A general relationship between extinction risk and carrying capacity

Andrew Balmford; Ben Balmford; Daniele Rinaldo; Piero Visconti; Rhys Green; Thomas S Ball

REVIEW 4 major objections 5 minor 1 cited by

A general relationship between extinction risk and carrying capacity

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Extinction risk versus carrying capacity collapses onto a single modified Gompertz curve.

desk verdict The simulations are impressive and the empirical modified-Gompertz pattern is credible, but Appendix F's theoretical derivation is internally inconsistent and does not support the claimed universality. read the letter →

arxiv 2411.13228 v2 pith:ZZOYZYEC submitted 2024-11-20 q-bio.PE

Thomas S Ball , Ben Balmford , Andrew Balmford , Daniele Rinaldo , Piero Visconti , Rhys Green This is my paper · ORCID

classification q-bio.PE MSC 92D2592D4060J70

keywords extinctionriskcarryingcapacitymodifiedGompertzcurvepopulationviabilityanalysisdensitydependencestochasticdynamicsOrnstein-Uhlenbeckapproximationconservationprioritization

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 sets out to establish a universal quantitative law: for any population whose growth is limited by density dependence, the probability of extinction within a fixed horizon is a modified Gompertz function of carrying capacity, $P_E(K) = 1 - \exp(-\exp(a + b K^{\gamma}))$. The claim is supported by roughly five billion individual-based simulations across four population models of increasing realism, with fits exceeding $r^2 = 0.995$ for every viable parameter combination, and by an analytic argument that starts from the logistic stochastic differential equation and reduces it near carrying capacity to an Ornstein-Uhlenbeck process. If the law holds, then the extinction-risk consequences of habitat loss can be read off a three-parameter curve, and conservation triage can be guided by where a population sits on that curve. The paper also shows that the standard Gompertz curve ($\gamma = 1$) fits well but systematically underestimates risk at high carrying capacity, and that the modified curve's inflection point consistently lies at extinction probabilities above 0.5, around 0.65 to 0.85.

What carries the argument

The load-bearing object is the modified Gompertz function $P_S(K) = \exp(-\exp(a + b K^{\gamma}))$, where $\gamma$ is a shape parameter that skews the curve and is consistently estimated below one across all models and most empirical datasets. The analytic machinery is the reduction of the logistic stochastic differential equation near carrying capacity to an Ornstein-Uhlenbeck process, followed by a first-eigenvalue approximation of the first-passage-time distribution to the extinction boundary; this turns an exponential decay in time into a double-exponential (Gompertz) dependence on $K$. The shape parameter $\gamma$ corrects the systematic deviation of the standard Gompertz at high $K$, and the paper argues this improvement is systematic rather than a dimensionality artifact because $\gamma$ clusters below one.

What would settle it

For a viable parameter set with $r/\sigma^2$ just above the extinction threshold and $K$ around 100, compute the exact first-passage-time survival probability of the logistic stochastic differential equation by summing the full spectral expansion rather than keeping only the first eigenvalue; if the exact curve deviates from the fitted modified Gompertz by more than the Monte Carlo error of 10,000 replicates, the claimed universality fails in that regime.

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Extended reading notes

Core claim

The central discovery is that survival probability as a function of carrying capacity takes the form $P_S(K) = \exp(-\exp(a + b K^{\gamma}))$, and therefore extinction probability follows a modified (asymmetric) Gompertz sigmoid. This pattern held across more than 1700 simulation settings spanning two-sex demography, biparental care, delayed maturation, and temporally autocorrelated environmental noise, and it also held when populations started away from carrying capacity or experienced Allee effects. The paper argues that the form is not empirical coincidence: linearizing the logistic stochastic differential equation around carrying capacity gives an Ornstein-Uhlenbeck process, and keeping only the slowest mode of its first-passage-time spectrum yields a Gompertz survival curve in $K$ under just two assumptions, negative density dependence and a viable steady state at large $K$. This is a broad universality claim: it does not depend on the specific mechanism of density dependence, and it implies that the extinction-risk-versus-capacity relationship for any extant population can be summarized by three parameters.

