastal ecosystem services, Science Advances6, eabc6434 (2020)

    2020

  37. [37]

    A. J. Wells, J. Harrington, and N. J. Balster, Seeding density alters the assembly of a restored plant commu- nity after the removal of a dam in southern wisconsin, usa, Environments11, 10.3390/environments11060115 (2024)

    work page doi:10.3390/environments11060115 2024

  38. [38]

    J. D. Corbin and K. D. Holl, Applied nucleation as a forest restoration strategy, Forest Ecology and Manage- ment265, 37 (2012)

    2012

  39. [39]

    M. L. E. Gr¨ afnings, J. H. T. Heusinkveld, N. Hijner, D. J. J. Hoeijmakers, Q. Smeele, M. Zwarts, T. van der Heide, and L. L. Govers, Spatial design improves effi- ciency and scalability of seed-based seagrass restoration, Journal of Applied Ecology60, 967 (2023)

    2023

  40. [40]

    E. L. Kjaer, G. R. Houseman, K. N. Luu, B. L. Foster, L. Laanisto, and A. J. Golubski, Spatial pattern of seed arrival has a greater effect on plant diversity than does soil heterogeneity in a grassland ecosystem, Plant Soil (2024)

    2024

  41. [41]

    Behrens, B

    F. Behrens, B. Hudcov´ a, and L. Zdeborov´ a, Backtrack- ing dynamical cavity method, Phys. Rev. X13, 031021 (2023)

    2023

  42. [42]

    Behrens, B

    F. Behrens, B. Hudcov´ a, and L. Zdeborov´ a, Dynamical phase transitions in graph cellular automata, Phys. Rev. E109, 044312 (2024)

    2024

  43. [43]

    H. Sun, S. Chen, J. Xie, and Y. Hu, Growth-induced percolation on complex networks, PNAS Nexus4, pgaf192 (2025)

    2025

  44. [44]

    Timonin, Statistics of geometric clusters in the ising model on a bethe lattice, Physica A: Statistical Mechan- ics and its Applications527, 121402 (2019)

    P. Timonin, Statistics of geometric clusters in the ising model on a bethe lattice, Physica A: Statistical Mechan- ics and its Applications527, 121402 (2019)

    2019

  45. [45]

    D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Network robustness and fragility: Perco- lation on random graphs, Phys. Rev. Lett.85, 5468 (2000)

    2000

  46. [46]

    Karrer and M

    B. Karrer and M. E. J. Newman, Message passing ap- proach for general epidemic models, Phys. Rev. E82, 016101 (2010)

    2010

  47. [47]

    Karrer, M

    B. Karrer, M. E. J. Newman, and L. Zdeborov´ a, Perco- lation on sparse networks, Phys. Rev. Lett.113, 208702 (2014)

    2014

  48. [48]

    M. E. J. Newman, Spread of epidemic disease on net- works, Phys. Rev. E66, 016128 (2002)

    2002

  49. [49]

    Cohen, K

    R. Cohen, K. Erez, D. ben Avraham, and S. Havlin, Resilience of the internet to random breakdowns, Phys. Rev. Lett.85, 4626 (2000)

    2000

  50. [50]

    J. Xie, X. Wang, L. Feng, J.-H. Zhao, W. Liu, Y. Moreno, and Y. Hu, Indirect influence in social networks as an induced percolation phenomenon, Pro- ceedings of the National Academy of Sciences119, e2100151119 (2022)

    2022

  51. [51]

    Shrestha, S

    M. Shrestha, S. V. Scarpino, and C. Moore, Message- passing approach for recurrent-state epidemic models on networks, Phys. Rev. E92, 022821 (2015)

    2015

  52. [52]

    Dobrinevski and E

    A. Dobrinevski and E. Frey, Extinction in neutrally sta- ble stochastic lotka-volterra models, Phys. Rev. E85, 051903 (2012)

    2012

  53. [53]

    Knebel, M

    J. Knebel, M. F. Weber, T. Kr¨ uger, and E. Frey, Evolutionary games of condensates in coupled birth–death processes, Nature Communications6, 10.1038/ncomms7977 (2015)

    work page doi:10.1038/ncomms7977 2015

  54. [54]

