François Rousset - présentation MEE2013

Fitness for the impatient
Fran¸cois Rousset
May 2013
Fran¸cois Rousset Fitness for the impatient May 2013 1 / 33

The message
To understand the forces operating in social evolution, particularly in
spatially structured populations:

The message
Relatedness concepts under localized dispersal

The message
Stability of kin recognition polymorphisms
Relationship between inclusive ﬁtness and evolutionary stability

The message
A fundamental impatience: reduce these problems to trivial building blocks

The message
A fundamental impatience: reduce these problems to trivial building blocks
Draw connections to other methods such as diﬀusion theory, multilocus
methods

Continuous evolutionary stability for the very impatient
Two alleles, a and A, inducing phenotypes za and zA

Two alleles, a and A, inducing phenotypes za and zA
FitnessofAallele
Phenotypes
za, zA
1
a
z1 zm z2

Classiﬁcation of diﬀerent cases:
Continuously stable
strategy CSS
Branching point
Evolutionarily stable
noninvasible
Invasible
Unattainable
Convergence stable
attainable
Eshel 1983,1996; Christiansen, 1991; Abrams et al., 1993

Implicit assumptions
Resident phenotype z, mutant z + δ
∆p ∼ δstuﬀ + δ2
blob (1)
assumed to be essentially of the form
∆p ∼ δp(1 − p)S(z) + δ2
p(1 − p)(1 − 2p)blob(z) (2)
where S(z) and blob(z) are of constant sign wrt p.

Implicit assumptions
Resident phenotype z, mutant z + δ
∆p ∼ δstuﬀ + δ2
blob (1)
assumed to be essentially of the form
∆p ∼ δp(1 − p)S(z) + δ2
p(1 − p)(1 − 2p)blob(z) (2)
where S(z) and blob(z) are of constant sign wrt p.
Actually S(z) is independent from p in many models (strong claim!).
Why?

A study of frequency (p) dependence
The ﬁrst-order term
A conceptual device
Fitness
The minimal algorithm
Two views of the Prisoner’s dilemma
Many views of inclusive ﬁtness
The more general logic: dominance, kin recognition
Kin recognition
Glimpses of second-order results
Continuous evolutionary stability
Kin recognition

A conceptual device
p =
parents i
Ai Xi
X indicator variable for an allele;
A frequency of copies of parental genes
Consider the function f giving (conditional) probabilities
E[p | . . .] =
i
ai (. . .)Xi
where ai is the probability that a descendant copy originates from parental
copy i.

A conceptual device
p =
gene copies g
Ag Xg
X indicator variable for an allele;
A frequency of copies of parental genes
Consider the function f giving (conditional) probabilities
E[p | . . .] =
i
ai (. . .)Xi
where ai is the probability that a descendant copy originates from parental
copy i.

Trivial example
Individuals share a resource in total amount R, their fecundity being equal
to the amount of resource they consume
Their share of resource is proportional to the value of some trait z:
sharei =
zi
i zi
= ai , fecundityi = R
zi
i zi

Trivial example
Individuals share a resource in total amount R, their fecundity being equal
to the amount of resource they consume
Their share of resource is proportional to the value of some trait z:
sharei =
zi
i zi
= ai , fecundityi = R
zi
i zi
Conﬂict between individual and group
fecundityi = (R − ¯z)
zi
i zi
but ai is still zi
i zi
.

Fitness
In terms of the number of adult offspring, or“fitness”W
p =
gene copies g Xg Wg
g Wg
= mean(Xg Wg )
Define fitness functions so that
E[p |p] =
1
Ntot g
Xg w(zg (p))
In a demographically stable population Ntota = w (e.g., w = zi
¯z ).
A (minimal: ¯W = 1) Price equation
p = ¯X = ¯W ¯X + Cov(wg , Xg ),
E[∆p|z] = Cov(wg (z), Xg ).

Minimal algorithm
Express fitness in terms of indicator variables
e.g., phenotype zi = z + Xi δ
Differentiate with respect to some measure of strength of selection
To first order in δ,
wf(zf, zp) ∼ linear combination of (X, X)
Collect products of indicator variables
p ∼ mean(wi Hi ) =
linear combination of means of (H2
, HH) =
p + δ
∂w
∂zf
mean(H2
− HH)
Take expectations E.g., a large population without (spatial) structure
p = p + δ
∂w
∂zf
(p − p2
)

(“dilemma”: T > R, P > S, R > P i.e. T > R > P > S)

Phenotype (z): probability of cooperating in a prisoner’s dilemma
Fecundityf ∝ 1 + Rzfz◦ + Szf(1 − z◦) + T(1 − zf)z◦ + P(1 − zf)(1 − z◦)
No spatial structure:
wf =
Fecundityf
Mean fecundity
phenotype zi = zres + Xi δ
∆p = Cov(wi , Xi ) = δpq[S−P+(z+pδ)(R−S+P−T)]+O[δ3
, (R, S, T, P)2
]
View 1: R, T, S, P are given ecological constraints; evolution of z ⇒
expansion in δ. ∆p ∼ δpq[S − P + z(R − S + P − T)]
View 2: expansion in R, T, S, P, not in δ.