Load-bearing premise

The universal curve rests on treating a population near carrying capacity as an Ornstein-Uhlenbeck process and on assuming that, over 100 years, survival is dominated by the slowest decaying mode of that process; if that spectral approximation fails at moderate carrying capacities, the analytic claim that every density-dependent population follows a Gompertz curve is unsupported even if the simulation fits remain excellent.

Editorial extensions

If this is right

A power-law or logistic curve misstates how extinction risk changes with carrying capacity; for populations at low risk, a power law underestimates the rate of change, and a logistic places the maximum benefit of intervention at $P_E = 0.5$ rather than at the observed $0.65$ to $0.85$.
For a population already at risk, the largest marginal reduction in extinction probability from increasing carrying capacity occurs when current extinction probability is roughly 0.65 to 0.85, which can inform where to target habitat protection or restoration.
The carrying capacity at which a population has a 10% chance of extinction in 100 years ($K_{10}$, aligned with IUCN Criterion E) can be estimated from growth rate and environmental stochasticity via linear models, at least for the simpler population structures, offering a rapid assessment tool.
The same Gompertz relationship holds when initial population size differs from carrying capacity and when Allee effects are present, so the curve can be applied to reintroductions, newly protected habitats, and populations with mate-finding or cooperative-breeding constraints.

Reading between the lines

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

If the modified Gompertz is truly universal, then comparing species reduces to comparing three parameters, so a promising next step is to model $a$, $b$, and $\gamma$ as functions of measurable life-history traits and phylogenetic relatedness; the paper only predicts the standard Gompertz parameters from traits.
The consistent $\gamma < 1$ across models and empirical datasets hints that extinction risk is a stretched-exponential function of $K$; whether $\gamma$ clusters by taxon or environmental regime is a testable extension not pursued here.
The analytic approximation should degrade for small carrying capacities or for parameter sets near the viability boundary, so a targeted simulation campaign at $K$ between 1 and 100 would map the curve's domain of validity; the paper does not report such a boundary.
Because the curve is asymmetric, the extinction-risk impact of a proportional habitat loss depends on where the population starts on the curve, implying that safe operating spaces for habitat retention should be defined relative to each population's inflection point.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Nick, the short version: the simulation work is substantial and the empirical result is worth knowing, but the theoretical appendix is not in a shape to support the universality claim. As it stands, the paper does not establish a general law.

What is actually new is a coordinated simulation campaign across four progressively realistic population models—billions of runs—all pointing to a modified Gompertz relationship PE(K)=1−exp(−exp(a+bK^γ)), with fitted γ consistently below 1. That systematic deviation from the standard Gompertz (Wolff et al. 2023) is a real finding, and the comparison against power-law and logistic alternatives is clear and useful. The practical regressions for K10 and related thresholds are a nice bonus, and the analyses of non-equilibrium initial sizes and Allee effects widen the scope. Code and data are deposited, which is exactly the kind of reproducibility that makes the empirical core credible.

The soft spot is Appendix F. It does not hold together as printed: the Fokker–Planck drift has a sign error, the diffusion term is written with x rather than x², the stated stationary Gamma distribution does not match the printed FPE, and the later 'Lamperti transform' is not a transform of the original SDE. With the correct noise amplitude, the OU approximation produces no K-dependence in survival over a fixed horizon; the K-dependence in t1(K) and equation (F15) appears to be an artifact of the rescaling. There is also no error bound for keeping only the first eigenvalue. More important, even a repaired derivation would at best yield the standard Gompertz (γ=1), not the modified form that the paper's empirical discovery hinges on. So the abstract's 'few assumptions suffice' claim overreaches. The authors admit γ is fitted and that they could not predict the modified Gompertz from life-history parameters, which is honest, but the framing in the introduction and abstract does not reflect that limitation.

The empirical content deserves a serious referee; the theory needs major repair or an explicit demotion to a provisional sketch. I would send it to review, but with a directive to fix Appendix F, provide bounds or drop the analytical claim, and reframe the paper as a well-supported empirical generalization with a tentative theoretical note.

