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Evolutionary Stability
Mixed strategy dynamics
,...,1 ,)( strategies mixed typesnow,
spanning strategies pure typesfar, so
NiSip
Se
n
ni
Mixed strategy dynamics
payoff, )()(
,...,1 ,)( strategies mixed typesnow,
spanning strategies pure typesfar, so
jApipu
NiSip
Se
Tij
n
ni
Mixed strategy dynamics
Sex ratio
usually near 50 percent
Sex ratio
usually near 50 percentmales XY, females XX
Sex ratio
other sex-determining mechanismsmealy bug:
Sex ratio
other sex-determining mechanismsmealy bug: if mating immediately 102:100
Sex ratio
other sex-determining mechanismsmealy bug: if mating immediately 102:100after 6 weeks 181:100
Sex ratio
other sex-determining mechanismsmealy bug: if mating immediately 102:100after 6 weeks 181:100after 10 weeks 991:100
Sex ratio
Sex ratio
m
p
m
pmpw
m
p
1
1),(
toalproportiont grandparen Gammas be toprob.
generation its within ratio-sex is and
boy) afor (prob. ratiosex sAlpha' is if
Sex ratio
21 if decreasing
21 if increasing ),(
)1(
)21(
1
1),(
toalproportiont grandparen Gammas be toprob.
generation its within ratio-sex is and
boy) afor (prob. ratiosex sAlpha' is if
m
mmpwp
mm
mmp
m
p
m
pmpw
m
p
Sex ratio
? ratio-sex invade ratio-sexcan when qp
Sex ratio
),(),(
iswhen
)1( ratio-sex ,populationin
? ratio-sex invade ratio-sexcan when
rqwrpw
qpr
qp
Sex ratio
better) do willlarger every smaller (for
ratio-sex stablerily evolutiona unique 21
),(),(
iswhen
)1( ratio-sex ,populationin
? ratio-sex invade ratio-sexcan when
pq
q
rqwrpw
qpr
qp
Mixed strategy dynamics
population of state
payoff, )()(
,...,1 ,)( strategies mixed typesnow,
spanning strategies pure typesfar, so
N
Tij
n
ni
Sx
jApipu
NiSip
Se
Mixed strategy dynamics
populationin strategy mean )()(
population of state
payoff, )()(
,...,1 ,)( strategies mixed typesnow,
spanning strategies pure typesfar, so
ipxxp
Sx
jApipu
NiSip
Se
i
N
Tij
n
ni
Mixed strategy dynamics
Mixed strategy dynamics
)]())()([())((
populationin strategy mean )()(
population of state
payoff, )()(
,...,1 ,)( strategies mixed typesnow,
spanning strategies pure typesfar, so
xApxpipxUxxUxxx
ipxxp
Sx
jApipu
NiSip
Se
Ti
Tiii
i
N
Tij
n
ni
Chicken with mixed strategy
Evolutionarily stable strategies
residentˆ invadecan minority no if ESS ˆ ppSp n
Evolutionarily stable strategies
)]ˆˆˆ)(1()ˆ()[1(
invader offrequency if
residentˆ invadecan minority no if ESS ˆ
pAppApxAppAppxxxx
x
ppSp
TTTT
n
Evolutionarily stable strategies
attractor 0 invasion no
)]ˆˆˆ)(1()ˆ()[1(
invader offrequency if
residentˆ invadecan minority no if ESS ˆ
pAppApxAppAppxxxx
x
ppSp
TTTT
n
Evolutionarily stable strategies
AppApp
ppAppAp
p
pAppApxAppAppxxxx
x
ppSp
TT
TT
TTTT
n
ˆ then equality, if (b)
and Nash) ˆ( ˆˆˆ (a)
iff ESS ˆ hence
attractor 0 invasion no
)]ˆˆˆ)(1()ˆ()[1(
invader offrequency if
residentˆ invadecan minority no if ESS ˆ
Evolutionarily stable strategies
by-close ˆfor ˆ iff ESS ˆ ppAppAppp TT
Evolutionarily stable strategies
!converselynot but
equ. replicatorfor attractor ˆ ESS ˆ
by-close ˆfor ˆ iff ESS ˆ
pp
ppAppAppp TT
Evolutionarily stable strategies
pxpxptxp
Nppcop
pp
pp
ppAppAppp TT
ˆ toclose )( with allfor ˆ))((
then))(),...,1((ˆ if i.e.
