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arxiv: 1906.12047 · v1 · pith:MA2WRZCKnew · submitted 2019-06-28 · 🧬 q-bio.PE · math.DS

Evolution of longevity, age at last birth and sexual conflict with grandmothering

Matthew H. Chan , Kristen Hawkes , Peter S. Kim This is my paper

Pith reviewed 2026-05-25 13:51 UTC · model grok-4.3

classification 🧬 q-bio.PE math.DS
keywords grandmother hypothesislongevity evolutionage at last birthsexual conflicttwo-sex modelhunter-gatherer longevitygreat ape longevitypartial differential equation
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The pith

A two-sex model finds grandmothering enables the shift from great-ape to hunter-gatherer longevities via two stable equilibria without extending the end of fertility.

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

The model tracks simultaneous evolution of adult longevity and age at last birth under the Grandmother Hypothesis. It identifies exactly two locally stable equilibria, one with great-ape longevities and one with hunter-gatherer longevities. Grandmothering support from post-fertile females to daughters allows the population to move from the low-longevity state to the high-longevity state. Male competition arising from a male-biased mating sex ratio controls whether the transition occurs and sets the final adult lifespan value. Grandmothering changes the equilibrium age at last birth but leaves the equilibrium adult lifespan largely unaffected.

Core claim

In the extended two-sex PDE model, grandmothering restricted to daughters' fertility produces only two locally stable equilibria for longevity and reproductive span: one matching great-ape life histories and one matching hunter-gatherer life histories. The grandmothering effect permits the transition between these equilibria without any extension of the age at last birth. Sensitivity analysis shows that the strength of male competition, set by mating sex-ratio skew, determines both whether the transition to higher longevity is reachable and the equilibrium value of average adult lifespan.

What carries the argument

The two-sex partial differential equation that couples evolving adult longevity, age at last birth, grandmothering support to daughters, and male competition driven by mating sex-ratio skew.

If this is right

  • Grandmothering alone is sufficient to produce the observed increase in human longevity from ape-like baselines.
  • The equilibrium adult lifespan depends on the intensity of male competition.
  • The equilibrium age at last birth is shifted by grandmothering while adult lifespan is not.
  • No change in the physiological end of fertility is required for the longevity increase.

Where Pith is reading between the lines

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

  • Social rules that channel grandmother help exclusively to daughters may be necessary for the longevity transition to remain stable.
  • Changes in mating systems that alter sex-ratio skew could block or enable similar longevity shifts in other species.
  • The model predicts that populations with strong grandmothering should show earlier age at last birth than expected from longevity alone.

Load-bearing premise

Post-fertile females provide reproductive support only to their daughters and male competition is driven solely by the mating sex-ratio skew.

What would settle it

A simulation or empirical comparison in which grandmothering support is extended to sons or other relatives and the two equilibria merge or the transition to higher longevity disappears.

Figures

Figures reproduced from arXiv: 1906.12047 by Kristen Hawkes, Matthew H. Chan, Peter S. Kim.

