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arxiv: 1907.01666 · v1 · pith:3ZPC4BPFnew · submitted 2019-07-02 · 🧬 q-bio.PE · math.DS

Modelling the evolution of traits in a two-sex population, with an application to grandmothering

M.H. Chan , K. Hawkes , P.S. Kim This is my paper

Pith reviewed 2026-05-25 10:04 UTC · model grok-4.3

classification 🧬 q-bio.PE math.DS
keywords grandmotheringtwo-sex populationevolutionary dynamicsfitness landscapePrice equationlongevitylife-history traitssex ratio
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The pith

Grandmothering alone can evolve longer lifespans for both sexes via a two-sex fitness landscape.

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

The paper introduces a simplified dynamical model for the evolution of heritable traits that control the timing of sex-specific life-history events, such as age at maturity and end of female fertility, each carrying distinct fitness tradeoffs for males and females. It derives an explicit formula for the population fitness landscape that extends the Price Equation to two sexes, age structure, and multiple traits. When applied to grandmothering, where post-fertile females provision grandchildren and thereby subsidize their daughters' fertility, the model shows this behavior lengthens female post-reproductive life, raises the proportion of fertile-age males, and produces a net fitness gain for both sexes at greater ages. The central result is that these fitness changes alone generate an evolutionary path toward increased longevity.

Core claim

The paper derives a formula for the fitness landscape of a heritable trait in a two-sex population and shows that grandmothering, by lengthening female post-fertile longevity and raising sex ratios in fertile ages, produces higher net fitness for both males and females at extended lifespans, thereby supplying an evolutionary trajectory to greater longevities.

What carries the argument

The fitness landscape formula obtained by treating trait evolution as a dynamical system that generalizes the Price Equation to two sexes and age structure.

If this is right

  • Grandmothering shifts the population toward higher overall sex ratios through extended female post-fertile life.
  • Males gain a longer fertile span as longevity increases.
  • Both sexes experience a net fitness increase at longer lifespans.
  • Grandmothering supplies a self-contained evolutionary route to higher longevities without requiring additional selective pressures.

Where Pith is reading between the lines

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

  • The same landscape formula could be used to compare grandmothering against other forms of kin help in different species.
  • Predictions about age-specific sex ratios offer a measurable signature that could be checked against demographic records from societies with varying grandmother involvement.
  • The model implies that once grandmothering appears, selection can favor further increases in longevity even if other life-history changes remain fixed.

Load-bearing premise

Post-fertile females' provisioning of grandchildren must lengthen female post-reproductive longevity in a manner that raises net fitness for both sexes.

What would settle it

Empirical data showing that grandmother provisioning fails to increase the reproductive success of daughters or their offspring enough to favor longer female post-reproductive life would refute the claimed trajectory.

Figures

Figures reproduced from arXiv: 1907.01666 by K. Hawkes, M.H. Chan, P.S. Kim.

Figure 1
Figure 1. Figure 1: An illustration of the process described in Eq. (7) for hypothetical functions F(x) and M(x). We note that the height of the individual curves is irrelevant; the important aspect is that they integrate to 1, since they describe distributions across trait-space. The black curve, representing un+1(x), is obtained by convolving the function given in red with a normal density function with zero mean and finite… view at source ↗
Figure 2
Figure 2. Figure 2: Figure (a) shows box plots of the proportion of fertile females with a surviving post-fertile mother when the population has reached stable age distribution in the ABM. The function G(x) = (x−23)a b+(x−23)c is used as a model for G(x), where a = 1.164, b = 43.83 and c = 1.36. Figure (b) shows fitness landscapes of the population with and without grandmothering. The dashed lines correspond to the maxima of … view at source ↗
Figure 3
Figure 3. Figure 3: Figure (a) shows plots of the first moment of un(L), as defined in Eq. (7), with and without grandmothering, with ε 2 = 0.025. When the population without grandmothering (blue curve) reached equilibrium, grandmothering was allowed for (orange curve). The equilibria match the maxima of the fitness landscapes shown in Figure 2b, as expected from the proof in Section 5.1. Figure (b) shows plots of the female … view at source ↗
read the original abstract

We present a mathematical simplification for the evolutionary dynamics of a heritable trait within a two-sex population. This trait is assumed to control the timing of sex-specific life-history events, such as the age of sexual maturity and end of female fertility, and each sex has a distinct fitness tradeoff associated with the trait. We provide a formula for the fitness landscape of the population and show a natural extension of the result to an arbitrary number of heritable traits. Our method can be viewed as a dynamical systems generalisation of the Price Equation to include two sexes{, age structure} and multiple traits. We use this formula to examine the effect of grandmothering, whereby post-fertile females subsidize their daughter's fertility by provisioning grandchildren. Grandmothering can drive a shift towards higher sex ratios due to lengthening female post-fertile longevity, leading to changes in fitness for both sexes. For males, increased longevity is accompanied by a substantially longer fertile lifespan resulting in higher sex ratios in the fertile ages. Our fitness landscapes show a net increase in fitness for both males and females at longer lifespans, and as a result, we find that grandmothering alone provides an evolutionary trajectory to higher longevities.

