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Transcript of Sex allocation theory Dominique Allainé UMR-CNRS 5558 « Biométrie et Biologie Evolutive »...
Sex allocation theorySex allocation theory
Dominique AllainéUMR-CNRS 5558« Biométrie et Biologie Evolutive »Université Lyon 1 France
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)
Formalization of the Fisher’s model
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
Formalization of the Fisher’s model
1. Formalization by Shaw and Mohler (1953)
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
Formalization of the Fisher’s model
1. Formalization by Shaw and Mohler (1953)
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 anda proportion F = 1-M of resources to females
Formalization of the Fisher’s model
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
2. The marginal value criterion
Formalization of the Fisher’s model
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
2. The marginal value criterion
Formalization of the Fisher’s model
It follows that:
*)(
)(
*)(
)(W M*)(M,
FN
F
MN
M
*
ˆ
*
ˆ W M*)(M,
f
f
m
m
2. The marginal value criterion
Formalization of the Fisher’s model
This is the Shaw and Mohler equationThis is the Shaw and Mohler equation
3. Model of inclusive fitness
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
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
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
3. Model of inclusive fitness
Formalization of the Fisher’s model
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 :
3. Model of inclusive fitness
Formalization of the Fisher’s model
This is the Shaw and Mohler equationThis is the Shaw and Mohler equation
The relative fitness of a mutant female is :
Formalization of the Fisher’s model
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)?
Solution with equal costs of production
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
Solution with equal costs of production
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
Solution with equal costs of production
Fisher’s model
Comments:
s ≠ r
s = r
best
resp
onse
(s
)
Sex ratio in the population (r)
Solution with equal costs of production
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
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
Fisher’s model
Model with different costs of production
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
Model with different costs of production
2
1q* 0 *2q-1 0
q
W
*
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
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:
n♂ x C♂ = n♀x C♀
Fisher’s model
Model with different costs of production
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
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
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
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)]
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
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
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
LRE: test in the alpine marmot
Biased sex ratio
Test of the Helper Repayment Hypothesis
LRE: test in the alpine marmot
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
Individual level: Burley (1981)
From Griffith et al. (2003)
On the blue tit Parus caeruleus
Assume the heritability ofUV coloration
Biased sex ratio
Individual level: Burley (1981)
Heritability confirmed but lowFrom Kölliker et al. (1999)
Test at the individual level: Great tit Parus major
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 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
Individual level: Burley (1981)
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]
LRE: test in the alpine marmot
Biased sex ratio
LRE: test in the alpine marmot
These results suggest that mothers are able to facultatively adjust the sex ratiofacultatively adjust the sex ratio of their offspring
Mechanism ???