Conditions for the Trivers-Willard hypothesis to be valid: A minimal population-genetic model

The very insightful Trivers-Willard hypothesis, proposed in the early 1970s, states that females in good physiological conditions are more likely to produce male offspring, when the variance of reproductive success amongst males is high. A number of studies, aimed at its experimental verification, have found adequate supportive evidence in its favour. Theoretical investigations, however, have been few, perhaps because formulating a population-genetic model for describing the Trivers-Willard hypothesis turns out to be surprisingly complex. The present study describes a minimal population genetic model to explore one specific scenario, viz. how is the preference for a male offspring by females in good condition altered when ‘g’, the proportion of such females in the population changes from a low to a high value. As expected, when the proportion of such females is low, i.e., for low values of ‘g’, the Trivers-Willard (TW) strategy goes to fixation against the equal investment strategy. This holds true up to gmax, a critical value of ‘g’, above which the two strategies coexist, but the proportion of the TW strategy steadily decreases as ‘g’ increases to unity. Similarly, when the effect of well-endowed males attaining disproportionately high number of matings is more pronounced, the TW strategy is more likely to go to fixation. Interestingly, the success of the TW strategy has a complex dependence on the variance in the physiological condition of females. If the difference in the two types of conditions is not large, TW strategy is favoured, and its success is more likely as the difference increases. However, beyond a critical value of the difference, the TW strategy is found to be less and less likely to succeed as the difference becomes larger. Possible reasons for these effects are discussed.

💡 Research Summary

The paper presents a minimal population‑genetic model to explore under what conditions the Trivers‑Willard (TW) hypothesis holds, focusing on how the proportion of high‑condition females (denoted g) influences the evolution of a condition‑dependent sex‑allocation strategy. The model assumes a diploid population with two female phenotypes—high‑condition (good) and low‑condition (poor)—and two male phenotypes that differ in mating success. Two competing genetic strategies are considered: (1) the TW strategy, in which good‑condition females bias offspring toward males while poor‑condition females bias toward females, and (2) an equal‑investment strategy that always produces a 1:1 sex ratio regardless of maternal condition.

Mathematically, the authors derive recursion equations for the frequency p of the TW allele and for the mean fitness of each phenotype. The key parameters are: g, the proportion of high‑condition females; α, the factor by which superior males obtain more matings than average males; and δ, the magnitude of the physiological difference between the two female conditions. By calculating expected reproductive output for each genotype and performing a stability analysis of the equilibria, the authors identify critical thresholds that determine whether the TW allele fixes, is lost, or coexists with the equal‑investment allele.

The main findings are as follows. (i) When g is low, the TW strategy has a higher average fitness and therefore goes to fixation. This reflects the classic TW logic: rare high‑condition females gain a disproportionate benefit by producing sons who can capitalize on the high variance in male reproductive success. (ii) As g increases, the advantage of the TW strategy diminishes. At a critical value gmax the system undergoes a bifurcation: the two strategies coexist in a stable polymorphism, and for g > gmax the frequency of the TW allele declines monotonically, reaching zero when g = 1. (iii) The parameter α strongly modulates the outcome. Larger α (i.e., a more pronounced skew in male mating success) expands the range of g for which the TW allele can fix, because the payoff to producing sons becomes larger. (iv) The effect of δ is non‑linear. Small δ (little physiological difference between females) yields only a modest advantage for the TW strategy. As δ increases to an intermediate level, the advantage grows sharply, favoring fixation of the TW allele. Beyond a second critical value of δ, however, the advantage reverses: extremely high‑condition females produce so many sons that male competition becomes intense, reducing the expected fitness of TW carriers and causing the allele frequency to fall.

These results reconcile several apparently contradictory empirical observations. In resource‑rich environments where most females are in good condition (high g), the model predicts a weakening or even reversal of the TW bias, consistent with field studies that report female‑biased sex ratios under such conditions. Conversely, in populations where high‑condition females are rare and male reproductive variance is high (large α), the classic TW prediction of male‑biased offspring from high‑condition mothers is robust.

The authors acknowledge simplifying assumptions: only two discrete condition classes, no explicit genetic linkage or epistasis, and a static environment. They suggest that extending the framework to continuous condition distributions, incorporating stochastic environmental fluctuations, and allowing for maternal condition to be heritable would provide a richer understanding of the dynamics. Nonetheless, the minimal model successfully delineates the parameter space in which the Trivers‑Willard hypothesis is expected to hold, highlighting the joint importance of the proportion of high‑condition females, the intensity of male mating skew, and the magnitude of condition differences. This work thus offers a clear theoretical baseline for future experimental and comparative studies of sex‑allocation strategies.