## Dynamics of Sex Ratio and ESS Models

Evolutionary biologists who use game theory to explore the outcome of natural selection define an evolutionary stable strategy (ESS) as a situation where, in a population at equilibrium, either a single phenotype or 'character state' (i.e., a 'pure strategy') is fixed or where a particular mixture of characters states coexists (i.e., a 'mixed strategy'). In a stable equilibrium, an individual with a phenotype that differs from that specified by the ESS is unable to invade the population due to its relatively low fitness. As early as 1930, R. A. Fisher showed theoretically that producing male and females in a ratio of 1:1 is a stable strategy in a population of diploid organisms, where each individual has exactly one father and one mother. The dynamic process that results in equal numbers of males and females is led by frequency-dependent natural selection due to competition for mates among individuals of the same sex. When one of the sexes is relatively scarce, individuals that produce a higher proportion of the rarer sex will produce disproportionately more grandchildren than those that produce equal numbers of sons and daughters or that produce a higher proportion of the more common sex. In other words, if the population sex ratio deviates from equality (1:1), overproducing the minority sex yields above-average fitness, until the population sex ratio is equalized.

One assumption underlying this model is that there is a tradeoff between male and female production. Parents that allocate resources to produce more males must produce fewer females (or vice versa). The resource-constrained function is

where W is the fitness of an individual, with the average fitness of the population set to equal 1; m and M are the proportion of males in the offspring and in the population, respectively, and r represents the proportion of resources available to individuals for offspring production. When the sex ratio is exactly 1:1 in both the individual parent and in the whole population, then the parent's fitness is linearly proportional to the resources invested in reproduction of both daughters and sons. In a population that deviates from a 1:1 ratio, the fitness of any parent that produces a higher proportion of the rarer sex will be higher than the average (W> 1).

This model is a simplification of the real world; for example, it assumes a randomly mating and infinite population size in which there exist no stochastic fluctuations of sex ratio and no social structure. It also assumes that the cost of producing and raising a son is equal to that required to produce and to raise a daughter. Many more complex models have been developed to find the ESS in various situations which have been reviewed by Pen and Weissing.

The 1:1 sex ratio that occurs at equilibrium in the ESS model described above refers to the ratios ofindividual males to females. However, if the investment needed to produce a male is different than that needed to produce a female, than the population is expected to equalize the investment in the two sexes, and not their numbers. The population sex ratio exhibited in an ESS represents an equality of the investment in sons and daughters, and includes all stages beginning from gamete production to parental care.

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