arxiv: 2604.11555 · v1 · submitted 2026-04-13 · 🧬 q-bio.PE

Recognition: no theorem link

Will a Large Complex System be Stable? Revisited

Michael Thorne

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification 🧬 q-bio.PE

keywords ecological stabilitycomplexity-stability debateMay's modelrandom matrix theoryecosystem dynamicsstability analysismathematical ecology

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The pith

A mathematical approach developed after the 1970s shows ecological systems can stay stable as they grow larger and more complex.

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

The paper revisits Robert May's 1972 result that larger ecological systems with more interactions tend to lose stability. May reached this using random matrix theory, which led to the long-running complexity-stability debate. By applying a later mathematical framework, the analysis skips random matrices and instead traces stability to particular mechanisms in the interaction structure. This produces a more detailed picture that weakens the broad claim of inevitable instability with added size and complexity. A sympathetic reader would care because the outcome affects how models treat biodiversity, food-web persistence, and ecosystem responses to change.

Core claim

Using a mathematical approach unavailable in the early 1970s, May's original argument can be re-examined without random matrix techniques. The revised analysis supplies concrete mechanisms that govern whether stability holds, showing that the earlier broad conclusion does not follow once those mechanisms are taken into account.

What carries the argument

A post-1970s mathematical framework for stability analysis that examines specific interaction mechanisms instead of the statistical properties of random matrices.

If this is right

Stability depends on identifiable structural features rather than on the sheer number of species or links.
The complexity-stability debate can be narrowed by replacing the statistical argument with mechanism-specific conditions.
Large systems remain stable when those conditions are met, even if they would appear unstable under random-matrix assumptions.
Critiques of May's model gain precision once the deciding mechanisms are isolated.

Where Pith is reading between the lines

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

Modelers could test the approach on measured food-web matrices to see whether real systems fall on the stable side of the new criteria.
Conservation planning might shift from limiting diversity to preserving the interaction patterns the method identifies as stabilizing.
The same framework could be checked against other complex networks whose stability is currently assessed only with random-matrix tools.

Load-bearing premise

That the post-1970s approach can be applied directly to ecological interaction models in a way that produces valid, mechanism-level results capable of overturning the random-matrix conclusion.

What would settle it

A side-by-side calculation on the same large interaction matrix in which the new method predicts stability while May's random-matrix criterion predicts instability, followed by numerical integration or empirical checks that confirm which prediction matches the actual dynamics.

read the original abstract

Over fifty years ago, Robert May applied random matrix theory to show that as ecological systems grow in size, stability decreases. What emerged from this and the critique that followed was decades of what has been called the complexity-stability debate. However, decades of critique over the assumptions that Robert May applied in carrying out his analysis have not been enough to fully dispel the strength of his conclusion and close the debate. Drawing on a mathematical approach that had not yet been fully developed in the early 70s, it is possible to revisit the argument without the use of random matrix techniques, and provide more detailed understanding of the mechanisms that play a deciding role in stability of ecological systems, countering the broad conclusion that led to the complexity-stability debate.

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.

Desk Editor's Note private letter to a colleague

The paper claims a mechanism-level counter to May's stability result but does not show it applies to the generic random matrix ensemble.

read the letter

The key point is that this paper tries to revisit May's 1972 result on ecological stability without using random matrix theory, but it is not clear from the abstract whether the new approach actually changes the stability probability for the standard ensemble of random interaction matrices. It does a decent job of highlighting the need for mechanism-level understanding beyond the broad probabilistic conclusion. The claim is that post-1970s math can provide more detailed insights into what drives stability in these systems. The soft spot is exactly the one in the stress test. May's argument is that the fraction of stable systems goes to zero as size grows for random matrices with certain scaling. Countering the broad conclusion requires showing a different result for that same random case. The abstract gives no indication that the alternative method derives a new stability criterion or probability for unstructured random matrices. It could be limited to cases with extra structure that are already known to be more stable. This makes the central claim hard to evaluate without the derivations. The paper might be useful for explaining specific mechanisms, but it risks not landing on the probabilistic heart of the debate. This work is aimed at researchers in the complexity-stability area of ecology and network science. Someone familiar with May's paper and the subsequent critiques could find value in the alternative framing if the math checks out. I would not cite it yet because it does not appear to deliver a new quantitative result on the random ensemble. It deserves peer review so that experts can verify whether the non-RMT approach truly counters the original finding or just applies to restricted cases.

Referee Report

1 major / 0 minor

Summary. The paper revisits Robert May's 1972 result that large complex ecological systems tend to be unstable, arguing that a mathematical approach developed after the early 1970s permits re-examination of the stability question without random matrix theory. This is claimed to yield more detailed, mechanism-level insights into the factors governing stability and to counter the broad conclusion that drove the complexity-stability debate.

Significance. If the non-RMT construction can be shown to recover or alter the stability probability for the same unstructured random interaction ensembles used by May, the work would supply a valuable alternative perspective on the mechanisms that control stability in large systems and could help narrow the decades-long debate.

major comments (1)

The central claim requires demonstrating that the post-1970s method produces a stability probability (or criterion) for the ensemble of random matrices with mean-zero entries and variance scaled by 1/N that differs from May's RMT result. No such derivation, eigenvalue analysis, or explicit probability calculation for this ensemble appears in the manuscript; without it the result applies only to already-known structured cases and does not counter the probabilistic claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and insightful review of our manuscript. We address the major comment below and describe the revisions we will undertake to address the concerns raised.

read point-by-point responses

Referee: The central claim requires demonstrating that the post-1970s method produces a stability probability (or criterion) for the ensemble of random matrices with mean-zero entries and variance scaled by 1/N that differs from May's RMT result. No such derivation, eigenvalue analysis, or explicit probability calculation for this ensemble appears in the manuscript; without it the result applies only to already-known structured cases and does not counter the probabilistic claim.

