pith. sign in

arxiv: 2606.22001 · v1 · pith:DFG4H5NKnew · submitted 2026-06-20 · 🧬 q-bio.PE · math.DS

Evolutionary Entropy Shapes Reproductive Lifespan in Age-Structured Populations

Jorge Buescu , Saber Elaydi , Henrique Oliveira This is my paper

Pith reviewed 2026-06-26 11:02 UTC · model grok-4.3

classification 🧬 q-bio.PE math.DS
keywords evolutionary entropyreproductive lifespanLeslie matricesage-structured populationsiteroparous animalsgeneration timeentropy maximizationdemographic models
0
0 comments X

The pith

Evolutionary entropy of the normalized post-maturity reproductive distribution shapes reproductive windows in age-structured populations

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

The paper develops a reduction principle showing that evolutionary entropy and generation time remain unchanged when survivorship and fertility are multiplicatively rescaled on the reproductive interval after Euler-Lotka normalization. This means the entropy value is fixed solely by the shape of the normalized distribution of reproductive contributions after maturity, independent of absolute rates or juvenile mortality. The authors derive explicit formulas for finite and geometric-tailed Leslie matrices and identify a critical threshold in the geometric case that separates populations with a single finite entropy-maximizing endpoint from those whose entropy approaches an asymptotic limit determined only by age at first reproduction. Tests across 130 animal species show that the entropy-maximizing ages computed from demographic matrices alone closely match observed reproductive medians, with exact coincidence in a majority of cases and over 90 percent within three age classes even after phylogenetic correction.

Core claim

Our central result is a reduction principle: under Euler-Lotka normalization, evolutionary entropy and generation time are invariant under multiplicative rescaling of survivorship and fertility on the reproductive interval. The relevant entropy is determined not by absolute survivorship, fertility, or juvenile mortality, but by the normalized post-maturity reproductive distribution. We derive explicit entropy functionals for finite and open-group Leslie models, including geometric reproductive tails. For the geometric regime we prove a sharp critical threshold separating populations with a unique finite entropy-maximizing endpoint from those whose entropy increases toward an asymptotic value

What carries the argument

The reduction principle for evolutionary entropy under Euler-Lotka normalization, which isolates entropy to the shape of the normalized post-maturity reproductive distribution in Leslie matrices

If this is right

  • Reproductive windows in iteroparous populations are frequently organized near the age classes that maximize entropy on the normalized post-maturity distribution.
  • In the geometric reproductive tail regime, a critical threshold separates populations with a unique finite entropy-maximizing endpoint from those whose entropy increases asymptotically with endpoint age.
  • Entropy-derived predictions from demographic matrices alone coincide exactly with observed reproductive medians in a majority of 130 tested species and fall within three classes for over 90 percent.
  • The associations between predicted and observed values remain strong after phylogenetic correction.

Where Pith is reading between the lines

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

  • The invariance result would allow reproductive lifespan predictions from partial demographic schedules without requiring full knowledge of juvenile survival rates.
  • Populations whose observed reproductive windows lie far from entropy-maximizing ages may be subject to additional unmodeled pressures such as seasonal resource constraints or predation risk.
  • The same normalization and reduction approach could be applied to stage-structured or size-structured models to test whether entropy continues to organize reproductive timing outside strict age-class Leslie frameworks.

Load-bearing premise

Iteroparous animal populations can be represented by Leslie-type demographic matrices whose post-maturity reproductive distributions are finite or geometric, and entropy maximization on these matrices selects observed reproductive windows without additional ecological or behavioral constraints.

What would settle it

Finding a substantial set of species in which entropy-maximizing ages computed from their Leslie matrices deviate by more than three reproductive classes from independently measured reproductive medians would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.22001 by Henrique Oliveira, Jorge Buescu, Saber Elaydi.