Referee Report

4 major / 5 minor

Summary. The paper claims that the probability of extinction of a population within 100 years, as a function of carrying capacity K, is universally well described by a modified Gompertz curve, PS(K) = exp(-exp(a + b K^γ)). This claim is supported by a very large simulation campaign across four population models of increasing demographic complexity, by fits to three empirical/observational datasets, and by an analytical argument in Supplementary Appendix F intended to show that a standard Gompertz curve follows from a logistic stochastic differential equation under two broad assumptions. The authors further use the fitted curve to discuss conservation triage, focusing on the inflection point and on the carrying capacity K10 corresponding to 10% extinction risk.

Significance. If the empirical relationship holds as a general pattern, this is a valuable contribution to conservation science: it provides a compact quantitative description of extinction risk versus carrying capacity that could inform status assessments and priority setting. The simulation effort is extensive and the data and code are made publicly available, which is a strength of the paper. The empirical fits are consistently very good (R² > 0.99 for simulated curves, and strong fits for the three empirical examples). However, the analytical derivation in Appendix F, which is the basis for the claim that the relationship is universal and follows from 'few assumptions', contains serious mathematical inconsistencies. As printed, the derivation does not bridge the gap from the stochastic logistic equation to the modified Gompertz law, and it does not predict the fitted shape parameter γ. The empirical law may still be a useful descriptive generalization, but the theoretical universality claim is not supported by the material presented.

major comments (4)

[Appendix F, Eq. (F3)] The Fokker-Planck equation does not match the stochastic differential equation (F1). For dX = rX(1 - K^{-1}X)dt + σX dW, the Kolmogorov forward equation should have drift -∂x[r x(1 - x/K)P] and diffusion (σ²/2)∂xx[x² P]. Equation (F3) instead has a plus sign before the K^{-1} term and a diffusion coefficient proportional to x rather than x². As a result, the stationary distribution (F4) is not the stationary distribution of (F1), and the transition density used in the subsequent first-passage calculation is not well defined.
[Appendix F, OU approximation after Eq. (F13)] The Ornstein-Uhlenbeck approximation is internally inconsistent. For X near K, with Y = X - K, the correct linearization of (F1) is dY ≈ -rY dt + σK dW, not dY = -ρY dt + σ dW with ρ = r/K. With the correct noise amplitude σK, the absorbing boundary at Y = -(K-1) is at a standardized distance approximately 1/σ, independent of K. Consequently the first-passage properties of the OU process do not depend on K, and the K-dependence in Eqs. (F14) and (F15) is an artifact of the inconsistent scaling. The claimed Gompertz dependence of survival on K is therefore not obtained from this approximation.
[Main text, 'The theoretical basis for the modified Gompertz' and Eq. (1)] The analytical derivation, even if corrected, yields at most a standard Gompertz curve (γ = 1), not the modified Gompertz curve with γ < 1 that is fitted to the simulations. The shape parameter γ is estimated from the same simulation data used to establish the law, and the paper provides no theoretical mechanism predicting its value. Thus the statement that 'few assumptions suffice to show that the relationship holds' for the modified Gompertz law is not supported by the derivation; the gap between the derived standard Gompertz and the empirically fitted modified Gompertz is acknowledged in the text but not resolved.
[Appendix F, Eqs. (F14)-(F15) and 'large K' claim] No error bound or quantified domain of validity is provided for the first-eigenvalue approximation, and Eq. (F15) has dimensional inconsistencies (the exponent contains products of r, K, and σ with units that do not cancel). The paper repeatedly invokes 'large K' and 'large t' without specifying how large is required, which is especially important because the simulations use a fixed 100-year horizon and K values from 1 to 3 million.

minor comments (5)

