Diffusion approximations for matrix games in group-structured populations

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Diffusion approximations for matrix games in group-structured populations

Sabin Lessard Universit´e de Montr´eal

1 Introduction Biology and Game Theory, Paris, November 4-6, 2010

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Fundamental Theorem of Natural Selection

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.(Fisher 1930, p.35)

I Increase of mean fitness under weak selection (Nagylaki 1976, 1977, 1987, 1989, 1993)

I Partial change in mean fitness

(Price 1972, Ewens 1989, Lessard 1997)

I Stochastic effects

(Wright 1931, Mal´ecot 1948, Kimura 1964)

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Group selection

The average adaptiveness of the species thus advances under intergroup selection, an enormously more effective process than intragroup selection.(Wright 1932)

By all odds the most important cases of interdeme selection are those in which the character that increases the probability of survival of the deme as a unit is itself being selectedagainst within the population. (Lewontin 1965)

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Kin selection

Species ... should tend to evolve behaviour such that each organism appears to be attempting to maximize its inclusive fitness.(Hamilton 1964)

˜

w_J =1+s

∑

I

ρ_J→Ia_J→I

for somecoefficient of relatednessρ_J→I

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Evolutionarily stable strategy

Roughly, an ESS is a strategy such that, if most of the members of a population adopt it, there is no ”mutant” strategy that would give higher reproductive fitness.(Maynard Smith and Price 1973)

Stable state of thereplicator dynamics(Taylor & Jonker 1978)

˙

x_k=x_k((Ax)_k−x·Ax)

for agame matrixA= [a_kl]ⁿ_k,l=1, and conversely ifAsymmetric.

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Evolutionary stability for finite populations

For finite N, we propose that B is an ESS ... if two conditions hold: (1) selection opposes A invading B,... and (2) selection opposes A replacing B.(Nowak et al. 2004)

P(fixation of singleAamongN)< 1 N

Selection favors a strategy if its abundance (average frequency) exceeds1/n ... in the stationary distribution of the

mutation-selection process. (Antal et al. 2009)

E(frequency of strategyAamongn)>1 n

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Questions

•Can we ascertaindiffusion approximationsfor matrix games in finite group-structured populations?

•What is the relationship withgame dynamicsin well-mixed populations?

•What are the roles ofrelatednessandgroup selection?

•Whatcoefficients of relatednesscome into play?

•Is aninclusive fitnessformulation possible?

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Population structure

DgroupsofNindividuals

nstrategiesS₁, ...,S_n

ordered group type(S_i,1, ...,S_i,N)

with strategy frequency vectorx_i= (x_1,i, . . . ,x_n,i) frequencyz_i(t)in generationt≥0

9 Population structure Biology and Game Theory, Paris, November 4-6, 2010

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Life cycle

Reproduction:infinite number of offspring

Migration:proportional dispersal of a fractionmof offspring

Selectionaftermigration : viabilityofS_kin a group of typei

w

_k,i

= 1 +

^(Ax_NDⁱ⁾^k

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Life cycle

Reproduction:infinite number of offspring

Migration:proportional dispersal of a fractionmof offspring

Selectionaftermigration : viabilityofS_kin a group of typei

w

_k,i

= 1 +

^(Ax_NDⁱ⁾^k

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Mutation:fromS_k toS_lwith probability

u

_kl

=

_ND^µ^kl

Sampling:group of typejfrom typeiwith probability

P

^D_ij

(z(t))

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Key Lemma

When D=∞,there isuniform convergenceof the frequency z_j(t+1) =∑iz_i(t)P_ij(z(t))

to

ˆ

z_j(x) =∑_rc_j(r)x^r₁¹·...·x^r_L^L

withc_j(r)the number of ways for a group of type j to have r_k ancestors of type S_kandx_k the overall frequency of S_k.

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Diffusion approximation

The strategy frequency processX(bNDτc)converges in lawas D→∞to a diffusion with infinitesimal covariances

v_kl(x,0) =Cx_k(δ_kl−x_l)

for somecoefficient of diffusion C,and infinitesimal means m_k(x,0) =

∑

l

µ_lkx_l−

∑

l

µ_klx_k+x_k((Ax)˜ _k−x·Ax)˜

for someeffective game matrixA˜.

