Sex allocation theory Dominique Allainé UMR-CNRS 5558 « Biométrie et Biologie Evolutive » Université Lyon 1 France [email protected]

Sex allocation theorySex allocation theory

Dominique AllainéUMR-CNRS 5558« Biométrie et Biologie Evolutive »Université Lyon 1 France

[email protected]

Allocation to sex

Sex allocation is the allocation of resources to male versus female reproductive function

Definition

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

4. In what condition hermaphrodism or dioecy is evolutionarily stable?

1. 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 ?

5. 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

Two comments

1. It is a frequency-dependant model

2. It is a verbal model

How to demonstrate the Fisher’s equal allocation principle?How to demonstrate the Fisher’s equal allocation principle?

Fisher’s model

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

Fisher’s model

ESS approach

The fitness of an individual playing the strategy s in a population where individuals play the strategy r is noted W(s,r)

Fisher’s model

ESS approach

W(r*,r*) > W(s,r*) r* is the unique best response to r*

r* is an ESS if:

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

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

We consider:

1. a continuous variable, for example the proportion of males produced2. a strategy r adopted by the females of the population3. a strategy s adopted by a mutant female4. an optimal strategy r*

When applied to sex allocation :

The offspring of the N+1 females, produce as a whole K offspring

The relative contribution of the mutant female to grandchildren through her sons is:

NCrS1CsS1

CsS1k

2

1

(1)

(2)


1. Formalization by Shaw and Mohler (1953)

r)S2NC(1s)S2C(1

s)S2C(1k

2

1

The relative contribution of the mutant female to grandchildren through her daughters is:

The relative contribution of the mutant female to genes of grandchildren,that is her relative fitness, is the sum of (1) and (2)

r)S2NC(1s)S2C(1

s)S2C(1

NCrS1CsS1

CsS1k

2

1 W r)(s,

(3)

If N is large, (3) can be approximated by:

r)S2NC(1

s)S2C(1

NCrS1

CsS1k

2

1W r)(s,

(4)

This is the Shaw and Mohler equationThis is the Shaw and Mohler equation



r)S2-C(1 f CrS1 m

s)S2-C(1 f̂ CsS1 m̂ :given

Then (4) becomes :

ˆˆ

W r)(s,

f

f

m

m (5)

This is a generalization of the Shaw and Mohler’s equationThis is a generalization of the Shaw and Mohler’s equation



2. The marginal value criterion


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 anda proportion F = 1-M of resources to females


The relative fitness of the mutant female in a population allocating M* is:

*)(

)(

*)(

)(W M*)(M,

FN

F

MN

M

(M) is the competitive ability of males having received an allocation M

(F) is the competitive ability of females having received an allocation F

*)(

)(

MN

M

is the genetic profit through males

*)(

)(

FN

F

is the genetic profit through females



In the Fisher’s model, competitive abilities ((M) and (F)) are

linear: (M) = aM and (F) = bF a and b being constants

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))

These competitive abilities are often measured by the number ofmales and females offspring surviving to the age of reproduction



It follows that:

*)(

)(

*)(

)(W M*)(M,

FN

F

MN

M

*

ˆ

*

ˆ W M*)(M,

f

f

m

m




3. Model of inclusive fitness





Consider S1 and S2 the proportions of males and females that survive to the age of first reproduction


the fitness W of a female is measured by:

W = number of adult daughters + number of females inseminated by her sons

r)S2-C(1 f CrS1 m

s)S2-C(1 f̂ CsS1 m̂ :given

2f Nm

mNf f W(r) and

m

m̂f f̂

Nm

m̂Nf f̂ W(s):Then



f

f̂

m

m̂

f

f̂

m

m̂

2

1

m

m̂ff̂

2f

1 r)W(s,

Then, the relative fitness of a mutant female is :




The relative fitness of a mutant female is :


The Shaw and Mohler (1953)’ equation

r)(1

s)(1

r

sW r)(s,

Solution with equal costs of production

The question is :

Does a mutant female contribute more to the next generation than a non mutantfemale?

In other words :

Is the fitness of the mutant female greater than the fitness of a non mutantfemale?

Or, does the « mutant » allele will invade the population ?

Or, is the relative fitness of the mutant female W(s,r) greater than 1 ?

Fisher’s model

What is the optimal sex ratio (allocation)?