Figure 1
Figure 1. Figure 1: Absolute prediction errors of the reproductive median B50, ordered by species percentile. Horizontal dashed lines indicate error thresholds of 1, 2, 3, 5 and 10 reproductive classes. However, reproductive replacement does not by itself describe the temporal organization of reproduction. Two species may have comparable net reproductive numbers but very different life histories if one concentrates reproducti… view at source ↗
Figure 2
Figure 2. Figure 2: Entropy curves for three representative long-lived species. Filled circles indicate entropy￾optimal classes D∗ and horizontal dotted lines indicate asymptotic entropy values H∞ [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Similarity between observed and entropy-predicted reproductive distributions across all analysed species measured by overlap coefficients. Representative examples spanning a broad range of agreement levels are shown in [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Similarity between observed and entropy-predicted reproductive distributions across all analysed species measured by Bhattacharyya coefficients. These results provide distribution-level support for the entropy-maximization principle. The extent to which entropy-derived quantities predict independent life-history variables is examined in the following sections. Generation time, which depends on the entire r… view at source ↗
Figure 5
Figure 5. Figure 5: Observed and entropy-predicted reproductive distributions for four representative species spanning a broad range of agreement levels. Overlap and Bhattacharyya coefficient are reported within each panel. 5.6. Correlations with independent life-history variables. The preceding analyses compared entropy-derived predictions with quantities computed directly from the demographic matrices. A substantially stron… view at source ↗
Figure 6
Figure 6. Figure 6: Phylogenetically corrected relationship between B pred 50 and observed mean reproductive age. Secondly, we compared the predicted generation time TP with the same observed mean reproductive age. Since TP depends on the entire reproductive distribution rather than on a single percentile, this provides an even broader test of the theory. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Phylogenetically corrected relationship between TP and observed mean reproductive age. From the perspective of evolutionary entropy, this observation is of particular relevance. As shown in Section 2.3, open-group populations generate reproductive distributions with geometric tails, and it is within this setting that finite entropy optima, asymptotic entropy regimes, and the critical-threshold criterion na… view at source ↗
Figure 8
Figure 8. Figure 8: Absolute prediction error of the reproductive median B50 as a function of observed generation time T. Prediction error increases strongly with generation time (r = 0.806, R2 = 0.649, p < 10−31). Finally, both demographic matrices and independent life-history variables are subject to measurement uncertainty and sampling variation. Despite these limitations, the empirical agreement between theory and observa… view at source ↗
read the original abstract

Evolutionary entropy measures the temporal organization of reproductive contributions along the life cycle of an age-structured population. We develop a mathematical and empirical framework showing that, in iteroparous animal populations represented by Leslie-type demographic matrices, reproductive windows are frequently organized near the age classes selected by entropy maximization. Evolutionary entropy complements the classical net reproductive number and asymptotic growth rate: whereas these measure lifetime replacement and growth, entropy measures the temporal dispersion of the growth-adjusted reproductive distribution. Our central result is a reduction principle: under Euler--Lotka normalization, evolutionary entropy and generation time are invariant under multiplicative rescaling of survivorship and fertility on the reproductive interval. The relevant entropy is determined not by absolute survivorship, fertility, or juvenile mortality, but by the normalized post-maturity reproductive distribution. We derive explicit entropy functionals for finite and open-group Leslie models, including geometric reproductive tails. For the geometric regime, governed by we prove a sharp critical threshold separating populations with a unique finite entropy-maximizing endpoint from those whose entropy increases toward an asymptotic value in terms solely of the age at first reproduction. The theory is tested on 130 animal species. Entropy-derived predictions, computed from the demographic matrices alone, are compared with independent life-history variables. Predicted and observed reproductive medians coincide exactly for a majority of species, over 90% are predicted within three reproductive classes, and associations remain strong after phylogenetic correction. These results identify a quantitative regularity across taxa, with geometric reproductive distributions playing a central role.