[Main text, Introduction] There is a typographical issue in the sentence 'Wolff et al. (2023) re-analysed Hilbers et al. (2016) data': the year should refer to the Hilbers et al. (2016) study consistently.
[Supplementary Appendix E, Section E] The text refers to 'Figure 4' and 'Figure 1(b)' in several places, but the actual figure numbers in the appendix are E1, E2, E3, and E4; these cross-references should be corrected.
[Supplementary Appendix G, Figures G2-G5] The captions of Figures G2-G5 mention 'the distribution of differences between the R2 values', but the main text does not state which baseline R² (modified vs. standard) is being compared; clarifying the definition of 'difference' would improve readability.
[Supplementary Appendix A, Eq. (A10)] Equation (A10) appears to be a rearrangement of Equation 17 of Niel and Lebreton (2005), but the relation between the annual survival rate S_a and the age at first breeding B is written without a derivation; providing the original equation or a reference would help the reader verify the formula.
[Main text, Figure 3 caption] The caption of Figure 3 states that the power-law curve 'takes the form 1 - PE ∝ (K + X0)^0.25', but the notation is not standard; the authors should specify that this is a fitted proportional relationship and define the horizontal offset X0 more explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the modified Gompertz curve is an empirical fit to independent simulations and external datasets, while the analytical derivation concerns the standard Gompertz; the fitted gamma is a transparent empirical refinement, not a disguised prediction.

full rationale

The paper's central claim is two-part. First, empirically, roughly 5 billion PVA simulations across Models A-D, plus three external case-study datasets (bighorn sheep, brown mudfish, grizzly bear), are closely fitted by a modified Gompertz curve. That is an in-sample curve-fitting claim; the shape parameter gamma is explicitly fitted to the simulations and is not presented as a theoretically predicted value. Second, analytically, Appendix F attempts to derive from the logistic SDE that survival probability is approximately a standard Gompertz function in K with gamma = 1. That derivation is a calculation, not a refit of the simulations, and the paper explicitly distinguishes the standard Gompertz result from the empirically fitted modified version. The paper states that the standard Gamma curve systematically deviates from the simulations and then modifies the curve by fitting gamma to the same simulated data; this is transparent and does not substitute the fitted curve back into the derivation as if it were derived. There is no equation in which the claimed output is equivalent to an input by construction, and no fitted parameter is renamed as a predicted value. The self-citations (Eyres et al. 2025; Green et al. 2020) are not load-bearing for the Gompertz claim: one supplies a power-law exponent for a comparison curve, and the other motivates autocorrelated environmental noise. There are serious mathematical-validity concerns in Appendix F: the Fokker-Planck equation as printed does not match the SDE in Eq. F1, the near-K rescaling appears inconsistent, and no error bound is given for keeping only the first eigenvalue of the first-passage spectrum. These are correctness risks, not circularity. Similarly, the fact that the analytical theory supports the standard Gompertz while the claimed universal law is the modified Gompertz with fitted gamma is a scope limitation and a possible overstatement, but not a circular reduction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical or biological entities are postulated. The paper contributes a fitted functional relationship and an approximate analytical explanation; the main ledger items are the fitted curve parameters and the viability and approximation assumptions used to make the claim universal.

free parameters (3)

gamma (modified Gompertz shape parameter) = median 0.85, 0.46, 0.34, 0.32 for Models A-D
Fitted to each simulated PE-K curve; not derived from the theory. Its systematic departure from 1 is the empirical core of the modified Gompertz claim.
a (Gompertz location parameter) = estimated per parameter set
Fitted when applying Equation (1); the paper offers no universal value, so the proposed law is a family of curves indexed by per-population parameters.
b (Gompertz rate parameter) = estimated per parameter set
Fitted when applying Equation (1); predicted only indirectly through linear models for the standard Gompertz, not for the modified curve.

assumptions (4)

domain assumption Population growth follows negative density dependence and cannot grow without limit.
Invoked in the simulation equations from Appendix A onward and stated as the first of the two assumptions sufficient for the analytical result.
domain assumption Extant species correspond to parameter sets for which PE reaches zero at large carrying capacity.
Used to discard simulation parameter sets that do not reach PE=0 by K=3 million (Supplementary Appendix B). This filter partly defines what counts as an extant population.
ad hoc to paper The first-passage survival probability of the logistic SDE can be approximated by the first eigenvalue of an Ornstein-Uhlenbeck process for large K and large time.
Used to obtain Eq. F15 in Appendix F; no error estimate is supplied, and the notation in the appendix is internally inconsistent.
domain assumption A 100-year time horizon and initialization at carrying capacity capture the population viability quantity of interest.
All base simulations run for 100 years from N0=K (Appendix A), and the analytical survival probability fixes time at T, making the curve horizon-dependent.

