Figure 1. Fitness and the sex ratio. Mothers that produce an equal number of sons and daughters will have a higher long-term fitness than mothers that produce an unequal sex ratio, even if all individuals have the same survival and fertility.

(a) For example, imagine a population that initially contains 80% females and 20% males. (b) Now consider a new type of mother that produces 50% daughters and 50% sons. If this new type represents X% of the mothers, their children will account for X% of the population in the next generation, but they will account for a much larger proportion of the males.

Because each offspring in generation 2 must have exactly one mother and one father, the new type of mother from generation 0 will have more grandchildren, on average, because her sons will make up a disproportionately large fraction of the rarer sex (males).

Figure 2. Geometric mean fitness. If the fitness of a type varies over time, its geometric mean fitness determines whether it will spread or disappear (Haldane and Jayakar, 1963).

(a) For example, consider two environments (wet and dry) that alternate from generation to generation, and two types of individuals (A, B), whose absolute fitness depends on the environment. (b) Starting with 100 A individuals in a wet environment, there will be 100 × 2 = 200 offspring produced. The environment then becomes dry, and these 200 individuals produce 200 × 1/2 = 100 offspring. This creates a seesaw pattern over time. (c) Starting with 100 B individuals, however, 100 × 3/2 = 150 offspring are produced in the first generation (in the wet environment). These produce 150 × 1 = 150 offspring in the next generation (in the dry environment). The cycle begins again, but now with more individuals.

In this example, types A and B have the same average fitness (5/4) over the two environments but different geometric mean fitnesses. Since the geometric mean of fitnesses, W_i, measured over T generations equals (W₁W₂...W_T)^1/T, the geometric mean fitness of type B is higher (1.2) than that of type A (1.0) and accurately predicts the spread of B.