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 claims that evolutionary entropy in iteroparous populations modeled by Leslie matrices is governed by a reduction principle: under Euler-Lotka normalization, entropy and generation time depend only on the normalized post-maturity reproductive distribution and are invariant under multiplicative rescaling of survivorship and fertility. It derives explicit entropy functionals for finite and geometric-tail cases (with a sharp threshold for the latter), and reports that entropy maximization on these matrices predicts observed reproductive medians exactly for a majority of 130 animal species and within three classes for over 90%, with associations persisting after phylogenetic correction.

Significance. If the reduction principle is correctly derived and the empirical matches prove robust to matrix construction details, the work supplies a parameter-free, falsifiable regularity for reproductive schedule organization that is independent of absolute rates and juvenile mortality, complementing classical metrics like r and R0. The geometric-tail threshold constitutes a clean mathematical contribution that could guide further tests.

major comments (2)
  1. [Empirical validation] Empirical validation section: the reported exact matches and >90% within-three-class accuracy for 130 species rest on Leslie matrices whose construction, exclusion rules, parameter error propagation, and independence from post-hoc selection of the entropy-maximizing age are not described; without these, it remains possible that the association simply recovers the shape of the input reproductive distributions rather than constituting an independent prediction from entropy maximization alone.
  2. [Discussion] Discussion: the central claim that entropy maximization on normalized Leslie matrices organizes observed reproductive windows without additional constraints is load-bearing for the interpretation, yet the 130-species comparison does not examine robustness when explicit trade-off functions or density-dependent terms are added to the underlying demographic model.
minor comments (2)
  1. [Abstract and methods] The abstract and methods should explicitly reference the prior literature on evolutionary entropy to clarify the precise novelty of the functionals derived here.
  2. [Figures] Figure captions for the species-level comparisons should report per-class sample sizes and the precise phylogenetic correction method employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. Below we respond point-by-point to the major comments and indicate planned revisions.

read point-by-point responses

  1. Referee: [Empirical validation] Empirical validation section: the reported exact matches and >90% within-three-class accuracy for 130 species rest on Leslie matrices whose construction, exclusion rules, parameter error propagation, and independence from post-hoc selection of the entropy-maximizing age are not described; without these, it remains possible that the association simply recovers the shape of the input reproductive distributions rather than constituting an independent prediction from entropy maximization alone.

    Authors: We agree that the Methods section requires expansion for reproducibility. The revised manuscript will add a dedicated subsection detailing: (i) the published life-table sources for the 130 species, (ii) explicit exclusion criteria for incomplete tables, (iii) the exact procedure for assembling Leslie matrices from age-specific l_x and m_x values, and (iv) any handling of parameter uncertainty. Regarding independence, the entropy-maximizing age is obtained by optimizing the entropy functional solely over the normalized post-maturity distribution; the observed median is compared only after this optimization. We will add clarifying text to emphasize that the prediction does not use the observed median as input, thereby addressing the concern that the result merely recovers the input shape. revision: yes

  2. Referee: [Discussion] Discussion: the central claim that entropy maximization on normalized Leslie matrices organizes observed reproductive windows without additional constraints is load-bearing for the interpretation, yet the 130-species comparison does not examine robustness when explicit trade-off functions or density-dependent terms are added to the underlying demographic model.

    Authors: The reduction principle demonstrates invariance of entropy under multiplicative rescaling of rates on the reproductive interval, which holds regardless of the absolute values set by trade-offs. We nevertheless accept that explicit robustness checks under density dependence or functional trade-offs would strengthen the interpretation. The revised Discussion will include a new paragraph noting this invariance while acknowledging that full numerical examination of such extensions lies outside the scope of the present Leslie-matrix framework and is reserved for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain or empirical predictions

