françois rousset - présentation mee2013
TRANSCRIPT
Fitness for the impatient
Francois Rousset
May 2013
Francois Rousset Fitness for the impatient May 2013 1 / 33
The message
To understand the forces operating in social evolution, particularly inspatially structured populations:
Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods
Francois Rousset Fitness for the impatient May 2013 2 / 33
The message
To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersal
Stability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods
Francois Rousset Fitness for the impatient May 2013 2 / 33
The message
To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stability
A fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods
Francois Rousset Fitness for the impatient May 2013 2 / 33
The message
To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocks
Draw connections to other methods such as diffusion theory, multilocusmethods
Francois Rousset Fitness for the impatient May 2013 2 / 33
The message
To understand the forces operating in social evolution, particularly inspatially structured populations:Relatedness concepts under localized dispersalStability of kin recognition polymorphismsRelationship between inclusive fitness and evolutionary stabilityA fundamental impatience: reduce these problems to trivial building blocksDraw connections to other methods such as diffusion theory, multilocusmethods
Francois Rousset Fitness for the impatient May 2013 2 / 33
Continuous evolutionary stability for the very impatient
Two alleles, a and A, inducing phenotypes za and zA
Francois Rousset Fitness for the impatient May 2013 3 / 33
Continuous evolutionary stability for the very impatient
Two alleles, a and A, inducing phenotypes za and zA
Fitn
ess
ofA
alle
le
Phenotypes
za, zA1
HaL
z1 zm z2
Francois Rousset Fitness for the impatient May 2013 3 / 33
Continuous evolutionary stability for the very impatient
Two alleles, a and A, inducing phenotypes za and zA
Fitn
ess
ofA
alle
le
Phenotypes
za, zA1
HaL
z1 zm z2
Francois Rousset Fitness for the impatient May 2013 3 / 33
Continuous evolutionary stability for the very impatient
Classification of different cases:
Continuously stablestrategy HCSSL
Branching point
Evolutionarily stable= noninvasible
Invasible
Unattainable
Convergence stable= attainable
Francois Rousset Fitness for the impatient May 2013 4 / 33
Eshel 1983,1996; Christiansen, 1991; Abrams et al., 1993
Implicit assumptions
Resident phenotype z , mutant z + δ
∆p ∼ δstuff + δ2blob (1)
assumed to be essentially of the form
∆p ∼ δp(1− p)S(z) + δ2p(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?
Francois Rousset Fitness for the impatient May 2013 5 / 33
Implicit assumptions
Resident phenotype z , mutant z + δ
∆p ∼ δstuff + δ2blob (1)
assumed to be essentially of the form
∆p ∼ δp(1− p)S(z) + δ2p(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?
Francois Rousset Fitness for the impatient May 2013 5 / 33
A study of frequency (p) dependence
The first-order termA conceptual deviceFitnessThe minimal algorithmTwo views of the Prisoner’s dilemmaMany views of inclusive fitnessThe more general logic: dominance, kin recognitionKin recognition
Glimpses of second-order resultsContinuous evolutionary stabilityKin recognition
Francois Rousset Fitness for the impatient May 2013 6 / 33
A conceptual device
p′ =∑
parents i
AiXi
p′ =∑
gene copies g
AgXg
X indicator variable for an allele;A frequency of copies of parental genesConsider the function f giving (conditional) probabilities
E[p′| . . .] =∑i
ai (. . .)Xi
where ai is the probability that a descendant copy originates from parentalcopy i .
Francois Rousset Fitness for the impatient May 2013 7 / 33
A conceptual device
p′ =∑
gene copies g
AgXg
X indicator variable for an allele;A frequency of copies of parental genesConsider the function f giving (conditional) probabilities
E[p′| . . .] =∑i
ai (. . .)Xi
where ai is the probability that a descendant copy originates from parentalcopy i .
Francois Rousset Fitness for the impatient May 2013 7 / 33
Trivial example
Individuals share a resource in total amount R, their fecundity being equalto the amount of resource they consumeTheir share of resource is proportional to the value of some trait z :
sharei =zi∑i zi
= ai , fecundityi = Rzi∑i zi
Conflict between individual and group
fecundityi = (R − z)zi∑i zi
but ai is still zi∑i zi
.
