Evolutionary dynamics of phenotypically structured...
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Evolutionary dynamics of phenotypically structuredpopulations in time-varying environments
Sepideh Mirrahimi
CNRS, IMT, Toulouse, France
Summer school on mathematical biology,Shanghai, July 2019
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
The influence of fluctuating temperature on bacteriaBacterial pathogen Serratia marcesens evolved in fluctuatingtemperature (daily variation between 24◦C and 38◦C, mean 31◦C),outperforms the strain that evolved in constant environments(31◦C):
Figure from: Fluctuation temperature leads to evolution of thermal generalism and preadaptation to
novel environments, Ketola et al. 20132 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
A typical selection-mutation model with time varyingenvironment
∂∂t nε − ε
2∆nε = nε(R(e(t), x)− κρε
),
nε(t = 0, ·) = nε,0(·),
ρε(t) =∫Rd nε(t, x) dx .
x ∈ Rd : phenotypical traitnε(t, x): density of trait xe(t): an environmental stateR(e, x): growth rate
ρε(t): size of the populationκ: intensity of thecompetitionε2: mutation effective size
Objective: to describe the dynamics of nε.In which situations a population evolved in a periodic environmentoutperforms a population evolved in a constant environment?
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
A typical selection-mutation model with time varyingenvironment
∂∂t nε − ε
2∆nε = nε(R(e(t), x)− κρε
),
nε(t = 0, ·) = nε,0(·),
ρε(t) =∫Rd nε(t, x) dx .
x ∈ Rd : phenotypical traitnε(t, x): density of trait xe(t): an environmental stateR(e, x): growth rate
ρε(t): size of the populationκ: intensity of thecompetitionε2: mutation effective size
Objective: to describe the dynamics of nε.In which situations a population evolved in a periodic environmentoutperforms a population evolved in a constant environment?3 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
Some references
The work presented in this part is based on:Figueroa Iglesias–M. (2018).
Some closely related works considering time-varyingenvironments:
Lorenzi–Chisholm–Desvillettes–Hughes (2015)
M.–Perthame–Souganidis (2015)
Almeida–Bagnerini–Fabrini–Hughes–Lorenzi (2019)
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
Assumptions
Notation:
R(x) =1T
∫ T
0R(e(t), x)dt.
Assumptions:e is T -periodic with respect to t.
R is bounded and takes negative values for large |x |.
There exists a unique xm ∈ Rd such that
maxx∈Rd
R(x) = R(xm) > 0.
ε is small.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
Assumptions
Notation:
R(x) =1T
∫ T
0R(e(t), x)dt.
Assumptions:e is T -periodic with respect to t.
R is bounded and takes negative values for large |x |.
There exists a unique xm ∈ Rd such that
maxx∈Rd
R(x) = R(xm) > 0.
ε is small.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Introduction
Table of contents
1 Introduction
2 Main results
3 Case studies
4 Adaptation in a shifting and fluctuating environment
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Table of contents
1 Introduction
2 Main results
3 Case studies
4 Adaptation in a shifting and fluctuating environment
5 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Long time behavior of the population’s phenotypicaldistribution
Proposition (Figueroa Iglesias and M., 2018 )
As t →∞, nε(t, x) converges to the unique periodic solution of{∂∂t nε,p − ε
2∆nε,p = nε,p(R( t, x)− κρε,p
),
ρε,p(t) =∫Rd nε,p(t, x)dx , nε,p(0, ·) = nε,p(T , ·).
Our objective is to describe such periodic solution for ε small.The Hopf-Cole transformation:
nε,p(t, x) =1
(2πε)d2
exp(uε,p(t, x)
ε
).
An expected asymptotic expansion:
uε,p(t, x) = u + εv + O(ε2).
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Long time behavior of the population’s phenotypicaldistribution
Proposition (Figueroa Iglesias and M., 2018 )
As t →∞, nε(t, x) converges to the unique periodic solution of{∂∂t nε,p − ε
2∆nε,p = nε,p(R( t, x)− κρε,p
),
ρε,p(t) =∫Rd nε,p(t, x)dx , nε,p(0, ·) = nε,p(T , ·).
Our objective is to describe such periodic solution for ε small.
The Hopf-Cole transformation:
nε,p(t, x) =1
(2πε)d2
exp(uε,p(t, x)
ε
).
An expected asymptotic expansion:
uε,p(t, x) = u + εv + O(ε2).
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Long time behavior of the population’s phenotypicaldistribution
Proposition (Figueroa Iglesias and M., 2018 )
As t →∞, nε(t, x) converges to the unique periodic solution of{∂∂t nε,p − ε
2∆nε,p = nε,p(R( t, x)− κρε,p
),
ρε,p(t) =∫Rd nε,p(t, x)dx , nε,p(0, ·) = nε,p(T , ·).