14 Diffusion approximation Biology and Game Theory, Paris, November 4-6, 2010

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Proof: Two-time scale MC (Ethier and Nagylaki 1980)

E_z(∆X_k) =m_k(x,y)

ND +o(D⁻¹)

E_z((∆X_k)(∆X_l)) =v_kl(x,y)

ND +o(D⁻¹) E_z((∆X_k)⁴) =o(D⁻¹)

E_z(∆Y_j) =d_j(x,y) +o(1),

Varz(∆Y_j) =o(1)

uniformly inz, wherey=z−ˆzandd_j(x,y) =∑_iz_iP_ij(z)−z_j

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Coefficient of diffusion

C = 1 − f f =

Nm(2−m)+(1−m)^(1−m)² ²

probability that 2 offspring in the same group after dispersal are ibd(identical-by-descent) with aninfinite number of groups and therefore in theabsence of selection and mutation

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Effective game matrix (1 − f ) a

_kl

−

^ρ₂^•

a

_kl

−

^ρ₂^•

a

_lk

+ f (1 − ρ

••

)a

_kk

ρ_••=

N(1−m) +2(N−1)(1−m)³

N²m(3−3m+m²) + (3N−2)(1−m)³

probability that an offspring is ibd to 2 others in the same group after dispersalgiven that these 2 are ibd

ρ•= 2f(1−ρ••) 1−f

probability that an offspring is ibd to either of 2 others in the same group after dispersalgiven that these 2 are not ibd

17 Effective game matrix Biology and Game Theory, Paris, November 4-6, 2010

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Approximations

ρ_••≈ρ_•≈ 2f 1+f

with equality asN→∞andm→0 withNmkept constant

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Additive case a

_kl

= −c

_k

+ b

_l

Same diffusion approximation withinclusive fitnessof typeS_k

˜

w_k=1+ a˜_k ND

˜

a_k = c_k

−1+ (1−f)ρ•

2 +fρ••

+ b_k

f−(1−f)ρ_•

2 −fρ_••

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Migration after selection

•Proportionaldispersal:

ρ_•• andρ_•calculated in offspring before dispersal and multiplied by(1−m)≤1

•Uniformdispersal:

ρ_•• andρ_•calculated in offspring before dispersal and multiplied by(1−m)²≤(1−m)

20 Other assumptions Biology and Game Theory, Paris, November 4-6, 2010

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•Local extinctionand recolonization:

like proportional dispersal butC= (2−m)(1−f)>(1−f)

•Fertilityselection : complete dispersal, regulation of groups

w

_k,i

= 1 +

^(Axⁱ⁾^k⁻

akk N

ND

effective game matrix ˜A=A−^A+A_N^T !

21 Other assumptions Biology and Game Theory, Paris, November 4-6, 2010

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Concluding remarks

I Adiffusion approximationfor selection, mutation and drift in group-structured populations as the number of groups increases can be ascertained by a two-time scale argument.

I With random pairwise interactions within groups, the effect of selection on the infinitesimal means are given by the replicator equation for someeffective game matrix.

I The entries of the effective game matrix are sums of effects weighted bycoefficients of relatedness.

22 Concluding remarks Biology and Game Theory, Paris, November 4-6, 2010

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Concluding remarks

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Concluding remarks

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Concluding remarks

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I Competition within groupswhich results from population regulation is reduced bydifferential contributions of groups which result from dispersal or colonization after selection

I Aninclusive fitness formulationis possible in the case of interactions having additive individual effects.

I Future work will deal with otherpopulation structuresand migration patterns.

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Diffusion approximations for matrix games in group-structured populations

Diffusion approximations for matrix games in group-structured populations

Fundamental Theorem of Natural Selection

Group selection

Kin selection

∑

Evolutionarily stable strategy

Evolutionary stability for finite populations

Questions

Population structure

Life cycle

w

= 1 +

Life cycle

w

= 1 +

u

=

P

(z(t))

Key Lemma

Diffusion approximation

∑

∑

Coefficient of diffusion

C = 1 − f f =

Effective game matrix (1 − f ) a

−

a

−

a

+ f (1 − ρ

)a

Approximations

Additive case a

= −c

+ b

Migration after selection

w

= 1 +

Concluding remarks

Concluding remarks

Concluding remarks

Concluding remarks

Thanks