Fisher’s model

Two conditions are needed:

(2) 0s

W

(1) 0s

W

*rrs

2

2

*rrs

To have an extremum

To have a maximum

r* is the value such that the fitness W is maximised for s=r=r*

2

1r* 0 *2r-1 0

s

W

*

rrs

r* = 0.5 is an ESS


Fisher’s model

r)r(1

2r1

s

W

1. The derived does not depend on s

2. If r = r* = 0.5, W’ = 0 whatever the value of s thus W(s,r*) = cte

3. If r = r*, s = r* is the best but not the unique best response to r*

4. If r < 0.5 W’ > 0 and s = 1 is the best response to r

5. If r > 0.5, W’ < 0 and s = 0 is the best response to r


Fisher’s model

Comments:

s ≠ r

s = r

best

resp

onse

(s

)

Sex ratio in the population (r)


Fisher’s model

The first model assumed that the energetic cost of production of both sexes was the same. Allocation was measured directly by the sex ratio.

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 »

What happens if the costs of production of the two sexes differ ?

Fisher’s model

Model with different costs of production


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

Fisher’s model


C2

q*)S2-R(1 *f

C1

S1*Rq m*

C2

q)S2-R(1 f

C1

RqS1 m

*f

f

*m

mW q*)(q,

and

then *q-1

q-1

*q

qW q*)(q,

Same form as the model with equal costs

Fisher’s model


2

1q* 0 *2q-1 0

q

W

*

qq

The optimal strategy is an equal allocation to males and females

This is the Fisher’s prediction !This is the Fisher’s prediction !

Fisher’s model


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:

n♂ x C♂ = n♀x C♀

Fisher’s model


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

Conclusion

Fisher’s model

Biased sex ratio

Local Mate Competition (LMC)

Patches of habitat

In some species of parasitoids, the environment is made of patches, each patch being occupied by fertilized females

Biased sex ratio

LMC

layingmating

laying

Offspring born on a patch mate on the patch

Then, males die and fertilized females disperse to vacant patches

♀

♂

♀♀ ♀

♂♂

♀

♀

♂♂

♀♀ ♀

♀♂

♂♀

♀

♂

♀ ♂

♀ ♂

♀

♀

♀

Biased sex ratio

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

LMC

Biased sex ratio

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




LMC : diploid species (Hamilton 1967)

s1)r(n

s1r)-1)(1(nss1

r)-2(1

1 r) W(s,

LMCBiased sex ratio

0 (nr*)

*nr

*nr

r*)n-(11- 0

s

W Then,

2*rrs

It comes : 2n

1 -

2

1

2n

1-n *r

f

f̂1)f(n

m̂1)m(n

m̂

f

f̂

2

1 r)W(s, :Then

r* is an ESS

Sex ratio in the population (r)

s best

resp

onse

to r

Unique best response

n = 3

0 s1)r-(n

s1)r-(n1

s1)r-(n

n-

s

W

2

2

22

2

LMC : diploid species

Biased sex ratio

r*

n

LMC : diploid species

Biased sex ratio

0 *r 1, n if Fisher 0.5 *r , n if

2n

1 -

2

1 *r

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

Biased sex ratio

LMC: test in parasitoid wasps

Local Resource Competition (LRC) Clark (1978)

Biased sex ratio

tt+1 Male dispersal

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

Biased sex ratio

Females are then the costly sex and a male-biased sex ratio is expected

Consider a population of N females of a diploid species, dioecious with discrete generations




Competition for resources affects females’ survival. Then the survival

of daughters will depend on sex ratio [(r) or (s)]

LRC

Biased sex ratio

LRC

r*)-(1*r

-1)*(r*)(2r (r*)' 0

s

W

*rs

r

However, ’(r*) > 0 => r* > 0.5

Biased sex ratio

(r)rr)(1

(r)r)-s(1(s)rs)(1

2

1W r)(s,

LRC: test in primates (Clark 1978)

Biased sex ratio

Galago crassicaudatus

From Clark (1978)

LRC: test in birds (Gowaty 1993)

Biased sex ratio

Dispersal female biased

Sex ratio female biased

Dispersal male biased

Sex ratio male biased

%males

Local Resource Enhancement (LRE) Emlen et al. (1986)

Biased sex ratio

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 excessbecause they help that helpers are produced in excess