Population structure
Color code: focal, neighbor, population
wf ∼ linear combination of (H, H, H)
so that
mean(H)t+1 ∼ mean(wi Hi ) =
p + linear combination of means of (H2
, HH, HH) =
p + δ
∂w
∂zf
(H2
− HH) +
∂w
∂zn
(HH − HH)
Traditional population genetic argument:
E[H|H, p] = FSTH + (1 − FST)p
for“relatedness”FST independent of p.

Genealogical interpretation: island model
past
. . . . . .
p p2
Lineages from distinct demes can be considered as draws of independent
genes copies, each A with frequency p.
Probability that ﬁrst event is coalescence: FST
Such coalescences are recent if migration“not too small”; p then considered
constant if selection is“weak”and total population size is“large”.

Inclusive ﬁtness under weak selection
Traditional argument about relatedness“r”(or FST) :
E[H|H, p] = rH + (1 − r)p
for“relatedness”r independent of p. Hence
E[HH − HH|p] = p(r + (1 − r)p) − p2
= rpq
hence (with E[HH − HH|p] = pq)
δ
∂w
∂zf
(H2
− HH) +
∂w
∂zn
(HH − HH) = pq δ
∂w
∂zf
+
∂w
∂zn
r
inclusive ﬁtness −c + rb
.
Selection gradient is independent of p.

Genealogical interpretation: “stepping stone”
past
. . . . . .
Lineages from distinct demes cannot be considered as draws of
independent genes copies, each A with frequency p

Frequency-dependence in the stepping-stone model
(Circular stepping-stone model with 200 demes of 10 haploid individuals,
dispersal rate 0.2, and a two allele model with mutation rate 10−5)

Weak selection under localized dispersal
Asymptotic results for large number of demes:
(Infinite) island model
∆p ∼ δpqsIF = δpq(1 − FST)φ
FST ≡
E(XX) − E(XX)
E(XX) − E(XX)
sIF: scaled inclusive fitness effect, selection gradient
φ: p-independent localized selection gradient

Weak selection under localized dispersal
Asymptotic results for large number of demes:
(Infinite) island model
∆p ∼ δpqsIF = δpq(1 − FST)φ
Localized dispersal
∆p ∼ δpqsIF(p) = δpq(1 − FST(p))φ
FST(p) same as FST but for conditional probabilities
FST ≡
E(XX) − E(XX)
E(XX) − E(XX)
FST(p) ≡
E(XX|p) − E(XX|p)
E(XX|p) − E(XX|p)
sIF: scaled inclusive fitness effect, selection gradient
φ: p-independent localized selection gradient

What does that mean ?

The localized selection gradient φ and local FST’s
The ﬁnite population meaning and computation of φ
φ ≡
∂π
∂δ
= −
k=f
∂w
∂zk
Tk
T0
Tk
T0
= lim
µ→0
E[p − XXk]
E[p − XX0]
= lim
µ→0
1
1 − FSTk
Localized interactions: short distances (k) only. Everything
understandable as the result of local interactions (global p doesn’t
matter!).
Expressed in terms of local population structure parameters relatively
easy to estimate using genetic markers, and with genealogical
interpretation(s)
Genealogical: FSTk(p) = FSTk at neutral genetic markers

Fitness costs and beneﬁts

Eﬀects B on neighbors’ fecundity and −C on focal’s fecundity
rb − c = rB − C? In general No!

Local competition ⇒“inclusive ﬁtness”= −C (Taylor, 1992).
Actually ∆p ∼ −pq(1 − FST)C in island model.

How to obtain rb − c ∝ rB − C?

Hamilton (1975): “groups break up completely and re-form in each
generation”“young animals take oﬀ to form a migrant pool”... [one type]
assort[s] positively with its own type in settling from the migrant pool(...)
to such a degree that the correlation of two separate randomly selected
members (...) is F”
⇒ ∆p ∼ (RB − C)pq

Version with spatially restricted dispersal
Individuals disperse as a group and compete as a group against other
groups for access to whole group breeding spots. The winners of such
group contests can then occupy whole demes
⇒ ∆p ∼ (1 − F)(RB − C)pq
(Gardner & West 2006; Lehmann et al. 2006).

More general logic: example of dominance
Phenotype zi = z + δ[2h(Xi1 + Xi2)/2 + (1 − 2h)Xi1Xi2]
To ﬁrst order,
wf(zf, zp) ∼ linear combination of (X1, X2, X1X2, ...)
so that
E[p ] = mean(wi Xi ) =
linear combination of means of (X2
1, X1X2, X1X, X1X1X2, . . . , X1X1X2) =
Triplets generally lead to frequency-dependence, although there are
intriguing exceptions:
Helping among diploid full sibs and among haplodiploid sisters, partial sib
mating (α)
∆p ∼ δpq
∂w
∂zf
+
∂w
∂zn
r f (h, α, /p)

Three gene lineages... island model
past
. . . . . .
p p2 p3
Lineages from distinct demes can be considered as draws of independent
genes copies, each A with frequency p.
Probability that ﬁrst event is coalescence: FST