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 / 1 minor

Summary. The paper derives a formula for the fitness landscape of a heritable trait controlling sex-specific life-history events (e.g., age at maturity and end of female fertility) in a two-sex population. It presents this as a dynamical-systems generalization of the Price equation that incorporates two sexes, age structure, and multiple traits, and applies the formula to grandmothering, where post-reproductive females provision grandchildren. The central claim is that grandmothering produces a net fitness increase for both sexes at longer post-fertile lifespans, thereby supplying an evolutionary trajectory toward higher longevities and altered sex ratios.

Significance. If the derivation is internally consistent and the grandmothering subsidy emerges from the trait dynamics rather than being inserted as an independent parameter, the work supplies a compact analytic tool for two-sex age-structured evolution that could be applied to other life-history questions. The explicit extension to multiple traits and the reported fitness-landscape results for grandmothering would constitute a concrete, falsifiable prediction about the direction of selection on longevity.

major comments (2)
  1. [Application section] Application section (grandmothering model): the subsidy is introduced as an additive benefit to daughter's fertility that lengthens post-fertile longevity. It is not shown whether this term is a derived function of the heritable trait that sets the end of female fertility or an externally parameterized quantity. If the latter, the reported net fitness gain at longer lifespans is an input to the landscape rather than an output of the Price-style dynamics, which directly affects the claim that grandmothering alone supplies the evolutionary trajectory.
  2. [Main derivation / fitness-landscape formula] Fitness-landscape formula (main derivation): the abstract states that the landscapes 'show' a net increase for both sexes, yet the provided description does not indicate how the two-sex, age-structured extension propagates the subsidy through the trait-controlled fertility schedule without additional free parameters. Clarification is required on whether the increase survives when the subsidy strength is tied to the same heritable trait that governs end-of-fertility age.
minor comments (1)
  1. [Abstract / multi-trait extension] The abstract mentions an extension to an arbitrary number of heritable traits but does not illustrate the multi-trait case with a concrete example or matrix notation; adding one would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the modeling choices in the grandmothering application. We address each major comment below with clarifications on how the subsidy enters the dynamics.

read point-by-point responses

  1. Referee: [Application section] Application section (grandmothering model): the subsidy is introduced as an additive benefit to daughter's fertility that lengthens post-fertile longevity. It is not shown whether this term is a derived function of the heritable trait that sets the end of female fertility or an externally parameterized quantity. If the latter, the reported net fitness gain at longer lifespans is an input to the landscape rather than an output of the Price-style dynamics, which directly affects the claim that grandmothering alone supplies the evolutionary trajectory.

    Authors: The heritable trait directly sets the age at end of female fertility and therefore the length of the post-fertile interval. The subsidy enters as an additive term to daughter fertility that is active only while the grandmother remains alive and post-fertile; its total integrated effect on the next generation is therefore a function of the trait value. The fitness landscape is obtained by substituting the resulting age-specific vital rates into the two-sex dynamical system and computing the selection gradient; the net gain at longer post-fertile lifespans is consequently an output of those dynamics rather than an exogenous input. We will add an explicit sentence in the application section stating that the subsidy duration is trait-controlled. revision: partial

  2. Referee: [Main derivation / fitness-landscape formula] Fitness-landscape formula (main derivation): the abstract states that the landscapes 'show' a net increase for both sexes, yet the provided description does not indicate how the two-sex, age-structured extension propagates the subsidy through the trait-controlled fertility schedule without additional free parameters. Clarification is required on whether the increase survives when the subsidy strength is tied to the same heritable trait that governs end-of-fertility age.

    Authors: The two-sex, age-structured extension inserts the grandmothering term into the fertility schedule of reproductive females; the magnitude of that term at each age is proportional to the number of post-fertile grandmothers, whose abundance is governed by the same trait that sets the end of fertility. No extra free parameters are required beyond the fixed per-year provisioning rate. The reported net fitness increase for both sexes is obtained directly from this construction. We acknowledge that making the provisioning rate itself a function of post-fertile duration would constitute a stronger test; the present model keeps the rate constant and still yields the directional result. We will insert a short paragraph after the landscape formula clarifying the propagation and note the constant-rate assumption as a modeling choice. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a two-sex age-structured generalization of the Price equation as a dynamical-systems formula for the fitness landscape of heritable traits controlling life-history timing, then applies the formula to a grandmothering model in which post-fertile females provide an explicit subsidy to daughter fertility. No equations or self-citations are exhibited that reduce the reported net fitness increase at longer lifespans to a fitted parameter, a self-defined quantity, or a load-bearing prior result by the same authors. The subsidy is an explicit modeling assumption whose consequences are computed; the evolutionary trajectory is therefore an output of the derived landscape rather than a tautology. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; therefore the ledger records only the assumptions explicitly invoked in the abstract. The model relies on standard population-genetic assumptions plus specific modeling choices about how grandmothering alters fertility.

free parameters (1)
  • grandmothering subsidy strength
    The magnitude of the fertility increase provided to daughters by post-fertile females is required to produce the reported fitness landscapes and longevity shift.
axioms (1)
  • domain assumption A heritable trait controls the timing of sex-specific life-history events and each sex experiences a distinct fitness tradeoff with that trait.
    This premise is stated at the opening of the abstract as the foundation for the entire modeling approach.

pith-pipeline@v0.9.0 · 5748 in / 1226 out tokens · 35575 ms · 2026-05-25T10:04:23.332879+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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