Francois Rousset Fitness for the impatient May 2013 8 / 33
Trivial example
Individuals share a resource in total amount R, their fecundity being equalto the amount of resource they consumeTheir share of resource is proportional to the value of some trait z :
sharei =zi∑i zi
= ai , fecundityi = Rzi∑i zi
Conflict between individual and group
fecundityi = (R − z)zi∑i zi
but ai is still zi∑i zi
.
Francois Rousset Fitness for the impatient May 2013 8 / 33
Fitness
In terms of the number of adult offspring, or “fitness” W
p′ =
∑gene copies g XgWg∑
g Wg= mean(XgWg )
Define fitness functions so that
E[p′|p] =1
Ntot
∑g
Xgw(zg (p))
In a demographically stable population Ntota = w (e.g., w = ziz ).
A (minimal: W = 1) Price equation
p′ = X ′ =W X + Cov(wg ,Xg ),
E[∆p|z] = Cov(wg (z),Xg ).
Francois Rousset Fitness for the impatient May 2013 9 / 33
Minimal algorithm
Express fitness in terms of indicator variablese.g., phenotype zi = z + XiδDifferentiate with respect to some measure of strength of selectionTo first order in δ,
wf(zf, zp) ∼ linear combination of (X ,X )
Collect products of indicator variables
p′ ∼ mean(wiH i ) =
linear combination of means of (H2,HH) =
p + δ∂w
∂zfmean(H2 − HH)
Take expectations E.g., a large population without (spatial) structure
p′ = p + δ∂w
∂zf(p − p2)
Francois Rousset Fitness for the impatient May 2013 10 / 33
Two views of the Prisoner’s dilemma
(“dilemma”: T > R, P > S , R > P i.e. T > R > P > S)
Francois Rousset Fitness for the impatient May 2013 11 / 33
Two views of the Prisoner’s dilemma
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 + Xiδ
∆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.
Francois Rousset Fitness for the impatient May 2013 12 / 33
Population structure
Color code: focal, neighbor, population
wf ∼ linear combination of (H,H,H)
so that
mean(H)t+1 ∼ mean(wiH i ) =
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.
Francois Rousset Fitness for the impatient May 2013 13 / 33
Genealogical interpretation: island model
past
. . . . . .
p p2
Lineages from distinct demes can be considered as draws of independentgenes copies, each A with frequency p.Probability that first event is coalescence: FST
Such coalescences are recent if migration“not too small”; p then consideredconstant if selection is “weak” and total population size is “large”.
Francois Rousset Fitness for the impatient May 2013 14 / 33
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.
Francois Rousset Fitness for the impatient May 2013 15 / 33
Genealogical interpretation: “stepping stone”
past
. . . . . .
Lineages from distinct demes cannot be considered as draws ofindependent genes copies, each A with frequency p
Francois Rousset Fitness for the impatient May 2013 16 / 33
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)
Francois Rousset Fitness for the impatient May 2013 17 / 33
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)
Francois Rousset Fitness for the impatient May 2013 17 / 33
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 gradientφ: p-independent localized selection gradient
Francois Rousset Fitness for the impatient May 2013 18 / 33
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 gradientφ: p-independent localized selection gradient
Francois Rousset Fitness for the impatient May 2013 18 / 33
What does that mean ?
Francois Rousset Fitness for the impatient May 2013 19 / 33
The localized selection gradient φ and local FST’s
The finite population meaning and computation of φ
φ ≡ ∂π
∂δ= −
∑k 6=f
∂w
∂zk
Tk
T0
Tk
T0=
(limµ→0
E[p − XX k ]
E[p − XX 0]
)=
(limµ→0
1
1− FSTk
)
Localized interactions: short distances (k) only. Everythingunderstandable as the result of local interactions (global p doesn’tmatter!).
Expressed in terms of local population structure parameters relativelyeasy to estimate using genetic markers, and with genealogicalinterpretation(s)
Genealogical: FSTk(p) = FSTk at neutral genetic markers
Francois Rousset Fitness for the impatient May 2013 20 / 33
Fitness costs and benefits
Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − 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?Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).