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Cite this review

Pith. "Pith review of A general relationship between extinction risk and carrying capacity." pith.science (2026). https://pith.science/paper/ZZOYZYEC

@misc{pith2026241113228,
  author       = {Pith},
  title        = {Pith review of: A general relationship between extinction risk and carrying capacity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZOYZYEC}},
  note         = {Machine review of arXiv:2411.13228}
}

read the original abstract

Understanding the relationship between a populations probability of extinction and its carrying capacity frames conservation status assessments and guides efforts to understand and mitigate the ongoing biodiversity crisis. Despite this, our understanding of the mathematical form of this relationship remains limited. We conducted ~5 billion population viability assessments that jointly converge on a modified Gompertz curve. This pattern is consistent across >1700 distinct model populations, representing different breeding systems and widely varying rates of population growth, levels of environmental stochasticity, adult survival rate, age at first breeding, and initial population size. Analytical treatment of the underlying dynamics shows that few assumptions suffice to show that the relationship holds for any extant population subject to density-dependent growth. Finally, we discuss the implications of these results and consider the practical use of our findings by conservationists.

Figures

Figures reproduced from arXiv: 2411.13228 by the authors.

**Figure 1.** Modelled probabilities of extinction (circles) for a range of carrying capacities and fitted modified Gompertz curves (lines) for Models A, B, C, and D, along with a diagrammatic representation of each model. Here, r 2 > 0.9999 for all four curves. In each diagram, the individuals contained within the sub-box(es) represent the ‘current’ population, with those outside and the various arrows representing the mechanism… view at source ↗

**Figure 3.** Comparison best fits for modified Gompertz, logistic, power [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 3.** Probability of extinction in a 100-year period Pext of grizzly bears Ursus arctos in relation to carrying capacity K. Results (circles) are from a simulation model in which demographic rates of bears were assumed to vary among years according to upon empirical observations of the variance of environmental stochasticity in demographic rates and assumptions about demographic stochasticity. The curve shows the modified… view at source ↗

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Works this paper leans on

3 extracted references · 2 canonical work pages · cited by 1 Pith paper

[1]

Barnosky, A., Matzke, N., Tomiya, S. et al. (2011). Has the Earth’s sixth mass extinction already arrived?. Nature 471, 51–57. https://doi.org/10.1038/nature09678 Berger, J. (1990) Persistence of Different-Sized Populations: An Empirical Assessment of Rapid Extinctions in Bighorn Sheep. Conservation Biology 4: 91 -98. Bird, J. P ., Martin, R., Akçakaya, H...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Barnosky, A., Matzke, N., Tomiya, S. et al. (2011). Has the Earth’s sixth mass extinction already arrived?. Nature 471, 51–57. https://doi.org/10.1038/nature09678 Berger, J. (1990) Persistence of Different-Sized Populations: An Empirical Assessment of Rapid Extinctions in Bighorn Sheep. Conservation Biology 4: 91 -98. Bird, J. P ., Martin, R., Akçakaya, H...

work page doi:10.1038/nature09678 2011

[2] [40]

Analytical survival analysis of the ornstein–uhlenbeck process

Mont. For. Conserv. Exp. Stn., University of Montana. Giorgini, Ludovico T., Woosok Moon, and John S. Wettlaufer. "Analytical survival analysis of the ornstein–uhlenbeck process. " Journal of Statistical Physics 181.6 (2020): 2404 -2414. Green R.E., Gilbert G., Wilson J.D., Jennings K . (2020). Implications of the prevalence and magnitude of sustained dec...

arXiv 2020

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Shaffer, M. L. (1978). Determining minimum viable population sizes: a case study of the grizzly bear(Ursus arctos L). Unpublished Ph.D. diss. Duke University, Durham. Virtanen, P ., Gommers, R., Oliphant, T. E, et al. (2020) SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods , 17(3), 261-272. White, R.S.A., Wintle, B.A., ...

work page arXiv 1978