Figure 1
Figure 1. Figure 1: Solid and dashed lines in plot (a) show the mortality function µ(a, L) and survivorship function l(a, L) respectively for L = 0.2 and L = 0.8. Plot (b) shows the life expectancy of a newborn e0(L) varying with L. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Age-specific mortality rates for three hunter-gatherer populations (!Kung, Ache, Hadza) and wild chimpanzees. !Kung data from Howell [23], Ache data from Hill & Hurtado [22], Hadza data from Blurton Jones et al. [3] and chimpanzee data (smoothed) from Hill et al. [21]. 2.3 Mating and offspring We assume that females randomly select a male to mate with, whereby the resultant offspring inherits the mean of t… view at source ↗
Figure 3
Figure 3. Figure 3: Male fertility-longevity tradeoff function φ(L). Unlike the female fertility-longevity tradeoff function, this tradeoff function does not have any underpinning empirical data. However, we find that the form of this function must be strict; we refer the reader to Section 4.1 where we provide justification for the form of Eq. (10) and also exploration of different forms in a sensitivity analysis. 2.4 Grandmo… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the mutation kernel N(x, y, M, L) with x = y = 0.5. The blue arrow represents (1 − p)δ(M − x)δ(L − y) and the box surrounding it has height p (2h) 2 and width h. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plots of the system without grandmothering at equilibrium. Plots (a) and (b) show R 1 0 uf dM and R 1 0 uf dL respectively. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of the average adult lifespan (solid line) and average age at last birth (dashed line), without grand￾mothering. 3.2 Time evolution with grandmothering 20 30 40 0 25 50 0 0.02 0.04 0.06 0.08 Life expectancy e0 Age Population density (a) 20 40 60 0 25 50 0 0.02 0.04 0.06 0.08 Age Age at last birth Population density (b) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plots of the system with grandmothering at equilibrium. Plots (a) and (b) show R 1 0 uf dM and R 1 0 uf dL respectively. 0 2 4 6 8 10 x 104 10 20 30 40 50 Time (years) Age [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time evolution of the average adult lifespan (solid line) and average age at last birth (dashed line), with grandmoth￾ering. Using the equilibrium from [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Male fertility-longevity tradeoff functions φ(L) (blue), φ1(L) (red), φ2(L) (black) and φ3(L) (green). Without grandmothering, the shape of the male fertility-longevity tradeoff function has a small effect on the equi￾librium value for average adult lifespan and average age at last birth, as shown in [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time evolutions of the system without grandmothering with φ(L) (blue), φ1(L) (red), φ2(L) (black) and φ3(L) (green). 0 2 4 6 8 10 x 104 30 35 40 45 50 Time (years) Average age at last birth (a) 0 2 4 6 8 10 x 104 10 15 20 25 30 35 40 Time (years) Average adult lifespan (b) [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Time evolutions of the system with grandmothering with φ(L) (blue), φ1(L) (red), φ2(L) (black) and φ3(L) (green). The initial condition used was the corresponding equilibria of the system without grandmothering, shown in [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time evolutions of the system with grandmothering with m = 2.5 (red), m = 3.0 (blue), m = 3.5 (black) and m = 4.0 (green). The initial condition used was the equilibrium of the system without grandmothering, shown in Figures 5-6. 0 20 40 60 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Age of daughter G(a,M,L)/u f(a,M,L) [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The proportion of females of any age with a live mother with age a ∈ [τ2(M), τ3] (that is, G(a, M, L)/uf (a, M, L)) at the hunter-gatherer equilibrium with m = 2.5 (red), m = 3.0 (blue), m = 3.5 (black) and m = 4.0 (green). From Figure 12b, differing strengths of grandmother effects have an insignificant effect on average adult lifespan. This can be surprising, since if one expects grandmothers to increas… view at source ↗
Figure 14
Figure 14. Figure 14: Time evolutions of the system without grandmothering with 0.8τ1(L) (red), 0.9τ1(L) (blue) and τ1(L) (black). 0 2 4 6 8 10 x 104 36 38 40 42 44 46 48 50 Time (years) Average age at last birth (a) 0 2 4 6 8 10 x 104 10 15 20 25 30 35 Time (years) Average adult lifespan (b) [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Time evolutions of the system with grandmothering with 0.8τ1(L) (red), 0.9τ1(L) (blue) and τ1(L) (black). The initial condition used was the corresponding equilibria of the system without grandmothering, shown in [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The proportion of females of any age with a live mother with age a ∈ [τ2(M), τ3] (that is, G(a, M, L)/uf (a, M, L)) at the hunter-gatherer equilibrium with with 0.8τ1(L) (red), 0.9τ1(L) (blue) and τ1(L) (black). 4.4 Late mortality rate In Figures 17 and 18 we examine the effect of the base rate of late mortality µs by comparing the equilibria generated when we use 0.9µs, 0.95µs, µs and 1.05µs. Without gra… view at source ↗
Figure 17
Figure 17. Figure 17: Time evolutions of the system without grandmothering with 0.9µs (red), 0.95µs (blue) and µs (black), 1.05µs (grey). The simulation with 1.05µs reaches an equilibrium with average age at last birth of 67.25 and average adult lifespan of 33.5, however the population goes extinct as the net reproductive rate is negative. 0 2 4 6 8 10 x 104 30 35 40 45 50 Time (years) Average age at last birth (a) 0 2 4 6 8 1… view at source ↗
Figure 18
Figure 18. Figure 18: Time evolutions of the system with grandmothering with 0.9µs (red), 0.95µs (blue) and µs (black). The initial condition used was the corresponding equilibria of the system without grandmothering, shown in [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Time evolutions of the system without grandmothering with µs (red), 1.1µs (blue) and 1.2µs (black), 1.25µs (green) and 1.3µs (grey). The male ferility-longevity tradeoff function used is φ3(L). The simulation with 1.3µs reaches an equilibrium with average age at last birth of 67.25 and average adult lifespan of 33.5, however the population goes extinct as the net reproductive rate is negative. 0 2 4 6 8 1… view at source ↗
Figure 20
Figure 20. Figure 20: Time evolutions of the system with grandmothering with with µs (red), 1.1µs (blue) and 1.2µs (black) and 1.25µs (green). The male ferility-longevity tradeoff function used is φ3(L). The initial condition used was the corresponding equilibria of the system without grandmothering, shown in [PITH_FULL_IMAGE:figures/full_fig_p017_20.png] view at source ↗
read the original abstract

We use a two-sex partial differential equation (PDE) model based on the Grandmother Hypothesis. We build on an earlier model by Kim et al. [27] by allowing for evolution in both longevity and age at last birth, and also assuming that post-fertile females support only their daughters' fertility. Similarly to Kim et al. [27], we find that only two locally stable equilibria exist: one corresponding to great ape-like longevities and the other corresponding to hunter-gatherer longevities. Our results show that grandmothering enables the transition between these two equilibria, without extending the end of fertility. Moreover, sensitivity analyses of the model show that male competition, arising from a skew in the mating sex ratio towards males, plays a significant role in determining whether the transition from great ape-like longevities to higher longevities is possible and the equilibrium value of the average adult lifespan. Whereas grandmothering effects have a significant impact on the equilibrium value of the average age at last birth and enable the transition to higher longevities, they have an insignificant impact on the equilibrium value of the average adult lifespan.

