Sex‐specific demography and generalization of the Trivers–Willard theory


The Trivers–Willard theory1 proposes that the sex ratio of offspring should vary with maternal condition when it has sex‐specific influences on offspring fitness. In particular, mothers in good condition in polygynous and dimorphic species are predicted to produce an excess of sons, whereas mothers in poor condition should do the opposite. Despite the elegance of the theory, support for it has been limited2,3. Here we extend and generalize the Trivers–Willard theory to explain the disparity between predictions and observations of offspring sex ratio. In polygynous species, males typically have higher mortality rates4, different age‐specific reproductive schedules and more risk‐prone life history tactics than females; however, these differences are not currently incorporated into the Trivers–Willard theory. Using two‐sex models parameterized with data from free‐living mammal populations with contrasting levels of sex differences in demography, we demonstrate how sex differences in life history traits over the entire lifespan can lead to a wide range of sex allocation tactics, and show that correlations between maternal condition and offspring sex ratio alone are insufficient to conclude that mothers adaptively adjust offspring sex ratio.

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Figure 1: Female reproductive value (RV, red line) and male RV (blue) depend on maternal condition.
Figure 2: Reproductive value differs with maternal condition and offspring sex in Columbian ground squirrels.
Figure 3: Size‐specific sensitivity of the Trivers–Willard effect to a 1% increase in male mortality in bighorn sheep lambs.


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We thank Y. Vindenes, S. Cubaynes, S. West, R. K. Kanda, J. A. Deere, J. Barthold, M. Brouard, R. A. Pozo, and E. G. Simmonds for comments. We thank M. Festa‐Bianchet and F. Pelletier for access to Bighorn sheep data and feedback. We acknowledge the use of the University of Oxford Advanced Research Computing facility. S.S. was funded by an ERC Advanced Grant to T.C., P.N. is funded by a Swiss National Science Foundation grant (SNF 3100AO‐109816), and L.T. was funded by grants from the European Commission (Marie Curie Fellowship 254442) and the Carnegie Corporation of New York (B8749.R01).

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S.S., J.M.G. and T.C. conceived and designed the study. S.S. developed the models and, with S.T., derived the formulas. S.S. and T.C. wrote the manuscript. S.S., A.G. and S.T. contributed to the mathematical formulation of the model. P.N. collated data on Columbian ground squirrels. L.T. parameterized data for bighorn sheep. T.C. parameterized data for squirrels. All authors edited the manuscript.

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Correspondence to Susanne Schindler.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Mean female age at reproduction affects optimal sex allocation.

a, Slope of the difference between male and female RV as a function of the size threshold above which females reproduce. The male size threshold is fixed at 279 g. Negative values indicate a reversed Trivers–Willard effect, positive values a Trivers–Willard effect. Dashed lines indicate a type 4 effect. When the population growth rate λ is greater than 1 (growing population), increasing female age at reproduction selects towards a Trivers–Willard effect. In contrast, when the population is shrinking (λ < 1), reproducing at a later age increases fitness36 and selects towards a reversed Trivers–Willard effect with increasing female age at reproduction. b, Mean maternal (red) and paternal (blue) age at reproduction as a function of the size threshold at which females reproduce. Dashed lines indicate the range of size thresholds that cause a type 4 effect.

Extended Data Figure 2 Strength of Trivers–Willard and reversed Trivers–Willard effects in squirrels as a function of the male to female survival ratio.

The x‐axis plots the ratio of male to female survival rate (independent of size and age) to the slope of the difference between male and female reproductive value, Δva(s). Positive values indicate a Trivers–Willard effect (grey background), negative values a reversed Trivers–Willard effect (red background). The more positive (or negative) the slope of Δva(s) the more the expected sex ratio in offspring to good‐condition mothers is biased towards males (or females). Solid line, no sex differences in mortality; dashed line, strength of mate selection has been increased from ρ = 0.1 to ρ = 0.25 (see Supplementary Table 1). Points highlighted with arrows indicate the settings that are used in Fig. 2 to plot Δva(s) against maternal size s.

Extended Data Figure 3 Trade‐off between survival and reproduction in squirrels.

a, Females (red) have higher survival rates than males (blue) at all ages. b, Small females are expected to produce more offspring than small males, while large females produce less offspring than large males.

Extended Data Figure 4 Sensitivity of the reversed Trivers–Willard effect to parameter perturbations in squirrels.

