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Sex allocation theory

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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 !!!

Biased sex ratio

LRE

Pen & Weissing demonstrated that:

!

Biased sex ratio

LRE

In the great majority of cooperatively breeding species, only one sex helps so:

- The ESS depends only on 2 parameters :
- Mean number of helpers
- Contribution of each helper

Remember that the model assumes that contributions are additive !

juvenile survival

We determined the winter survival

of 198 juveniles from 53 litters

Winter survival of juveniles = 0.78

(95% confidence interval : 0.72-0.84)

number of

subordinate

females

number of

subordinate males

Biased sex ratio

LRE: test in the alpine marmot

Test of the Helper Repayment Hypothesis

Subordinate males may be considered

as helpers and may reimburse parental

investment by warming juveniles during

winter

Five females in captivity gave birth to 22 sexed neonates

13 were males giving an overall sex ratio of 0.59

Biased sex ratio

LRE: test in the alpine marmot

Sex ratio at emergence

Complete sex composition at emergence was determined for 53 litters

representing a total of 207 juveniles

The overall sex ratio was 0.578 and significantly departed from 0.5

(95% confidence interval [0.511; 0.643])

Sex ratio at birth

Biased sex ratio

LRE: test in the alpine marmot

Test of the Helper Repayment Hypothesis

Mean number of helpers = 0,836

Mean effect of a helper = mean

percentage of increase in survival

b = 0,107

Sex ratio predicted = 0,541

Observed sex ratio = 0.578

[0.511; 0.643]

Biased sex ratio

Individual level

What individual strategy should be ?

Should all females have

the same strategy ?

Or not ?

Biased sex ratio

Individual level

Since the selection is only for the total expenditure, only the mean sex

ratio is fixed and there is no effect on the variance, that is, a population

can have any degree of heterogeneity so long as the totals expended on

the production of each sexes are equal (Kolman 1960)

Individuals producing offspring in sex ratios that deviate from 50/50

are not selected against as long as these deviations exactly cancel out

and result in a sex ratio at conception of 50/50 for the local breeding

population (Trivers and Willard 1973)

Biased sex ratio

Individual level

Parents should overproduce offspring of the most profitable sex in

term of fitness return (Trivers and Willard 1973)

Facultative sex ratio adjustment

Biased sex ratio

Individual level: Trivers and Willard (1973)

Assumptions of the Trivers and Willard’s hypothesis

- The condition of the young at the end of PI depends on the
- condition of the mother during PI
- 2. Differences in condition of young at the end of PI endure into
- adulthood
- 3. A slight advantage in condition has disproportionate effects on
- male reproductive success compared to the effects on female RS

3. => especially designed for polygynous species

Biased sex ratio

Individual level: Trivers and Willard (1973)

Predictions of the Trivers and Willard’s hypothesis

Females in relatively better condition tend to produce males and

females in relatively poor condition tend to produce females

Many studies aimed to test the TW model especially in ungulates

=> inconsistent results probably because:

1. assumptions not respected

2. predictions not clear (Leimar 1996)

Biased sex ratio

Individual level: LRC

Prediction of the LRC hypothesis

Females in a low quality environment should produce

more offspring of the dispersing sex

=> inconsistent results probably because :

The prediction may be the opposite of TW prediction: for example, in many

primates, daughters are philopatric. So, dominant females (in good situation)

should overproduce daughters and this is the opposite prediction of TW.

Biased sex ratio

Individual level: Burley (1981)

Assumptions and prediction

In species where some males are more attractive to females than

others, thereby leading to variation in male mating and reproductive

success, and where male attractiveness has a genetic basis, females

mated to attractive males should produce male-biased litters.

Biased sex ratio

Individual level: Burley (1981)

From Griffith et al. (2003)

On the blue tit Parus caeruleus

Assume the heritability of

UV coloration

Biased sex ratio

Individual level: Burley (1981)

Test at the individual level: Great tit Parus major

From Kölliker et al. (1999)

Heritability confirmed but low

Biased sex ratio

Individual level: Burley (1981)

In many species (especially in mammals), the attractiveness is hard to define

But if EPP occurs, and assuming that EPM are more attractive to females

than cuckolded males, we predict that the sex ratio should increase with the

proportion of EPY in the litter and that EPY should be more often males than

their half-sib WPY

Most studies in birds failed to show that the sex ratio of EPY was more

male-biased than the sex ratio of their half-sib WPY

Biased sex ratio

Individual level: Burley (1981)

Test at the individual level: alpine marmots

In the alpine marmot we found

that the sex ratio in mixed litters

was more male-biased as the

proportion of EPY increased

EPY were more likely males

(SR = 0.62 ± 0.09) than their

half-sib WPY (0.44 ± 0.08) but

the difference was not significant

(p = 0.2) => lack of power ?

Biased sex ratio

Individual level: LRE

Prediction of the LRE hypothesis

Females should produce more offspring of the helping sex when

helpers are absent in the family group

Ten mothers remained several years in their territory

They produced a sex ratio according to their social environment (p = 0.002):

Helpers absent: sr = 0.65 [0.54;0.74]helpers present: sr = 0.46 [0.36; 0.56]

Biased sex ratio

LRE: test in the alpine marmot

Test across females in the population

Only the presence of helpers

had a significant effect on sr

(2 = 8.74, df =1, p = 0.003)

sr = 0.66

sr = 0.49

Test in individual females across multiple years

Biased sex ratio

LRE: test in the alpine marmot

These results suggest that mothers are able to

facultatively adjust the sex ratio of their offspring

Mechanism ???