  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.