full rationale

The reduction principle is a direct mathematical consequence of applying the standard Euler-Lotka normalization to the definition of evolutionary entropy on the post-maturity reproductive distribution; this constitutes a valid invariance result rather than a self-definitional loop where an output is presupposed. Empirical claims derive entropy-maximizing endpoints from Leslie matrices constructed from demographic schedules and compare them to observed reproductive medians across 130 species, with explicit statements that predictions use matrices alone and associations hold after phylogenetic correction. No fitted parameters are relabeled as predictions, no load-bearing self-citations appear, and no ansatz or uniqueness result is imported from prior author work. The framework is self-contained against external species data benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the Euler-Lotka equation as background, the representation of populations by Leslie matrices, and the definition of evolutionary entropy as a functional on the normalized reproductive distribution; no free parameters are explicitly fitted in the abstract, but the geometric tail model introduces an implicit shape parameter.

axioms (2)
  • domain assumption Populations are accurately described by Leslie-type age-structured matrices with either finite support or geometric reproductive tails.
    Invoked throughout the abstract as the modeling choice for iteroparous animals.
  • standard math The Euler-Lotka equation provides the correct normalization for growth-adjusted reproductive distributions.
    Used to state the invariance under rescaling.
invented entities (1)
  • Evolutionary entropy functional on Leslie matrices no independent evidence
    purpose: Measure of temporal dispersion of reproductive contributions
    Newly defined quantity whose maximization is claimed to select reproductive windows.

pith-pipeline@v0.9.1-grok · 5804 in / 1688 out tokens · 49060 ms · 2026-06-26T11:02:35.576417+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

67 extracted references

  1. [1]

    J. F. Alves, A. Bravo, and H. Oliveira. Kneading determinants of infinite order linear recurrences.Linear Algebra and its Applications, (475):306–325, 2015

    2015

  2. [2]

    J. F. Alves, A. Bravo, and H. Oliveira. Population dynamics with infinite Leslie matrices: finite time properties.Journal of Difference Equations and Applications, 20(9):1307–1318, 2015

    2015

  3. [3]

    Animal diversity web, 2024

    Animal Diversity Web. Animal diversity web, 2024

    2024

  4. [4]

    Baudisch.Inevitable Aging?: Contributions to Evolutionary-Demographic Theory, volume 4 ofDemographic Research Monographs

    A. Baudisch.Inevitable Aging?: Contributions to Evolutionary-Demographic Theory, volume 4 ofDemographic Research Monographs. Springer, Berlin and Heidelberg, 1 edition, 2008

    2008

  5. [5]

    Bernard, C

    C. Bernard, C. de Vries, O. R. Jones, and R. Salguero-G´ omez. Response to discretising and validating keyfitz’ entropy for any demographic classification.Methods in Ecology and Evolution, 15(8):1286–1289, 2024

    2024

  6. [6]

    North american bird banding program, 2024

    Bird Banding Laboratory. North american bird banding program, 2024

    2024

  7. [7]

    Buescu, H

    J. Buescu, H. M. Oliveira, and M. Sousa. Growth rate, evolutionary entropy and ageing across the tree of life.Journal of Biological Dynamics, 17(1):2256766, 2023

    2023

  8. [8]

    J. R. Carey.Longevity: The Biology and Demography of Life Span. Princeton University Press, Princeton, NJ, 2003

    2003

  9. [9]

    Caswell.Matrix Population Models: Construction, Analysis, and Interpretation

    H. Caswell.Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer and associates, 2001

    2001

  10. [10]

    Charlesworth.Evolution in Age-Structured Populations, volume 2

    B. Charlesworth.Evolution in Age-Structured Populations, volume 2. Cambridge University Press, Cambridge, 1994

    1994

  11. [11]

    Colchero, R

    F. Colchero, R. Rau, O. R. Jones, J. A. Barthold, D. A. Conde, A. Lenart, L. Nemeth, A. Scheuerlein, J. Schoeley, C. Torres, V. Zarulli, J. Altmann, D. K. Brockman, A. M. Bronikowski, L. M. Fedigan, A. E. Pusey, T. S. Stoinski, K. B. Strier, and A. Baudisch. The emergence of longevous populations.Proceedings of the National Academy of Sciences, 113(48):E7...