This hypothesis was formalized by Emlen et al. in 1986

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




Helpers effect is expressed by a multiplicative coefficient H in the production of offspring

Biased sex ratio

LRE

(s)hb (s)hb 1 H(s) ffmm

Helpers’ effects are assumed to be additive !!!

femalea of and malea of onscontributi respective thebet b

helpers female ofnumber mean the(s)h

helpers male ofnumber mean the(s)h :given

fm

f

m

Helpers’ effect depends on the sex ratio produced by the mother

Biased sex ratio

LRE

r

s

r)-(1

s)-(1

H(r)

H(s)

2

1 r)W(s,

Biased sex ratio

*r-1

hb -

*r

hb 2-

*r-1

1 -

*r

1 0

s

W ffmm

*rr s

h2b 1

h2b 1

*r-1

*r : Then

ff

mm

Pen & Weissing demonstrated that:

LRE

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

h2b 1 sex helping non n

sex helping n

The ESS depends only on 2 parameters :

1. Mean number of helpers2. Contribution of each helper

! Remember that the model assumes that contributions are additive !

Biased sex ratio

Biased sex ratio

Test of the Helper Repayment Hypothesis

0

1

2

34

0

1

2

0.4

0.5

0.6

0.7

0.8

0.9

1

number of subordinate males

number of subordinate females

juve

nile

sur

viva

l

We determined the winter survival of 198 juveniles from 53 litters

Winter survival of juveniles = 0.78 (95% confidence interval : 0.72-0.84)

Subordinate males may be consideredas helpers and may reimburse parentalinvestment by warming juveniles duringwinter

LRE: test in the alpine marmot

Biased sex ratio

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 emergence

Sex ratio at birth

Five females in captivity gave birth to 22 sexed neonates

13 were males giving an overall sex ratio of 0.59


Biased sex ratio

Test of the Helper Repayment Hypothesis


0,4

0,6

0,8

1

0 1 2 3 4 5

Number of males ≥ 2 years-old

Juve

nile

Surv

ival

53

5425

2 Mean number of helpers = 0,836

Mean effect of a helper = mean percentage of increase in survivalb = 0,107

Sex ratio predicted = 0,541

Observed sex ratio = 0.578 [0.511; 0.643]

Biased sex ratio

What individual strategy should be ?

Individual level

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 populationcan have any degree of heterogeneity so long as the totals expended onthe production of each sexes are equal (Kolman 1960)

Individuals producing offspring in sex ratios that deviate from 50/50are not selected against as long as these deviations exactly cancel outand result in a sex ratio at conception of 50/50 for the local breedingpopulation (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

1. The condition of the young at the end of PI depends on thecondition of the mother during PI

2. Differences in condition of young at the end of PI endure intoadulthood

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

Predictions of the Trivers and Willard’s hypothesis

Females in relatively better condition tend to produce males andfemales 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 respected2. predictions not clear (Leimar 1996)

Individual level: Trivers and Willard (1973)

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

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.

=> inconsistent results probably because :

Biased sex ratio

Individual level: Burley (1981)

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.

Assumptions and prediction

Biased sex ratio


From Griffith et al. (2003)

On the blue tit Parus caeruleus

Assume the heritability ofUV coloration

Biased sex ratio


Heritability confirmed but lowFrom Kölliker et al. (1999)

Test at the individual level: Great tit Parus major

Biased sex ratio


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 theproportion 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


In the alpine marmot we foundthat the sex ratio in mixed litterswas more male-biased as the proportion of EPY increased

EPY were more likely males (SR = 0.62 ± 0.09) than theirhalf-sib WPY (0.44 ± 0.08) butthe difference was not significant(p = 0.2) => lack of power ?

Test at the individual level: alpine marmots

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

Biased sex ratio

Test across females in the population

0

0.2

0.4

0.6

0.8

1

-1 0 1

absence/presence of helpersse

x ra

tio

25

57sr = 0.66sr = 0.49

Only the presence of helpershad a significant effect on sr(2 = 8.74, df =1, p = 0.003)

Test in individual females across multiple years

Ten mothers remained several years in their territoryThey 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


These results suggest that mothers are able to facultatively adjust the sex ratiofacultatively adjust the sex ratio of their offspring

Mechanism ???

Documents

Sex allocation theory Dominique Allainé UMR-CNRS 5558 « Biométrie et Biologie Evolutive » Université Lyon 1 France [email protected]