    M. M. Desai and D. S. Fisher, Beneficial muta- tion–selection balance and the effect of linkage on pos- itive selection, Genetics176, 1759 (2007)

    2007

  55. [55]

    Stauffer and A

    D. Stauffer and A. Aharony,Introduction to percolation theory, 2nd ed. (Taylor & Francis, Philadelphia, PA, 1992)

    1992

  56. [56]

    Erd˝ os and A

    P. Erd˝ os and A. R´ enyi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci. (1960)

    1960

  57. [57]

    C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science284, 1826–1828 (1999)

    1999

  58. [58]

    M´ ezard and A

    M. M´ ezard and A. Montanari,Information, Physics, and Computation(2009)

    2009

  59. [59]

    M. E. J. Newman, Threshold effects for two pathogens spreading on a network, Phys. Rev. Lett.95, 108701 (2005)

    2005

  60. [60]

    V. Ros, F. Roy, G. Biroli, and G. Bunin, Quenched complexity of equilibria for asymmetric generalized lotka–volterra equations, Journal of Physics A: Mathe- matical and Theoretical56, 305003 (2023)

    2023

  61. [61]

    Jankola, F

    M. Jankola, F. Behrens, C. Koller, and L. Zdeborov´ a, Minority takeover in majority dynamics: Searching for rare initializations via the history passing algorithm, Phys. Rev. E113, 064303 (2026)

    2026

  62. [62]

    Koller, F

    C. Koller, F. Behrens, and L. Zdeborov´ a, Counting and hardness-of-finding fixed points in cellular automata on random graphs, J. Phys. A Math. Theor.57, 465001 (2024)

    2024

  63. [63]

    Zdeborov´ a and F

    L. Zdeborov´ a and F. Krzakala, Statistical physics of in- ference: thresholds and algorithms, Advances in Physics 65, 453 (2016)

    2016

  64. [64]

    M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E64, 026118 (2001)

    2001

  65. [65]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Brad- bury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. K¨ opf, E. Z. Yang, Z. De- Vito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, Pytorch: An imper- ative style, high-performance deep learning library., in NeurIPS(2019) pp. 8024–8035. 12 A...

    2019

  66. [66]

    We show in Figure 4 that when the system is simulated on a grid (each site has 4 neighbors, right, left, top and bottom) more complex spatial structures can emerge

    Discrete gLV on a grid. We show in Figure 4 that when the system is simulated on a grid (each site has 4 neighbors, right, left, top and bottom) more complex spatial structures can emerge. In particular we see the formation of ring-like patterns of occupied sites, which is consistent with what is observed in real ecosystems [16], and we graphically observ...

  67. [67]

    normalized

    The discrete dynamic in the limit of largeK. We show that for largeKthe discrete gLV model in Eq. (2) approximates very well the continuous model in Eq. (1). We carry out numerical integration of the differential equations for continuous dynamical rule in Eq. (1) 13 0 250 500 750 1000 0.00 0.25 0.50 0.75 1.00 Discrete (Top) vs Continuous (Bottom) gL V α= ...

  68. [68]

    We simulate our dynamics for heterogeneous interaction strengths and measure the observablesρ 0 andϕ LC, the results are shown in Figure 6

    Numerical simulations on tree-like graphs with non-uniform interactions. We simulate our dynamics for heterogeneous interaction strengths and measure the observablesρ 0 andϕ LC, the results are shown in Figure 6. In particular, we consider the following distributions for the couplingsα ij. For the discrete model we fix the carrying capacity toK= 100, and ...

  69. [69]

    Symmetric Gaussian.α ij sampled from a truncated Gaussian distribution with the constraintα ij =α ji (symmetric interactions). We consider distinct values of the varianceσ, while we callµthe average interaction strength 14 0.0 0.2 0.4 0.6 ρ0 Symmetric Gaussian Nonreciprocal Gaussian Directed Gaussian 0.0 0.5 1.0 µ (Avg. αij) 0.00 0.25 0.50 0.75 1.00 φLC C...

  70. [70]

    Nonreciprocal Gaussian.α ij sampled from a truncated Gaussian distribution without any symmetry con- straint (asymmetric interactions)

  71. [71]

    converge

    Directed Gaussian.α ij sampled from a truncated Gaussian distribution with the constraint that ifα ij >0 thenα ji = 0, corresponding to a directed network (i.e., interactions are one-way). For comparison, we also simulate the continuous gLV model on systems of the same size. We find that the behavior ofρ 0 andϕ LC is qualitatively similar across all sampl...