More general logic: example of kin recognition
Demes of N individuals; two loci; indicator variables R, H for alleles
Conditional helping:
fecundityf = 1 +
1
N − 1
neighbours k
[RfRk + (1 − Rf)(1 − Rk)](−CHf + BHk)
(two recognition alleles case)
wf =
(1 − d)ff
(1 − d)fdeme + dfothers
+ d
ff
fothers
(dispersal probability d; regulation after dispersal)
mean(H)t+1 =
mean(wi Hi )
mean(wi )
= mean(wi Hi ) over individuals i.
(strict regulation)

Expectedly...
To ﬁrst order in C and B,
wf ∼ linear combination of (H, H, RH, RH, RRH, RH, RRH,
H, RH, RRH, RH)
so that
mean(H)t+1 ∼ mean(wi Hi ) =
linear combination of expectations of (H2
, HH, RH2
, RH2
, RRH2
,
RHH, RRHH, HH, RHH, RRHH, RHH)
RH RH R RH H

Meaning
An act of helping always involves the conﬁguration
HR R
−C B

Meaning
HR R
−C B
RH RH
describes the probability that the receiver bears the helping allele;
can be computed in terms of the probability of joint coalescence within the
deme

Meaning
HR R
−C B
R RH H
describes the probability that a third individual bears the helping allele.
The ﬁtness of this individual is reduced in proportion to (B-C), that is, in
proportion to the increase in ﬁtness of the pair of interacting individuals;
can be computed in terms of the probability of joint coalescence within the
deme

Kin recognition: results
∆pH ∼ − C (1 − F) (1 − 2pqR) pqH
+ −C F + B φ − (1 − m)2
(B − C)
1
N
(F + φ) + 1 −
1
N
γ
∆pR ∼pHpqR(1 − 2pR)
B − C
N
[Z(N, m) < 0]
Polymorphism lost at the recognition locus.

Synthesis and developments
Many results follow mechanically from a trivial description of
allele-frequency changes in terms of fitness functions:
* Write a properly defined fitness function in terms of individual
behaviour
* Express behaviour in term of genotypes (indicator variables)
* Expand wi Xi to appropriate order, and take expectations of products
of indicator variables (special case: “direct fitness”method, Taylor &
Frank 1996)

behaviour
Frank 1996)
Deterministic“multi”locus models, with a recombination step:
∆(mean(any thing X)) = ∆sel(mean(X)) + ∆recomb(mean(X)).
Multilocus models often in terms of“centered”associations that are 0
in expectation in neutral models, e.g. E[(R − pR)(H − pH)].

behaviour
Frank 1996)
Diffusion methods (first order) e.g. approximation for fixation
probability
π ∼
1 − e−2Ntotφpini
1 − e−2Ntotφ
for φ taken in an infinite-deme limit.

behaviour
Frank 1996)
Diffusion methods (first order)
Diffusion with p-dependence (in first order, e.g. dominance)

behaviour
Frank 1996)
Diffusion methods (first order)
Diffusion with p-dependence (in first order, e.g. dominance)
Evolutionary stability

Continuous evolutionary stability in the island model
Second-order computation for any p feasible.
Simpler computation for extreme p, and connexion to“number of
successful emigrants”approach (Chesson 1984; Metz and Gyllenberg 2001)
B = ∂zf,zf
w +2F∂zf,zn
w +K∂zn,zn
w +4F(N −1) K∂zn
wp + F∂fwp wp∂zn
w
where wp: local offspring
In contrast to formula proposed by Day and Taylor (1998):
(1) three-genes coefficient K;
(2) products of identity coefficients, product of derivatives: joint effect of
first-order change in number of offspring and first-order change in parental
population structure.

Kin recognition: searching for stable polymorphisms
Second-order computation involves 30 associations, for up to 7 gene copies
in 5 individuals: R RH/R RH/R
Search for conditions for stable polymorphism: low migration and low
recombination
Convergent orbits plus drift:

“Conclusions”
First order: Simple formalism provides connections between different
modelling approaches (inclusive fitness, adaptive dynamics, diffusion,
coalescence). Relatively simple pattern of frequency-dependence.
Second order: Essentially the same formalism previously used for
multilocus models; Algebraically (and algorithmically) messy in
spatially structured populations; still allows an analysis of the forces
acting on a trait.

More general coalescent interpretation of relatedness

Quasi equilibrium
Quasi linkage equilibrium
D(t + 1) = (1 − r)D(t) + O(δ) ⇒ ˆD =
O(s)
r
understood as
O(s)
r
=
O(s)
1 − (1 − r)
= (1 − r)O(s) + (1 − r)2
O(s) + (1 − r)3
O(s) . . .
Quasi equilibrium
D(t + 1) − D◦
= λ(D(t) − D◦
) + O(δ) ⇒ D(t + 1) − D◦ =
O(s)
1 − λ
Actually
D(t + 1) − D◦
= A(D(t) − D◦
) + O(δ) ⇒ D − D◦ = (I − A)−1
O(s)

François Rousset - présentation MEE2013

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