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Sex Differences in Fitness Sex Differences in Fitness Variance and the Variance and the

Evolution of Mating Evolution of Mating SystemsSystems

Stephen M. ShusterNorthern Arizona University

Two Issues

1. The Source of Sexual Selection.2. The Intensity

of Sexual Selection.

Parental Investment and Sexual Selection

(Trivers 1972)

•The source of sexual selection is the

difference in initial parental investment between the sexes.

••“Females do all of the “Females do all of the investing; males do investing; males do

none of it.”none of it.”

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Problems with PIT

•The sex difference in initial parental investment is

difficult to compare within and among species.

•The correlation between sex differences in

gametic/parental investment and sexual dimorphism is

poor.•Measures of selection

intensity are indirect (e.g, PRR).

Measuring PRR

•Allow members of each sex unlimited

opportunities to mate in controlled setting.

•PRR=Xoffspring/unit time.

Mating Systems and Sexual Selection (Emlen & Oring 1977)

•Male reproduction is limited by the spatial

distribution of resources and by the temporal

distribution of sexually receptive females.

•The intensity of sexual selection on males depends

on the degree to which females are rare.

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The Operational Sex Ratio

OSR = Nmature males/Nreceptive

females

A reproductive competition coefficient.

OSR>1 = females are rare, competition for mates is

intense. OSR<1 = females are

abundant, competition for mates is relaxed.

The Environmental Potential for

Polygamy (EPP)

The degree to which the social and

ecological environment allows males to monopolize

females as mates.

Effects of the Spatial and Temporal

Distributions of Females

on EPP

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Problems with Estimating EPP

• The methods for measuring female distributions are not described.

•The scale for estimating EPP is undefined; selection intensity is not measured.

•Comparisons within and among species are imprecise.

Two Issues: Again

1. The Source of Sexual Selection.2. The Intensity

of Sexual Selection.

The Source of Sexual Selection

A sex difference in the variance in

fitness.(Wade 1979; Wade and

Arnold 1983; Shuster and Wade 2003)

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The Mean and The Mean and Variance in Variance in Offspring Offspring NumbersNumbers

NOtotal = 25

NOtotal = 25

O♀♀= 5; VO♀♀= 0; N=5.

O♂♂=5; VO♂♂= 0; N=5.

The Mean and The Mean and Variance in Variance in Offspring Offspring

Numbers, Numbers, by Sexby Sex

NOtotal = 25

O♀♀= 5; VO♀♀= 1010; N=5.

O♂♂=5; VO♂♂= 0; N=5.

If One Female If One Female Mates Twice...Mates Twice...

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NOtotal = 25

O♀♀= 5; VO♀♀= 2020; N=5.

O♂♂=5; VO♂♂= 0; N=5.

If One Female If One Female Mates Five Mates Five

Times...Times...

NOtotal = 25

O♀♀= 5; VO♀♀= 0; N=5.

O♂♂=5; VO♂♂= 2020; N=5.

If One Male If One Male Mates Five Mates Five

Times...Times...

When some individuals are excluded from mating, the

variance in offspring numbers is increased.

ThisThis is the source of sexual is the source of sexual selection.selection.

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The Intensity of Sexual SelectionThe variance in fitness

is proportional to the strength of selection.

A sex difference in the variance in fitness

determines whether and to what degree the

sexes will diverge.

Intensity of Sexual Selection“If each male

secures two or more females, many

males would not be able to pair.”

C. Darwin, 1871, p. 266.

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0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

Average donuts per student w/ donuts (D)

Prop

ortio

n of

exc

lude

d st

uden

ts (p

0)Graphically,Graphically, pp00 = = 11--(1/(1/DD))

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

Harem Size (H)

Prop

ortio

n of

non

mat

ing

mal

es (p

0)

pp00 = = 11--(1/(1/HH))Shuster and Wade 2003Shuster and Wade 2003

Sexual Selection is a Sexual Selection is a Powerful Evolutionary Powerful Evolutionary

Force Because: Force Because: For every male who sires young with with several females, there

must be several males who fail to reproduce at all.

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Quantitative Analysis of

Mating Systems

•The Opportunity for Selection.•Analysis of

Variance.

The Opportunity for SelectionThe Opportunity for SelectionCrow (1958, 1962)

I = VI = VWW/W/W22

Compares the fitness of breeding parents relative to the population before selection.

The variance in fitness, VW, places an upper limit on the change in mean fitness from one generation to

the next.

An Upper Bound Because,

Heritability (h2) is usually less than 1.

The correlation between the variance in fitness and phenotypic change, i.e., the

relationship between Vw and ∆Z, is usually less than 1.

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How Can We Measure I = VW/W2

In a Natural Population?

Begin with:The Mean and Variance

in Offspring Numbers Among Females

X = Σ xj / Nfemaleswhere xj the brood size of the j-th

female

VX = Σ pj (X - xj)2

Where pj = the proportion of females with brood size j

Next:The Mean and Variance

in Offspring Numbers Among Males

Average Male Mating Success

= Nfemales / Nmales= the Sex Ratio (R)

X = Average Offspring/Female

RX = Average Offspring/Male

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The Distribution of Females with Males

A Key PointThe mean and variance in

offspring numbers for each mating class of males equals the

products, jX and jVfemalesrespectively,

wherej = harem size ; X = average offspring/female,

Vfemales = variance in offspring/female

This Means That:The mean and variance in offspring

numbers among mating males will far exceed the mean and variance in

offspring numbers among females.

Males with no mates will produce no offspring at all; thus unsuccessful

males produce fewer offspring than the average female.