  72. [72]

    variational

    The Backtracking Dynamical Cavity Method Detailed expressions of the probability distributions needed to computeαatyp,α ext and observables. We now detail how to compute the entropy Φ (p/c) and the average values of the observables with respect to the following two probability distributions, adopting the same procedure originally introduced in [41, 42]. F...

  73. [73]

    X h∈∂i\j αihN t h X h∈∂i\j αihN t′ h # −E {N h∼ψh}h∈∂i\j

    High degree limit of the BDCM equations Derivation of the expression of the messages in the high degree limit.We derive now the high degree limit of the BDCM equations. For generality, let’s considerα ij to be i.i.d. with varianceσ/ √ dand expectation α/d, for ad-regular graph. The goal is to take the limitd→ ∞, and we will later focus on the caseσ= 0 (un...

  74. [74]

    Message passing for dynamics-dependent percolation In this section, we develop the general message passing approach that we use to study the percolation of an attractor of the gLV dynamic. We stress from the start that the equations derived below can be applied to any dynamical system, including stochastic ones, whose underlying probability measure admits...

  75. [75]

    Simplifying the general percolation recursion with an approximation.The approach we developed in the previous section is asymptotically exact, but solving the recursion in Eq

    (one can do exactly the same argument for bond percolation and find the recursion reported in [47]). Simplifying the general percolation recursion with an approximation.The approach we developed in the previous section is asymptotically exact, but solving the recursion in Eq. (D46) may still be complicated. Thus, we introduce an approximation that signifi...

  76. [76]

    ˜η (i) l (N p+1 i ,{N p+1 j }j∈∂i) is an indicator function that equals 1 if siteiis occupied and connected toloccupied neighbors, and 0 otherwise

    Computing the structure functionsη(ℓ)with FFT To compute this observable, we introduce the following edge localized observable ˜ηl(Np+1) = 1 S SX i=1 1[N p+1 i >0]1 hX j∈∂i 1[N p+1 j >0] =l i = 1 S SX i ˜η(i) l (N p+1 i ,{N p+1 j }j∈∂i).(E2) Remember that if a site is occupied at timep+ 1 it will be occupied over the full attractor. ˜η (i) l (N p+1 i ,{N ...

  77. [77]

    smallest possible initialization

    General properties about the percolation fixed point recursion and discussion about the initialization We start by firstly defining the shorthandq iℓ→i N iℓ ,N i =H iℓ→i N iℓ ,N i (1) and to write the percolation recursion in Eq. (D47), specialized to the BDCM case, as qiℓ→i N iℓ ,N i =    1,ifA iℓ = 0, P {N u}u∈∂iℓ \i Aiℓ N iℓ , N i,{N u}u∈∂iℓ\i Q...

  78. [78]

    (D24), which allow us to compute the extinction fractionρ 0 (via Eq

    Computing the critical extinction threshold and the atypical threshold atd=∞ We present in Figure 9 (K= 2) and Figure 10 (K= 3) the numerical solutions of Eq. (D24), which allow us to compute the extinction fractionρ 0 (via Eq. (D26)) in the fully connected limit (d=∞), and thus determine the corresponding extinction thresholdα d→∞ ext . These results sup...

  79. [79]

    This shows that the dominant attractor is a 2-cycle for any value of alpha, excluding very large alpha

    Is the dominant attractor a cycle or a fixed point? We show in Figure 12, forK= 2 andK= 3, the comparison of the entropies of the (p/c)-backtracking attractors withc= 2 andc= 1. This shows that the dominant attractor is a 2-cycle for any value of alpha, excluding very large alpha. Indeed, at largeα, the dominant attractor is actually an independent set fi...

  80. [80]

    We show also the value ofϕ LC obtained with the approximation discussed in Appendix D 3, and compare it with the exact result

    Additional plots forK= 2andd≥3 We show in Figure 13 the comparison between numerical simulations and theory for higherd. We show also the value ofϕ LC obtained with the approximation discussed in Appendix D 3, and compare it with the exact result. This supports the claim that the approximation works very well for almost alld, as the differences between ap...

    1950