(Darwin's theory of the sex ratio) A population of males and females mate pairwise to produce offspring. Suppose that each offspring is male with probability p and female with probability 1 - p. Then there is a steady state in which the fraction p of the population is male and the fraction 1 - p is female. If p ∗= 1 then males and females have different numbers of offspring (on average). Is such an equilibrium evolutionarily stable? Denote the number of children born to each female by n, so that the number of children born to each male is (p/(1 - p))n. Suppose a mutation occurs that produces boys and girls each with probability 1 .
Assume for simplicity that the mutant trait is dominant: if one partner in a couple has it, then all the offspring of the couple have it. Assume also that the number of children produced by a female with the trait is n, the same as for "normal" members of the population. Since both normal and mutant females produce the same number of children, it might seem that the ?tness of a mutant is the same as that of a normal organism. But compare the number of grand children of mutants and normal organisms. How many female offspring does a normal organism produce? How many male offspring? Use your answers to ?nd the number of grandchildren born to each mutant and to each normal organism. Does the mutant invade the population? Which value (values?) of p is evolutionarily stable?