:stablestrongly ˆ ESS ˆ
equ. replicatorfor attractor ˆ ESS ˆ
by-close ˆfor ˆ iff ESS ˆ
pxpxptxp
Nppcop
pp
ˆ toclose )( with allfor ˆ))((
then))(),...,1((ˆ if i.e.
:stablestrongly ˆ ESS ˆ
Adaptive dynamics
Adaptive dynamics
0),(),(:),( iff invade?it can
),( payoff ,minority mutant
all s,homogeneou pop.resident
.)escalate.. toprob. ratio,-(sex trait some be let
xxAxyAxhW
xyAhxy
x
Rx
Adaptive dynamics
limited)-(mutation sequenceon substitutitrait
0),(),(:),( iff invade?it can
),( payoff ,minority mutatant
all s,homogeneou pop.resident
.)escalate.. toprob. ratio,-(sex trait some be let
xxAxyAxhW
xyAhxy
x
Rx
Adaptive dynamics
direction favorable towardspoints
),0(
limited)-(mutation sequenceon substitutitrait
0),(),(:),( iff invade?it can
),( payoff ,minority mutatant
all s,homogeneou pop.resident
.)escalate.. toprob. ratio,-(sex trait some be let
xh
Wx
xxAxyAxhW
xyAhxy
x
Rx
Adaptive dynamics
),0( x
h
Wx
Adaptive dynamics
0)ˆ,( if stablerily evolutionalocally ˆ
),0(
xhWx
xh
Wx
Adaptive dynamics
)ˆ(),( if stable-econvergenc ˆ
0)ˆ,( if stablerily evolutionalocally ˆ
),0(
xxhxhWx
xhWx
xh
Wx
Adaptive dynamics
ESS toeconvergenc ... ratio-sex Chicken,for
)ˆ(),( if stable-econvergenc ˆ
0)ˆ,( if stablerily evolutionalocally ˆ
),0(
xxhxhWx
xhWx
xh
Wx
Adaptive dynamics
le!unattainab becan ESSan but
ESS toeconvergenc ... ratio-sex Chicken,for
)ˆ(),( if stable-econvergenc ˆ
0)ˆ,( if stablerily evolutionalocally ˆ
),0(
xxhxhWx
xhWx
xh
Wx
Adaptive dynamics
points branching
le!unattainab becan ESSan but
ESS toeconvergenc ... ratio-sex Chicken,for
)ˆ(),( if stable-econvergenc ˆ
0)ˆ,( if stablerily evolutionalocally ˆ
),0(
xxhxhWx
xhWx
xh
Wx
Population Genetics
Mendelian population
diploid-haploid life cycle
Population Genetics
nji
NN
SijpAA
xxAA
)( uses ),( genotype individual
,..., sfrequencie ,..., alleles
populationMendelian
11
Population Genetics
ji
i
ji
nji
NN
xijppA
xxijpxpp
SijpAA
xxAA
)( strategy uses allele
)()(mean population
)( uses ),( genotype individual
,..., sfrequencie ,..., alleles
populationMendelian
11
Population Genetics
])[(
dynamics replicator
)( strategy uses allele
)()(mean population
)( uses ),( genotype individual
,..., sfrequencie ,..., alleles
populationMendelian
11
Apppxx
xijppA
xxijpxpp
SijpAA
xxAA
Tiii
ji
i
ji
nji
NN
Population Genetics
)ˆ( if feasible ˆ
}ˆ)(:{)ˆ(
pointrest point rest
))(()(
pSp
pxpSxpS
px
txptx
n
Population Genetics
ptxp
UtxVxVxU
pSx
p
n
ˆ))(( and
)( s.t. ˆ of nbhd if
and )ˆ(ˆpoint rest i.e.
:stablelly strategica then feasible, and ESS ˆ if
:3
Linkage and Cross-over
Linkage and Cross-over
With recombination, fitness need not increase
Population genetics
frequency-dependent selection evolutionary game dynamics short-term vs. long-term evolution adaptive dynamics approximations vs. simulations