Our objective is to describe such periodic solution for ε small.The Hopf-Cole transformation:
nε,p(t, x) =1
(2πε)d2
exp(uε,p(t, x)
ε
).
An expected asymptotic expansion:
uε,p(t, x) = u + εv + O(ε2).6 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
The main resultTheorem (Figueroa Iglesias and M., 2018 )
(i) As ε→ 0, nε,p(t, x)−−⇀ρp(t) δ(x − xm),
with ρp the unique T -periodic solution to
dρpdt
(t) = ρp(t) (R(t, xm)− κρp(t)).
(i) As ε→ 0, uε,p converges locally uniformly to the uniqueviscosity solution of{
−|∇u|2(x) = R(x)− κρp,maxx∈Rd u(x) = 0, ρp = 1
T
∫ T0 ρp(t)dt.
For x ∈ R, this solution is indeed smooth and classical and can becomputed explicitly.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
The main resultTheorem (Figueroa Iglesias and M., 2018 )
(i) As ε→ 0, nε,p(t, x)−−⇀ρp(t) δ(x − xm),
with ρp the unique T -periodic solution to
dρpdt
(t) = ρp(t) (R(t, xm)− κρp(t)).
(i) As ε→ 0, uε,p converges locally uniformly to the uniqueviscosity solution of{
−|∇u|2(x) = R(x)− κρp,maxx∈Rd u(x) = 0, ρp = 1
T
∫ T0 ρp(t)dt.
For x ∈ R, this solution is indeed smooth and classical and can becomputed explicitly.7 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Some heuristics: an asymptotic expansion
Replacing the Hopf-Cole transformation in the equation on nε,p weobtain
1ε∂tuε,p − ε∆uε,p = |∇uε,p|2 + R(t, x)− κρε,p(t).
Asymptotic expansions:
uε,p(t, x) = u(t, x) + εv(t, x) + ε2w(t, x) + o(ε2),
ρε,p(t) = ρ(t) + εK (t) + o(ε),
with time periodic coefficients.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Some heuristics: derivation of the Hamilton-Jacobi equation
ε−1 order terms:∂tu(t, x) = 0
⇒ u(x , t) = u(x).
ε0 order term:
∂tv(t, x) = |∇u|2 + R(t, x)− κρ(t).
Integrating in t ∈ [0,T ], leads to
−|∇u|2 =1T
∫ T
0(R(t, x)− κρ(t))dt.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Some heuristics: derivation of the Hamilton-Jacobi equation
ε−1 order terms:∂tu(t, x) = 0
⇒ u(x , t) = u(x).
ε0 order term:
∂tv(t, x) = |∇u|2 + R(t, x)− κρ(t).
Integrating in t ∈ [0,T ], leads to
−|∇u|2 =1T
∫ T
0(R(t, x)− κρ(t))dt.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Some heuristics: the equations on the corrector
ε order terms:
∂tw −∆u = 2∇u · ∇v − κK (t),
and again integrating in [0,T ] we find
−∆u =2T∇u∫ T
0∇vdt − κK , with K =
1T
∫ T
0K (t)dt.
Evaluating the above equation at xm we obtain that
∆u(xm) = κK .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
Some heuristics: the equations on the corrector
ε order terms:
∂tw −∆u = 2∇u · ∇v − κK (t),
and again integrating in [0,T ] we find
−∆u =2T∇u∫ T
0∇vdt − κK , with K =
1T
∫ T
0K (t)dt.
Evaluating the above equation at xm we obtain that
∆u(xm) = κK .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Main results
The corrector term and the approximation of the momentsof the phenotypic distribution
Combining the above equations:∂tv(t, x) = R(t, x)− R(x)− κρ(t) + κρ,
−∆u =2T
∫ T
0∇u · ∇vdt − κK .