    2016

  12. [12]

    R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth. On the Lambert W Function.Advances in Computational Mathematics, 5(4):329–359, 1996

    1996

  13. [13]

    T. M. Cover and J. A. Thomas.Elements of Information Theory. Wiley-Interscience, Hoboken, New Jersey, 2 edition, 2006

    2006

  14. [14]

    J. M. Cushing.An Introduction to Structured Population Dynamics. SIAM, 1998

    1998

  15. [15]

    J. P. de Magalh˜ aes. Anage: The animal ageing and longevity database.https://genomics.senescence.info/species/, 2026. Accessed 2026-06-08

    2026

  16. [16]

    J. P. de Magalh˜ aes and J. Costa. A database of vertebrate longevity records and their relation to other life-history traits.Journal of Evolutionary Biology, 22(8):1770–1774, 2009

    2009

  17. [17]

    de Vries, C

    C. de Vries, C. Bernard, and R. Salguero-G´ omez. Discretising keyfitz’ entropy for studies of actuarial senescence and comparative demography.Methods in Ecology and Evolution, 14(5):1312–1322, 2023

    2023

  18. [18]

    E. S. Deevey. Life tables for natural populations of animals.The Quarterly Review of Biology, 22(4):283–314, 1947

    1947

  19. [19]

    Demetrius

    L. Demetrius. On an infinite population matrix.Mathematical Biosciences, 13(1):133–137, 1972

    1972

  20. [20]

    Demetrius

    L. Demetrius. Multiplicative processes ii.Mathematical Biosciences, 20(3):345–357, 1974

    1974

  21. [21]

    Demetrius

    L. Demetrius. Natural selection and age-structured populations.Genetics, 79(3):535–544, 1975

    1975

  22. [22]

    Demetrius

    L. Demetrius. Adaptive value, entropy and survivorship curves.Nature, (275):213 – 214, 1978

    1978

  23. [23]

    Demetrius

    L. Demetrius. Statistical mechanics and population biology.Journal of Statistical Physics, 30(3):709–753, 1983

    1983

  24. [24]

    Demetrius

    L. Demetrius. Directionality principles in thermodynamics and evolution.Proceedings of the National Academy of Sciences, 94(8):3491– 3498, 1997

    1997

  25. [25]

    Demetrius

    L. Demetrius. Chronon and directional evolution.Physica A: Statistical Mechanics and its Applications, 340(1–3):1–11, September 2004

    2004

  26. [26]

    Demetrius

    L. Demetrius. Boltzmann, Darwin and directionality theory.Physics Reports, 530(1):1 – 85, 2013. Boltzmann, Darwin and Directionality Theory

    2013

  27. [27]

    S. N. Elaydi and J. M. Cushing.Discrete Mathematical Models in Population Biology: Ecological, Epidemic, and Evolutionary Dynamics. Springer Undergraduate Texts in Mathematics and Technology. Springer Nature Switzerland, Cham, 2024

    2024

  28. [28]

    Euring data bank and longevity records, 2024

    EURING. Euring data bank and longevity records, 2024. European Union for Bird Ringing

    2024

  29. [29]

    Felsenstein

    J. Felsenstein. Phylogenies and the comparative method.American Naturalist, 125:1–15, 1985

    1985

  30. [30]

    O. E. Fern´ andez and H. Beltr´ an-S´ anchez. The entropy of the life table: A reappraisal.Theoretical Population Biology, 104:26–45, 2015

    2015

  31. [31]

    Finch.Longevity, Senescence, and the Genome

    C. Finch.Longevity, Senescence, and the Genome. University of Chicago Press, Chicago, 1994

    1994

  32. [32]

    R. P. Freckleton, P. H. Harvey, and M. Pagel. Phylogenetic analysis and comparative data.American Naturalist, 160:712–726, 2002

    2002

  33. [33]