Francois Rousset Fitness for the impatient May 2013 21 / 33
Fitness costs and benefits
Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − 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?Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).
Francois Rousset Fitness for the impatient May 2013 21 / 33
Fitness costs and benefits
Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − 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?Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).
Francois Rousset Fitness for the impatient May 2013 21 / 33
Fitness costs and benefits
Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − 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?
Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).
Francois Rousset Fitness for the impatient May 2013 21 / 33
Fitness costs and benefits
Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − 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 eachgeneration”“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 selectedmembers (...) is F”⇒ ∆p ∼ (RB − C )pq
Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).
Francois Rousset Fitness for the impatient May 2013 21 / 33
Fitness costs and benefits
Effects B on neighbors’ fecundity and −C on focal’s fecundityrb − 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?Version with spatially restricted dispersalIndividuals disperse as a group and compete as a group against othergroups for access to whole group breeding spots. The winners of suchgroup contests can then occupy whole demes⇒ ∆p ∼ (1− F )(RB − C )pq(Gardner & West 2006; Lehmann et al. 2006).
Francois Rousset Fitness for the impatient May 2013 21 / 33
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 (X 1,X 2,X 1X2, ...)
so that
E[p′] = mean(wiX i ) =
linear combination of means of (X 21,X 1X 2,X 1X ,X 1X 1X 2, . . . ,X 1X1X2) =
Triplets generally lead to frequency-dependence, although there areintriguing exceptions:Helping among diploid full sibs and among haplodiploid sisters, partial sibmating (α)
∆p ∼ δpq(∂w
∂zf+∂w
∂zn
r
)f (h, α, /p)
Francois Rousset Fitness for the impatient May 2013 22 / 33
Three gene lineages... island model
past
. . . . . .
p p2 p3
Lineages from distinct demes can be considered as draws of independentgenes copies, each A with frequency p.Probability that first event is coalescence: FST
Francois Rousset Fitness for the impatient May 2013 23 / 33
More general logic: example of kin recognition
Demes of N individuals; two loci; indicator variables R,H for allelesConditional 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
fffothers
(dispersal probability d ; regulation after dispersal)
mean(H)t+1 =mean(wiHi )
mean(wi )= mean(wiHi ) over individuals i .
(strict regulation)
Francois Rousset Fitness for the impatient May 2013 24 / 33
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(wiH i ) =
linear combination of expectations of (H2,HH,RH2,RH2,RRH2,
RHH,RRHH,HH,RHH,RRHH,RHH)
RH RH R RH H
Francois Rousset Fitness for the impatient May 2013 25 / 33
Meaning
An act of helping always involves the configurationHR R−C B
Francois Rousset Fitness for the impatient May 2013 26 / 33
Meaning
An act of helping always involves the configurationHR 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 thedeme
Francois Rousset Fitness for the impatient May 2013 26 / 33
Meaning
An act of helping always involves the configurationHR 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, inproportion to the increase in fitness of the pair of interacting individuals;can be computed in terms of the probability of joint coalescence within thedeme
Francois Rousset Fitness for the impatient May 2013 26 / 33
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.
Francois Rousset Fitness for the impatient May 2013 27 / 33
Synthesis and developments
Many results follow mechanically from a trivial description ofallele-frequency changes in terms of fitness functions:
* Write a properly defined fitness function in terms of individualbehaviour
* Express behaviour in term of genotypes (indicator variables)
* Expand wiXi to appropriate order, and take expectations of productsof indicator variables (special case: “direct fitness” method, Taylor &Frank 1996)
Deterministic “multi”locus models, with a recombination step:
Diffusion methods (first order)
Diffusion with p-dependence (in first order, e.g. dominance)
Evolutionary stability
Francois Rousset Fitness for the impatient May 2013 28 / 33
Synthesis and developments
Many results follow mechanically from a trivial description ofallele-frequency changes in terms of fitness functions:
* Write a properly defined fitness function in terms of individualbehaviour
* Express behaviour in term of genotypes (indicator variables)
* Expand wiXi to appropriate order, and take expectations of productsof indicator variables (special case: “direct fitness” method, Taylor &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 0in expectation in neutral models, e.g. E[(R − pR)(H − pH)].