We will use these formal expansions to provide analyticapproximations for the moments of the population’s distributionusing the Laplace’s method of integration, in terms of derivatives ofu and v at xm.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Table of contents
1 Introduction
2 Main results
3 Case studies
4 Adaptation in a shifting and fluctuating environment
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Some notationsAverage size of the population over a period of time:
ρε,p =1T
∫ T
0ρε,p(t)dt
Mean phenotypic trait of the population:
µε,p(t) =1
ρε,p(t)
∫Rd
x nε,p(t, x)dx
Phenotypic variance of the population’s distribution:
vε,p(t) =1
ρε,p(t)
∫Rd
(x − µε,p(t))2nε,p(t, x)dx
Mean fitness of the population in an environment with constantstate e0:
Fε,p(e0) =
∫Rd
R(e0, x)1T
∫ T
0
nε,p(t, x)
ρε,p(t)dtdx
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Some notationsAverage size of the population over a period of time:
ρε,p =1T
∫ T
0ρε,p(t)dt
Mean phenotypic trait of the population:
µε,p(t) =1
ρε,p(t)
∫Rd
x nε,p(t, x)dx
Phenotypic variance of the population’s distribution:
vε,p(t) =1
ρε,p(t)
∫Rd
(x − µε,p(t))2nε,p(t, x)dx
Mean fitness of the population in an environment with constantstate e0:
Fε,p(e0) =
∫Rd
R(e0, x)1T
∫ T
0
nε,p(t, x)
ρε,p(t)dtdx
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Some notationsAverage size of the population over a period of time:
ρε,p =1T
∫ T
0ρε,p(t)dt
Mean phenotypic trait of the population:
µε,p(t) =1
ρε,p(t)
∫Rd
x nε,p(t, x)dx
Phenotypic variance of the population’s distribution:
vε,p(t) =1
ρε,p(t)
∫Rd
(x − µε,p(t))2nε,p(t, x)dx
Mean fitness of the population in an environment with constantstate e0:
Fε,p(e0) =
∫Rd
R(e0, x)1T
∫ T
0
nε,p(t, x)
ρε,p(t)dtdx
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
An example with a constant environment
R(e, x) = R(x) = r − g(x − θ)2.
r : maximal growth rate, g : selection pressure, θ: optimal trait.
An explicitly steady solution:
nε,c = ρε,cg1/4√2πε
exp(−√g(x − θ)2
2ε).
The size of the population, the mean phenotypic trait and thevariance:
ρε,c = r − ε√g , µε,c = θ, vε,c =ε√g,
and the mean fitness in such constant environment:
Fε,c = r − ε√g .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
An example with a constant environment
R(e, x) = R(x) = r − g(x − θ)2.
r : maximal growth rate, g : selection pressure, θ: optimal trait.
An explicitly steady solution:
nε,c = ρε,cg1/4√2πε
exp(−√g(x − θ)2
2ε).
The size of the population, the mean phenotypic trait and thevariance:
ρε,c = r − ε√g , µε,c = θ, vε,c =ε√g,
and the mean fitness in such constant environment:
Fε,c = r − ε√g .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
An example with a constant environment
R(e, x) = R(x) = r − g(x − θ)2.
r : maximal growth rate, g : selection pressure, θ: optimal trait.
An explicitly steady solution:
nε,c = ρε,cg1/4√2πε
exp(−√g(x − θ)2
2ε).
The size of the population, the mean phenotypic trait and thevariance:
ρε,c = r − ε√g , µε,c = θ, vε,c =ε√g,
and the mean fitness in such constant environment:
Fε,c = r − ε√g .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
An example with a constant environment
R(e, x) = R(x) = r − g(x − θ)2.
r : maximal growth rate, g : selection pressure, θ: optimal trait.
An explicitly steady solution:
nε,c = ρε,cg1/4√2πε
exp(−√g(x − θ)2
2ε).
The size of the population, the mean phenotypic trait and thevariance:
ρε,c = r − ε√g , µε,c = θ, vε,c =ε√g,
and the mean fitness in such constant environment:
Fε,c = r − ε√g .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 1: when the fluctuations act on the optimal trait
R(e, x) = r − g(x − θ(e))2, θ(e) = e, e(t) = c sin(2πbt).
The average size of the population, the mean phenotypic trait andthe mean variance:
ρε,p ≈ r − gc2
2− ε√g , µε,p(t) ≈
εcb√g
πsin
(2πb
(t − b
4)
),
vε,p(t) ≈ ε√g.
Comparison between the mean fitness of this population in anenvironment with constant state (e0 = 0) and the mean fitness of apopulation evolved in the constant environment e0 = 0:
Fε,c(e0) ≈ Fε,p(e0) ≈ r − ε√g .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 1: when the fluctuations act on the optimal trait
R(e, x) = r − g(x − θ(e))2, θ(e) = e, e(t) = c sin(2πbt).
The average size of the population, the mean phenotypic trait andthe mean variance:
ρε,p ≈ r − gc2
2− ε√g , µε,p(t) ≈
εcb√g
πsin
(2πb
(t − b
4)
),
vε,p(t) ≈ ε√g.
Comparison between the mean fitness of this population in anenvironment with constant state (e0 = 0) and the mean fitness of apopulation evolved in the constant environment e0 = 0:
Fε,c(e0) ≈ Fε,p(e0) ≈ r − ε√g .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 1: when the fluctuations act on the optimal trait
R(e, x) = r − g(x − θ(e))2, θ(e) = e, e(t) = c sin(2πbt).