    Froese and D

    R. Froese and D. Pauly. Fishbase, 2024. World Wide Web electronic publication

    2024

  34. [34]

    S. Giaimo. Discretizing and validating keyfitz’ entropy for any demographic classification.Methods in Ecology and Evolution, 15, 2024

    2024

  35. [35]

    Gompertz

    B. Gompertz. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies.Philosophical Transactions of the Royal Society of London, 115:513–583, 1825

  36. [36]

    Gosselin and J.-D

    F. Gosselin and J.-D. Lebreton. Asymptotic properties of infinite Leslie matrices.Journal of Theoretical Biology, 256(2):157 – 163, 2009

    2009

  37. [37]

    A. Grafen. The phylogenetic regression.Philosophical Transactions of the Royal Society B, 326:119–157, 1989

    1989

  38. [38]

    R. M. Gray.Entropy and Information Theory. Information Systems and Data Science. Springer, New York, 2 edition, 2011

    2011

  39. [39]

    W. D. Hamilton. The moulding of senescence by natural selection.Journal of Theoretical Biology, 12(1):12–45, 1966

    1966

  40. [40]

    P. H. Harvey and M. D. Pagel.The Comparative Method in Evolutionary Biology. Oxford University Press, 1991

    1991

  41. [41]

    C. E. e. a. Hinchliff. Synthesis of phylogeny and taxonomy into a comprehensive tree of life.PNAS, 112:12764–12769, 2015

    2015

  42. [42]

    H. F. Inman and E. L. J. Bradley. The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities.Communications in Statistics – Theory and Methods, 18(10):3851–3874, 1989

    1989

  43. [43]

    E. T. Jaynes. Information theory and statistical mechanics.Physical Review, 106(4):620–630, 1957

    1957

  44. [44]

    E. T. Jaynes. Information theory and statistical mechanics. ii.Physical Review, 108(2):171–190, 1957

    1957

  45. [45]

    K. E. Jones, J. Bielby, M. Cardillo, S. A. Fritz, J. O’Dell, C. D. L. Orme, K. Safi, W. Sechrest, E. H. Boakes, C. Carbone, C. Connolly, M. J. Cutts, J. K. Foster, R. Grenyer, M. Habib, C. A. Plaster, S. A. Price, E. A. Rigby, J. Rist, A. Teacher, O. R. P. Bininda-Emonds, J. L. Gittleman, G. M. Mace, and A. Purvis. Pantheria: A species-level database of l...

    2009

  46. [46]

    K. E. Jones et al. Pantheria: Species-level database of life history, ecology, and geography of mammals. https://esapubs.org/archive/ ecol/E090/184/, 2026. Accessed 2026-06-08

    2026

  47. [47]

    O. R. Jones, A. Scheuerlein, R. Salguero-G´ omez, C. G. Camarda, R. Schaible, B. B. Casper, J. P. Dahlgren, J. Ehrl´ en, M. B. Garc´ ıa, E. S. Menges, P. F. Quintana-Ascencio, H. Caswell, A. Baudisch, and J. W. Vaupel. Diversity of ageing across the tree of life.Nature, 505(7482):169–173, 2014

    2014

  48. [48]

    Keyfitz.Introduction to the Mathematics of Population

    N. Keyfitz.Introduction to the Mathematics of Population. Addison-Wesley, Reading, Massachusetts, 1968

    1968

  49. [49]

    T. B. L. Kirkwood. Evolution of ageing.Nature, 270(5635):301–304, 1977

    1977

  50. [50]

    T. B. L. Kirkwood and S. N. Austad. Why do we age?Nature, 408(6809):233–238, 2000

    2000

  51. [51]

    D. Lepage. Avibase – the world bird database, 2024. Accessed 2026-06-08. 32

    2024

  52. [52]

    P. H. Leslie. On the use of matrices in certain population mathematics.Biometrika, 33(3):183–212, 1945

    1945

  53. [53]