Diffusion methods (first order)
Diffusion with p-dependence (in first order, e.g. dominance)
Evolutionary stability
Francois Rousset Fitness for the impatient May 2013 28 / 33
Synthesis and developments
Many results follow mechanically from a trivial description ofallele-frequency changes in terms of fitness functions:
* Write a properly defined fitness function in terms of individualbehaviour
* Express behaviour in term of genotypes (indicator variables)* Expand wiXi to appropriate order, and take expectations of products
of indicator variables (special case: “direct fitness” method, Taylor &Frank 1996)
Deterministic “multi”locus models, with a recombination step:Diffusion methods (first order) e.g. approximation for fixationprobability
π ∼ 1− e−2Ntotφpini
1− e−2Ntotφ
for φ taken in an infinite-deme limit.
Diffusion with p-dependence (in first order, e.g. dominance)Evolutionary stability
Francois Rousset Fitness for the impatient May 2013 28 / 33
Synthesis and developments
Many results follow mechanically from a trivial description ofallele-frequency changes in terms of fitness functions:
* Write a properly defined fitness function in terms of individualbehaviour
* Express behaviour in term of genotypes (indicator variables)
* Expand wiXi to appropriate order, and take expectations of productsof indicator variables (special case: “direct fitness” method, Taylor &Frank 1996)
Deterministic “multi”locus models, with a recombination step:
Diffusion methods (first order)
Diffusion with p-dependence (in first order, e.g. dominance)
Evolutionary stability
Francois Rousset Fitness for the impatient May 2013 28 / 33
Synthesis and developments
Many results follow mechanically from a trivial description ofallele-frequency changes in terms of fitness functions:
* Write a properly defined fitness function in terms of individualbehaviour
* Express behaviour in term of genotypes (indicator variables)
* Expand wiXi to appropriate order, and take expectations of productsof indicator variables (special case: “direct fitness” method, Taylor &Frank 1996)
Deterministic “multi”locus models, with a recombination step:
Diffusion methods (first order)
Diffusion with p-dependence (in first order, e.g. dominance)
Evolutionary stability
Francois Rousset Fitness for the impatient May 2013 28 / 33
Continuous evolutionary stability in the island model
Second-order computation for any p feasible.Simpler computation for extreme p, and connexion to “number ofsuccessful emigrants” approach (Chesson 1984; Metz and Gyllenberg 2001)
B = ∂zf,zfw +2F∂zf,znw +K∂zn,zn
w +4F (N−1)(K∂zn
wp + F∂fwp
)wp∂zn
w
where wp: local offspringIn contrast to formula proposed by Day and Taylor (1998):(1) three-genes coefficient K ;(2) products of identity coefficients, product of derivatives: joint effect offirst-order change in number of offspring and first-order change in parentalpopulation structure.
Francois Rousset Fitness for the impatient May 2013 29 / 33
Kin recognition: searching for stable polymorphisms
Second-order computation involves 30 associations, for up to 7 gene copiesin 5 individuals: R‖RH/R‖RH/RSearch for conditions for stable polymorphism: low migration and lowrecombinationConvergent orbits plus drift:
Francois Rousset Fitness for the impatient May 2013 30 / 33
“Conclusions”
First order: Simple formalism provides connections between differentmodelling approaches (inclusive fitness, adaptive dynamics, diffusion,coalescence). Relatively simple pattern of frequency-dependence.
Second order: Essentially the same formalism previously used formultilocus models; Algebraically (and algorithmically) messy inspatially structured populations; still allows an analysis of the forcesacting on a trait.
Francois Rousset Fitness for the impatient May 2013 31 / 33
More general coalescent interpretation of relatedness
Francois Rousset Fitness for the impatient May 2013 32 / 33
Quasi equilibrium
Quasi linkage equilibrium
D(t + 1) = (1− r)D(t) + O(δ)⇒ D =O(s)
r
understood as
O(s)
r=
O(s)
1− (1− r)= (1− r)O(s) + (1− r)2O(s) + (1− r)3O(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)−1O(s)
Francois Rousset Fitness for the impatient May 2013 33 / 33