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 / 0 minor

Summary. The manuscript presents a two-sex PDE model extending the Grandmother Hypothesis from Kim et al. (2019), allowing evolution in both longevity and age at last birth while restricting post-fertile female support to daughters' fertility only. It reports that only two locally stable equilibria exist—one with great ape-like longevities and one with hunter-gatherer longevities—with grandmothering enabling the transition between them without extending the end of fertility. Sensitivity analyses indicate that male competition (via mating sex ratio skew) significantly affects transition feasibility and equilibrium adult lifespan, while grandmothering strongly affects equilibrium age at last birth but not adult lifespan.

Significance. If the equilibria and transition results hold under the stated assumptions, the work provides a mechanistic account of how grandmothering can shift populations from ape-like to human-like longevity without fertility extension, while quantifying the modulating role of male competition. The use of a two-sex PDE framework and sensitivity analyses on the mating skew parameter are strengths that allow exploration of sexual conflict effects.

major comments (2)
  1. [Abstract and model description] The central claim that only two locally stable equilibria exist (and that grandmothering enables the transition) is load-bearing but presented without the explicit PDE system, functional forms for fertility/mortality, or the numerical method used to locate and classify the equilibria. This prevents independent verification that the reported equilibria are reached from realistic initial conditions.
  2. [Sensitivity analyses] The sensitivity results on male competition and grandmothering strength are reported without specific parameter values, ranges, or error analysis (e.g., how the mating sex ratio skew parameter and grandmothering support strength to daughters are varied). This makes it impossible to assess the robustness of the claim that male competition determines transition possibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment below and have made revisions to improve clarity and verifiability of the results.

read point-by-point responses

  1. Referee: [Abstract and model description] The central claim that only two locally stable equilibria exist (and that grandmothering enables the transition) is load-bearing but presented without the explicit PDE system, functional forms for fertility/mortality, or the numerical method used to locate and classify the equilibria. This prevents independent verification that the reported equilibria are reached from realistic initial conditions.

    Authors: The explicit two-sex PDE system, functional forms for fertility and mortality (including the grandmothering term restricted to daughters), and the numerical methods (finite-difference discretization, continuation to locate equilibria, and linear stability analysis) are fully specified in the Methods section, along with the initial conditions corresponding to ape-like demography. To make these details more immediately accessible and address the verification concern, we will add a concise presentation of the core PDE equations and numerical approach to the main text in the revised manuscript. revision: yes

  2. Referee: [Sensitivity analyses] The sensitivity results on male competition and grandmothering strength are reported without specific parameter values, ranges, or error analysis (e.g., how the mating sex ratio skew parameter and grandmothering support strength to daughters are varied). This makes it impossible to assess the robustness of the claim that male competition determines transition possibility.

    Authors: The parameter ranges and variation protocol for the mating-sex-ratio skew (varied systematically from 1 to 4) and grandmothering support strength (varied from 0 to 0.5) are documented in the supplementary materials and associated table. In the revision we will add an explicit subsection in Results that states these ranges, the discrete steps used, the criterion for declaring a transition feasible, and any robustness checks performed across initial conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends an existing two-sex PDE model from the cited Kim et al. reference by adding evolution in both longevity and age at last birth plus a restriction on grandmothering support. The reported finding of two locally stable equilibria is obtained by direct analysis of the extended system; the equilibria are not obtained by fitting parameters to data and relabeling the fit as a prediction, nor does any load-bearing step reduce to a self-citation that substitutes for an independent derivation. The functional forms and structural choices are carried forward explicitly as modeling decisions, but the extension itself supplies independent content within the stated framework. No self-definitional, fitted-input, or ansatz-smuggling patterns are present in the abstract or described claims.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on standard population-dynamics assumptions plus several modeling choices whose values are not derived from first principles. Free parameters include fertility schedules, mortality rates, and the strength of the mating-sex-ratio skew. Axioms include the restriction that only daughters receive grandmothering help and the form of the two-sex PDE system. No new invented entities are introduced.

free parameters (2)
  • mating sex ratio skew parameter
    Controls male competition strength and is varied in sensitivity analysis; its value determines whether transition occurs and the final lifespan equilibrium.
  • grandmothering support strength to daughters
    Fitted or chosen to produce the observed transition between equilibria.
axioms (2)
  • domain assumption Post-fertile females support only their daughters' fertility
    Stated in abstract as the modification to the earlier model; this choice structures who receives the benefit and is required for the reported equilibria.
  • standard math Two-sex PDE system with continuous age and time
    Inherited from Kim et al. [27] and used without re-derivation.

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Reference graph

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