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This Is Why:

Sexual selection is one of the most

powerful evolutionary

forces known,

It produces such rapid changes in

phenotype.

Darwin on Sexual SelectionDarwin on Sexual Selectionsexual selection "…depends, not on a

struggle for existence, but on a struggle between males for

possession of the females; the result is not death of the unsuccessful

competitor, but few or no offspring. Sexual selection is, therefore, less rigorous than natural selection”

(1859, p. 88).

The Quantitative Paradoxof Sexual Selection

Sexual selection CAN be much stronger than natural selection

because:

Males with no mates will produce no offspring at all.

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ANOVA:The Distribution of Females with Males

As in ANOVA,

VVtotaltotal = = VVwithinwithin + + VVamongamong

= The average of the variances in offspring numbers within the classes of males who mate

+The variance of the averages in offspring

numbers among the classes of mating and nonmating males

Vmales = Σ pj (jVfemales ) + Σ pj (jX-RX )2

= RVfemales + X 2Vmates

The Total Opportunity for Selection on Males and Females

(Wade 1979; Wade & Arnold 1980)

Recall that RX = average number of offspring/male;

dividing Vmales by [RX]2 gives

Vmales / [RX]2 = [RVfemales +X 2Vmates ] / [RX]2

which givesImales = (1/R)Ifemales + Imates

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Note That,

Imales = 1/R (Ifemales) + Imates

when R = 1,

Imales - Ifemales = Imates

In general, the sex difference in the opportunity for selection is due to differences in mate numbers

between the sexes.

The Opportunity for SelectionThe Opportunity for Selection•The sex difference in the variance in mate numbers causes sexual selection.

•Provides a direct measure of the intensity of sexual

selection.•Allows calculation of

confidence limits.•Provides an empirical

estimate of the degree to which the sexes will diverge.

How Can We Measure Imates

in a Natural Population?

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SyngnathusSyngnathus typhletyphle(Berglund et al. 1989)

0

5

10

15

20

0 0.5 1 1.5 2 2.5 3 3.5 4

N Filled Males

N F

emal

es

0

5

10

15

20

25

30

35

0 1 2 3 4 5 6

N Matings

N M

ales

Males and females have multiple mates; sexual selection occurs in

both sexes.

Factors Influencing Factors Influencing the Intensity of the Intensity of Sexual SelectionSexual Selection 1. The spatial

distribution of mates.

2. The temporal distribution of

mates.

A Model PopulationNmales = Nfemales = 20

a b

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Mean CrowdingMean Crowding(Lloyd 1967)(Lloyd 1967)

m* = the number of other individuals the average

individual experiences as competitors.

m* = [Σ mi (mi - 1)]/ (Σ mi).

P = “Patchiness”; the ratio of the mean crowding to the

average number of individuals per patch (= m*/m).

Mean Crowding and Resource Based Mating Systems

(Wade 1995)

If we letM = the number of resource patches

containing at least 1 female;mi = the number of females in the i-th

patch; thenthe average density of females per

patch ism = Σ mi / M

The Mean Crowding of Femalesin Space

The mean crowding of females on resources defended by males can be expressed as,

m* = m + [(Vm / m) - 1]

In this context, m* represents the number of other females the average female experiences on

her resource patch.

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The Patchiness of Females in Space

Lloyd’s (1967) estimator of patchiness, P, measures the patchiness of females on resources

P = m*/mor by substitution,

P = 1 + [(Vm / m2) - (1/m)]Thus, P measures the degree to which females

are patchily distributed in space.

Average Male Mating Success

If p0 = the proportion of non-

mating males, (1 - p0) = the proportion of mating males,

and R = (Nfemales / Nmales), then

H = R / (1 - p0) 0 1 2 3 4 5 60

20

40; ; ; ; ;

Number of females (i)

Num

ber

of m

ales

RH

Variance in Male Mating Success

When females are clumped in space, two components of variance in male mating

success exist:(1) the variance in mating success within the class of males that actually mate by

defending resources. (2) the variance in mating success among

the mating and non-mating males.

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Total Fitness VarianceThe largest fraction

exists among mating and non-mating males.

Measures of selection that

exclude p0 (e.g., BSR, Q) under

estimate selection's intensity.

0 1 2 3 4 5 60

20

40; ; ; ; ;

Number of females (i)

Num

ber

of m

ales

RH

The Variance in Mating Success Within and Among Males

Vmates = Vwithin + Vamong

Vwithin = (p0 ) (0) + (1 - p0) Vharem

Vamong = (H - 0)2 (p0 ) (1 - p0)

Vmates = (1 - p0) Vharem + H2(p0 ) (1 - p0)

The Opportunity for Sexual Selection (Imates)

Recall that I = VW/W2,and that R (avg. mates/male) = H (1 - p0).

Dividing Vmates by R2 yields:

Imates = [1/(1 - p0)][ (Vharem) / (H2)]+ (p0 ) / (1 - p0)

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When Females Are Clumped in Space:

and R is assumed to equal 1, then H = 1/(1 - p0) = m,

and(Vharem) / (H2) = (Vm) / (m2)

Then by substitution,

Imates = [1/(1 - p0)][ (Vharem) / (H2)]+ (p0 ) / (1 - p0)

becomesImates = (m) [ (Vm) / (m2)] + 1 - (1/m)

or more simply,Imates = (m) (P) = m*

What Does This Mean?The mean spatial crowding of sexually receptive

females, m*, provides a direct estimate of the opportunity for sexual selection, Imates, arising from the

distribution of females in space.