Sex allocation theory

Sex allocation theory Dominique Allainé UMR-CNRS 5558 « Biométrie et Biologie Evolutive » Université Lyon 1 France allaine@biomserv.univ-lyon1.fr

Allocation to sex Definition Sex allocation is the allocation of resources to male versus female reproductive function Charnov, E.L. 1982. The theory of sex allocation

Introduction • Two types of reproduction concerned: • Dioecy: individuals produce only one type of gamete during their lifetime • Hermaphrodism: individuals produce the two types of gametes during their lifetime • sequential hermaphrodism • simultaneous hermaphrodism

Introduction Problematic • In dioecious species, what is the sex ratio maintained by natural selection? 2. In sequential hermaphrodites, what is the order of sexes and the time of sex change ? 3. In simultaneous hermaphrodites, what is, at equilibrium, the resource allocated to males and females at each reproductive event ? • In what condition hermaphrodism or dioecy is evolutionarily stable? • In what condition, natural selection favors the ability of individuals to modify • their allocation to sexes ?

Introduction An old problem « … I formerly thought that when a tendency to produce the two sexes in equal numbers was advantageous to the species, it would follow from natural selection, but I now see that the whole problem is so intricate that it is safer to leave its solution for the future. » Charles Darwin, 1871. The descent of man, and selection in relation to sex. 2nd Edition 1874

Fisher’s model The first solution « If we consider the aggregate of an entire generation of such offspring it is clear that the total reproductive value of the males in this group is exactly equal to the total value of all the females, because each sex supply half the ancestry of all future generations of the species. From this it follows that the sex ratio will so adjust itself, under the influence of Natural Selection, that the total expenditure incurred in respect of children of each sex, shall be equal » R.A. Fisher. 1930. The genetical theory of natural selection

Fisher’s model Two comments • It is a frequency-dependant model • It is a verbal model How to demonstrate the Fisher’s equal allocation principle?

Fisher’s model ESS approach Concept adapted to ecology by J. Maynard-Smith An ESS is a strategy, noted r* that, if played in the population, cannot be invaded by an alternative strategy s, played by a mutant individual introduced in the population The fitness of an individual playing the strategy s in a population where individuals play the strategy r is notedW(s,r)

r* is an ESS if: W(r*,r*) > W(s,r*)  r* is the unique best response to r* Fisher’s model ESS approach Or: W(r*,r*) = W(s,r*) r* is not the unique best response to r*  and W(r*,r) > W(r,r) but r* is a better response to r than r

Formalization of the Fisher’s model When applied to sex allocation : • We consider: • a continuous variable, for example the proportion of males produced • a strategy r adopted by the females of the population • a strategy s adopted by a mutant female • an optimal strategy r* Consider a population of N females of a dioecious species with discrete generations Each female produces C offspring at each reproductive event Consider S1 and S2 the proportions of males and females that survive to the age of first reproduction Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons

The relative contribution of the mutant female to grandchildren through her sons is: The relative contribution of the mutant female to grandchildren through her daughters is: Formalization of the Fisher’s model 1. Formalization by Shaw and Mohler (1953) The offspring of the N+1 females, produce as a whole K offspring (1) (2)

The relative contribution of the mutant female to genes of grandchildren, that is her relative fitness, is the sum of (1) and (2) (3) (4) If N is large, (3) can be approximated by: Formalization of the Fisher’s model 1. Formalization by Shaw and Mohler (1953) This is the Shaw and Mohler equation

Then (4) becomes : (5) Formalization of the Fisher’s model 1. Formalization by Shaw and Mohler (1953) This is a generalization of the Shaw and Mohler’s equation

Formalization of the Fisher’s model 2. The marginal value criterion Consider a population of N females of a dioecious species with discrete generations Each female allocates an optimal proportion M* of resources to males and an optimal proportion F* = 1-M* of resources to females Consider a mutant female that allocates a proportion M of resources to males and a proportion F = 1-M of resources to females

(M) is the competitive ability of males having received an allocation M (F) is the competitive ability of females having received an allocation F is the genetic profit through males is the genetic profit through females Formalization of the Fisher’s model 2. The marginal value criterion The relative fitness of the mutant female in a population allocating M* is:

Formalization of the Fisher’s model 2. The marginal value criterion In the Fisher’s model, competitive abilities ((M) and (F)) are linear: (M) = aM and (F) = bF a and b being constants These competitive abilities are often measured by the number of males and females offspring surviving to the age of reproduction for example: (M) = CsS1 and (F) = C(1-s)S2 with a = CS1 and b = CS2 and allocation is measured by sex ratio (M = s and F = (1-s))

 Formalization of the Fisher’s model 2. The marginal value criterion It follows that: This is the Shaw and Mohler equation

Formalization of the Fisher’s model 3. Model of inclusive fitness Consider a population of N females of a dioecious species with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider S1 and S2 the proportions of males and females that survive to the age of first reproduction Consider a mutant female that produces a proportion s of sons

Formalization of the Fisher’s model 3. Model of inclusive fitness the fitness W of a female is measured by: W = number of adult daughters + number of females inseminated by her sons

Formalization of the Fisher’s model 3. Model of inclusive fitness Then, the relative fitness of a mutant female is : This is the Shaw and Mohler equation

Formalization of the Fisher’s model The Shaw and Mohler (1953)’ equation The relative fitness of a mutant female is :

In other words : Is the fitness of the mutant female greater than the fitness of a non mutant female? Or, is the relative fitness of the mutant female W(s,r) greater than 1 ? Or, does the « mutant » allele will invade the population ? Fisher’s model Solution with equal costs of production The question is : Does a mutant female contribute more to the next generation than a non mutant female?