Authors: We agree that an explicit demonstration for the unstructured ensemble is essential to counter the probabilistic aspect of May's claim. Although our current manuscript emphasizes the mechanistic insights afforded by the post-1970s approach in various structured settings, the underlying framework is general and not limited to those cases. In the revised version, we will add a derivation of the stability probability for the mean-zero, variance 1/N ensemble. This will include an analysis showing the conditions under which stability holds and how the mechanisms identified provide a more detailed understanding than the RMT approach alone. We will also discuss how this recovers or modifies the original result, thereby directly addressing the broad conclusion. revision: yes

Circularity Check

0 steps flagged

No circularity identified; alternative non-RMT approach claimed without equations or self-referential reductions

full rationale

The provided abstract and context describe a revisit of May's 1970s result using a post-1970s mathematical approach that avoids random matrix techniques, aiming to supply mechanism-level insights that counter the broad complexity-stability conclusion. No equations, derivations, fitted parameters, self-citations, or uniqueness theorems appear in the text. Without visible load-bearing steps that reduce by construction to inputs (e.g., no stability metric defined in terms of itself or renamed from prior results), the derivation chain cannot be shown to collapse. This matches the default expectation that most papers are non-circular when no specific reduction is quotable; the claim remains an assertion of independent content until full text equations are inspected.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the central claim rests on the unstated validity of the alternative mathematical framework.

pith-pipeline@v0.9.0 · 5409 in / 1095 out tokens · 35944 ms · 2026-05-10T15:29:03.796463+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references

[1]

MacArthur, R. H. (1955). Fluctuations of animal populations and a measure of community stability.Ecology, 36(3), 533–536

1955
[2]

(1927).Animal Ecology

Elton, C. (1927).Animal Ecology. Macmillan Co

1927
[3]

Odum, E. P. (1953).Fundamentals of Ecology. W.B. Saunders Co

1953
[4]

R., & Ashby, W

Gardner, M. R., & Ashby, W. R. (1970). Connectance of large dynamic (cybernetic) systems: Critical values for stability.Nature, 228, 784

1970
[5]

May, R. M. (1972). Will a large complex system be stable?Nature, 238, 413–414

1972
[6]

May, R. M. (1973).Stability and Complexity in Model Ecosystems. Princeton University Press

1973
[7]

E., & Stephens, D

Cohen, J. E., & Stephens, D. W. (1978).Food Webs and Niche Space. Princeton University Press

1978
[8]

Pimm, S. L. (1979). The structure of food webs.Theoretical Population Biology. 16, 144-158

1979
[9]

Pimm, S. L. (1982).Food Webs. Chapman & Hall

1982
[10]

E., & Newman, C

Cohen, J. E., & Newman, C. M. (1985). A stochastic theory of community food webs I. Models and aggregated data. Proc. R. Soc. Lond. B 224, 421-448

1985
[11]

C., Neutel, A.-M., & Moore, J

De Ruiter, P. C., Neutel, A.-M., & Moore, J. C. (1995). Energetics, patterns of interaction strengths, and stability in real ecosystems.Science, 269, 1257-60

1995
[12]

McCann, K., Hastings, A., & Huxel, G. R. (1998). Weak trophic interactions and the balance of nature.Nature, 395, 794-798

1998
[13]

Neutel, A.-M., Heesterbeek, J. A. P., & De Ruiter, P. C. (2002). Stability in real food webs. Science, 296, 1120-3

2002
[14]

Neutel, A.-M., Heesterbeek, J., van de Koppel, J., et al. (2007). Reconciling complexity with stability in naturally assembling food webs.Nature, 449, 599–602

2007
[15]

Allesina, S., & Tang, S. (2012). Stability criteria for complex ecosystems.Nature, 483, 205-8

2012
[16]

Lyapunov, A. M. (1892).The General Problem of the Stability of Motion. English translation, Taylor & Francis, 1992

1992
[17]

LaSalle, J. P. (1960). Some Extensions of Liapunov’s Second Method.IRE Transactions on Circuit Theory, 7(4), 520–527. 4

1960
[18]

E., & Yakubovich, V

Kalman, R. E., & Yakubovich, V. A. (1963). On the Stability of Linear Systems.Proceedings of the National Academy of Sciences, 49(2), 260–263

1963
[19]

Karmarkar, N. (1984). A New Polynomial-Time Algorithm for Linear Programming.Com- binatorica, 4(4), 373–395

1984
[20]

(1994).Interior-Point Polynomial Algorithms in Convex Programming

Nesterov, Y., & Nemirovskii, A. (1994).Interior-Point Polynomial Algorithms in Convex Programming. SIAM studies in applied mathematics, vol. 13

1994
[21]

(1994).Linear Matrix Inequalities in System and Control Theory

Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994).Linear Matrix Inequalities in System and Control Theory. SIAM studies in applied mathematics, vol. 15. 5

1994