    P. B. Medawar.An Unsolved Problem of Biology: An Inaugural Lecture Delivered at University College, London, 6 December 1951. Published for University College London by H. K. Lewis and Company, London, 1952

    1951

  54. [54]

    Open tree of life, 2025

    Open Tree of Life Project. Open tree of life, 2025. https://tree.opentreeoflife.org, accessed Month Year

    2025

  55. [55]

    M. Pagel. Inferring the historical patterns of biological evolution.Nature, 401:877–884, 1999

    1999

  56. [56]

    Pastore and A

    M. Pastore and A. Calcagn` ı. Measuring distribution similarities between samples: A distribution-free overlapping index.Frontiers in Psychology, 10:1089, 2019

    2019

  57. [57]

    Pearl.The Rate of Living

    R. Pearl.The Rate of Living. University of London Press, London, 1928

    1928

  58. [58]

    Salguero-G´ omez et al

    R. Salguero-G´ omez et al. Comadre animal matrix database.https://www.comadre-db.org, 2026. Accessed 2026-06-08

    2026

  59. [59]

    Salguero-G´ omez et al

    R. Salguero-G´ omez et al. Compadre plant matrix database.https://compadre-db.org, 2026. Accessed 2026-06-08

    2026

  60. [60]

    Salguero-G´ omez, O

    R. Salguero-G´ omez, O. R. Jones, C. R. Archer, C. Bein, H. de Buhr, C. Farack, F. Gottschalk, A. Hartmann, A. Henning, G. Hoppe, G. R¨ omer, J. Runge, T. Ruoff, V. Sommer, J. Wille, J. Voigt, D. Vieregg, Y. C. Buckley, J. Che-Castaldo, D. Hodgson, A. Scheuerlein, H. Caswell, J. W. Vaupel, and A. Baudisch. Comadre: A global data base of animal demography....

    2016

  61. [61]

    Salguero-G´ omez, O

    R. Salguero-G´ omez, O. R. Jones, C. R. Archer, Y. C. Buckley, J. Che-Castaldo, H. Caswell, D. Hodgson, A. Scheuerlein, D. A. Conde, E. Brinks, H. de Buhr, C. Farack, F. Gottschalk, A. Hartmann, A. Henning, G. Hoppe, G. R¨ omer, J. Runge, T. Ruoff, J. Wille, S. Zeh, D. Vieregg, A. Baudisch, and J. W. Vaupel. The compadre plant matrix database: An open onl...

    2015

  62. [62]

    C. E. Shannon. A mathematical theory of communication.Bell System Technical Journal, 27:379–423, 623–656, 1948

    1948

  63. [63]

    S. C. Stearns.The Evolution of Life Histories. Oxford University Press, Oxford and New York, 1 edition, 1992

    1992

  64. [64]

    Tacutu, D

    R. Tacutu, D. Thornton, E. Johnson, A. Budovsky, D. Barardo, T. Craig, E. Diana, G. Lehmann, D. Toren, J. Wang, V. E. Fraifeld, and J. P. de Magalh˜ aes. Human ageing genomic resources: New and updated databases.Nucleic Acids Research, 46(D1):D1083–D1090, January 2018

    2018

  65. [65]

    J. W. Vaupel, A. Baudisch, M. D¨ olling, D. A. Roach, and J. Gampe. The case for negative senescence.Theoretical Population Biology, 65(4):339–351, 2004

    2004

  66. [66]

    W. Weibull. A statistical distribution function of wide applicability.Journal of Applied Mechanics, 18(3):293–297, 1951

    1951

  67. [67]

    G. C. Williams. Pleiotropy, natural selection, and the evolution of senescence.Evolution, 11(4):398–411, 1957. Email address:jsbuescu@ciencias.ulisboa.pt Email address:selaydi@trinity.edu Email address:henrique.m.oliveira@tecnico.ulisboa.pt

    1957