Two conditions are needed: To have an extremum To have a maximum Fisher’s model Solution with equal costs of production What is the optimal sex ratio(allocation)? r* is the value such that the fitness W is maximised for s=r=r*

Fisher’s model Solution with equal costs of production r* = 0.5 is an ESS

Fisher’s model Solution with equal costs of production Comments: • The derived does not depend on s • If r = r* = 0.5, W’ = 0 whatever the value of s thus W(s,r*) = cte • If r = r*, s = r* is the best but not the unique best response to r* • If r < 0.5 W’ > 0 and s = 1 is the best response to r • If r > 0.5, W’ < 0 and s = 0 is the best response to r

s ≠ r s = r Fisher’s model Solution with equal costs of production best response (s) Sex ratio in the population (r)

Fisher’s model Model with different costs of production Fisher (1930) « From this it follows that the sex ratio will so adjust itself, under the influence of Natural Selection, that the total expenditure incurred in respect of children of each sex, shall be equal » The first model assumed that the energetic cost of production of both sexes was the same. Allocation was measured directly by the sex ratio. What happens if the costs of production of the two sexes differ ?

Fisher’s model Model with different costs of production Consider a population of N females of a dioecious species with discrete generations Each female has a quantity R of resources to allocate at each reproductive event Each female allocates a proportion q* of resources to the production of males Consider a mutant female allocating a proportion q of resources to the production of males

then Same form as the model with equal costs Fisher’s model Model with different costs of production and

Fisher’s model Model with different costs of production The optimal strategy is an equal allocation to males and females This is the Fisher’s prediction !

n♂x C♂=n♀x C♀ Fisher’s model Model with different costs of production Because each female has a quantity R of resources to allocate at each reproductive event and because young of the two sexes are not equally costly to produce, q* = 0.5 implies that the Fisher’s equal allocation principle can be written as:

Fisher’s model Conclusion Fisher’s model does not predict a sex ratio equal to 0.5 in the population if costs of production of the two sexes differ. Costs of production should be used sensu Trivers (1972) that is to say in term of fitness cost and not only in term of energetic cost (cf. Charnov 1979). Fisher’s principle should be rephrased in terms of equal investment rather than of equal allocation

Biased sex ratio Local Mate Competition (LMC) In some species of parasitoids, the environment is made of patches, each patch being occupied by fertilized females Patches of habitat

♀ ♀ ♀ ♂ ♂ ♂ ♀ ♂ ♀ ♀ ♀ ♀ ♀ ♂ ♀ ♂ ♂ ♂ ♀ ♀ ♀ ♂ ♀ ♀ ♂ ♀ ♀ Biased sex ratio LMC laying Offspring born on a patch mate on the patch Then, males die and fertilized females disperse to vacant patches laying mating

Biased sex ratio LMC In this kind of species, there is a local competition between males to fertilize females on the birth patch Males are then the costly sex and a female-biased sex ratio is expected

Biased sex ratio LMC : diploid species (Hamilton 1967) To predict the sex ratio in this situation, Hamilton has relaxed one assumption of the Fisher’s model: the hypothesis of a panmictic reproduction Consider a population of n females of a diploid species, dioecious with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons

It comes : Biased sex ratio LMC

s best response to r n = 3 Unique best response Sex ratio in the population (r) Biased sex ratio LMC : diploid species r* is an ESS

r* n Biased sex ratio LMC : diploid species

Biased sex ratio LMC: test in parasitoid wasps Many studies on parasitoid species give evidence that the sex ratio may be extremely biased towards females in these species.

Biased sex ratio Local Resource Competition (LRC) Clark (1978) Male dispersal t+1 t

Biased sex ratio LRC In some primate species, males disperse early while females stay with their mother beyond sexual maturity. Daughters compete with each other (and with their mother if alive) for resources. There is a local competition for resources between related females Females are then the costly sex and a male-biased sex ratio is expected

Biased sex ratio LRC Consider a population of N females of a diploid species, dioecious with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons Competition for resources affects females’ survival. Then the survival of daughters will depend on sex ratio [(r) or (s)]

Biased sex ratio LRC However, ’(r*) > 0 => r* > 0.5

Galago crassicaudatus Biased sex ratio LRC: test in primates (Clark 1978) From Clark (1978)

Biased sex ratio LRC: test in birds (Gowaty 1993) %males Dispersal female biased Sex ratio female biased Dispersal male biased Sex ratio male biased

Biased sex ratio Local Resource Enhancement (LRE) Emlen et al. (1986) In cooperative breeders, the sex ratio seems biased towards the helping sex Initially, we thought that helpers help because they are in excess in the population. Being in excess for an unknown reason, individuals of the helping sex do not find mate and they can increase their fitness by helping. However, Gowaty and Lennartz (1985) proposed an alternative interpretation. They argued that it is because they help that helpers are produced in excess This hypothesis was formalized by Emlen et al. in 1986

Biased sex ratio LRE In cooperatively breeding species, offspring of one sex generally stay in the family group and help parents in raising young. For example, helpers are provisioning food for young (local resource enhancement). The helping sex is less costly in fitness term because it provides a fitness benefit to parents by increasing reproductive success or decreasing the workload of parents. So, helpers reimburse parental investment. ➨ Helper repayment model

Biased sex ratio LRE Consider a population of N females of a diploid species, dioecious with discrete generations Each female produces C offspring at each reproductive event Each female produces a proportion r of sons Consider a mutant female that produces a proportion s of sons Helpers effect is expressed by a multiplicative coefficient H in the production of offspring

Helpers’ effect depends on the sex ratio produced by the mother Biased sex ratio LRE Helpers’ effects are assumed to be additive !!!

Sex allocation theory