When bars lie in the positive (or negative) range, then a change in parameters works towards a Trivers–Willard effect (or strengthening the reversed Trivers–Willard effect). The horizontal dashed line shows the difference in slope needed to neutralise the reversed Trivers–Willard effect. The bar above the dashed line indicates a Trivers–Willard effect (black); bars below indicate a reversed Trivers–Willard effect (red). Filled fractions of the bars indicate the contribution of change caused by parameter perturbation owing to change of female RV, and, in the white fractions, to change in male RV (see also Extended Data Fig. 5). Bars 1 to 4 show the sensitivity of the Trivers–Willard effect in squirrels to perturbations of the following parameters by 1% downwards (which corresponds to higher mortality in the sex affected): (1) female survival intercept; (2) female survival slope; (3) male survival intercept, parameter change resulted in curved Δva which we indicate with ‘TW effect type 3’ and omit the bar; (4) male survival slope. Bars from 5 to 14 show the sensitivity of the Trivers–Willard effect in squirrels to perturbations of the following parameters by 1% upwards (which corresponds to higher rates in the affected sex): (5) female growth (mean intercept); (6) female growth (mean slope); (7) female growth variance; (8) male growth (mean intercept); (9) male growth (mean slope); (10) male growth variance; (11) inheritance (mean intercept); (12) inheritance (mean slope); (13) inheritance variance; (14) expected offspring number. All parameters are listed in Supplementary Table 1.

Extended Data Figure 5 Female (red) and male (blue) variance in RV in original model (bar 0) and when parameters are perturbed (bars 1–14) in squirrels.

Number 1 above bars indicates a Trivers–Willard effect, number 2 a reversed Trivers–Willard effect, and number 3 a type 3 Trivers–Willard effect. Bars 1 to 4 show variances in RV when the survival parameters are perturbed by 1% downwards (which corresponds to higher mortality in the affected sex): (1) female survival intercept; (2) female survival slope; (3) male survival intercept; and (4) male survival slope. Bars 5 to 14 show variances in RV when the following parameters are perturbed by 1% upwards (which corresponds to higher rates in the affected sex): (5) female growth (mean intercept); (6) female growth (mean slope); (7) female growth variance; (8) male growth (mean intercept); (9) male growth (mean slope); (10) male growth variance; (11) inheritance (mean intercept); (12) inheritance (mean slope); (13) inheritance variance; and (14) expected offspring number. All parameters are listed in Supplementary Table 1.

Extended Data Figure 6 Reproductive value (RV) of female (solid line) and male (dashed line) offspring in bighorn sheep.

For small mothers, daughters have higher RV than sons. For large mothers, sons have higher RV than daughters. We scaled the RV of females and males such that the female RV of the smallest reproductive size class is 1.

Extended Data Figure 7 Sensitivity of the Trivers–Willard effect to parameter perturbations in sheep.

Bars in the positive range indicate that the Trivers–Willard effect is strengthened; bars in the negative range indicate a weakened Trivers–Willard effect. The horizontal dotted line marks the sensitivity needed to reverse the Trivers–Willard effect. The parameters are: 1–8 female survival (2 parameters each for the stages lamb, yearling, adult, and senescent); 9–16 male survival (2 parameters for each stage); 17–36 female growth (5 parameters for each stage); 37–56 male growth (5 parameters for each stage); 57–60 female inheritance (inh) (2 intercepts for mean and variance, 2 slopes for female contribution to mean and variance); 61–62 male inheritance (male contributions to mean and variance); and 63–68 fecundity (fecund). All parameters are listed in Supplementary Tables 2 and 3.

Extended Data Figure 8 Sensitivity of Trivers–Willard effect to size‐specific male mortality increases of 1% in sheep.

ad, The survival probability of each size class in each stage (lamb (a), yearling (b), adult (c), and senescent (d)) has been independently lowered by 1%. The vertical dashed black line denotes the mean body weight of male sheep in the corresponding stage. In the early stages (a and b, lamb and yearling) we find that male‐mortality increases in small size classes strengthen the Trivers–Willard effect, whereas male‐mortality increases in heavy size classes weaken the Trivers–Willard effect. In the later stages (c and d, adult and senescent), mortality increases hardly affect the Trivers–Willard effect (note that adult rams usually weigh above 60 kg).

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Schindler, S., Gaillard, J., Grüning, A. et al. Sex‐specific demography and generalization of the Trivers–Willard theory. Nature 526, 249–252 (2015).

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