The average size of the population, the mean phenotypic trait andthe mean variance:
ρε,p ≈ r − gc2
2− ε√g , µε,p(t) ≈
εcb√g
πsin
(2πb
(t − b
4)
),
vε,p(t) ≈ ε√g.
Comparison between the mean fitness of this population in anenvironment with constant state (e0 = 0) and the mean fitness of apopulation evolved in the constant environment e0 = 0:
Fε,c(e0) ≈ Fε,p(e0) ≈ r − ε√g .14 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 1: numerical computations for ε small
R(e, x) = r − g(x − θ(e))2, θ(e) = e, e(t) = c sin(2πbt).
r = 2, c = g = 1, b = 2π, ε = 0.01.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (t)
-4
-2
0
2
4
6
8
10
Mo
me
nts
of
the
dis
trib
utio
n o
f th
e p
op
ula
tio
n
10-3
approx
2
2approx
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
Optimal Trait
Left: comparison between the analytical and the numericalapproximations of the moments of the population’s density.Right: comparison between the mean phenotypic trait (numerical andanalytical approximations) and the (rescaled) optimal trait. Delay= 0.24.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 2: when the fluctuations act on the pressure of theselection
R(t, x) = r − g(e(t))x2, g =
∫ 1
0g(e(s)
)ds.
with e 1-periodic and g a positive function.
The size of the population, the mean phenotypic trait and themean variance:
ρε,p ≈ r − ε√g , µε,p(t) ≈ 0, vε,p(t) ≈ ε√
g.
and the mean fitness in an environment with constant state e0
Fε,c(e0) ≈ r − ε√
g(e0)< Fε,p(e0) ≈ r − ε
g(e0)
√g, if g > g(e0).
⇒ the population evolved in a periodic environment mayoutperform the population evolved in a constant environment.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 2: when the fluctuations act on the pressure of theselection
R(t, x) = r − g(e(t))x2, g =
∫ 1
0g(e(s)
)ds.
with e 1-periodic and g a positive function.The size of the population, the mean phenotypic trait and themean variance:
ρε,p ≈ r − ε√g , µε,p(t) ≈ 0, vε,p(t) ≈ ε√
g.
and the mean fitness in an environment with constant state e0
Fε,c(e0) ≈ r − ε√g(e0)< Fε,p(e0) ≈ r − ε
g(e0)
√g, if g > g(e0).
⇒ the population evolved in a periodic environment mayoutperform the population evolved in a constant environment.
16 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 2: when the fluctuations act on the pressure of theselection
R(t, x) = r − g(e(t))x2, g =
∫ 1
0g(e(s)
)ds.
with e 1-periodic and g a positive function.The size of the population, the mean phenotypic trait and themean variance:
ρε,p ≈ r − ε√g , µε,p(t) ≈ 0, vε,p(t) ≈ ε√
g.
and the mean fitness in an environment with constant state e0
Fε,c(e0) ≈ r − ε√g(e0)< Fε,p(e0) ≈ r − ε
g(e0)
√g, if g > g(e0).
⇒ the population evolved in a periodic environment mayoutperform the population evolved in a constant environment.
16 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 2: numerical computations for ε small
R(t, x) = r − g(e(t))x2, g(e) = e, e(t) = 1.5 + cos(2πt),
r = 2, ε = 0.01.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (t)
0
1
2
3
4
5
6
7
8
9
Mo
me
nts
of
the
dis
trib
utio
n o
f th
e p
op
ula
tio
n
10-3
approx
2
2approx
Comparison between the analytical and the numerical approximations ofthe mean phenotypic trait nad the variance of the phenotypicdistribution.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
What if the mutations have larger effects?
In the above examples, semi-explicit solutions can be indeedcomputed (Almeida et al, 2019) for arbitrary ε.
A study of such semi-explicit solutions confirms that, when themutations have larger effect:
• the population follows better the environment
• the fluctuations acting on the pressure of the selection can leadto a population with better performance.
18 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 1: numerical computations for ε large
R(e, x) = r − g(x − θ(e))2, θ(e) = e, e(t) = c sin(2πbt).
r = 2, c = g = 1, b = 2π, ε = 1.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (t)
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Mo
me
nts
of
the
dis
trib
utio
n o
f th
e p
op
ula
tio
n
approx
2
2approx
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Optimal Trait
Left: comparison between the analytical and the numericalapproximations of the moments of the population’s density.Right: comparison between the mean phenotypic trait (numericaland analytical approximations) and the optimal trait. Delay= 0.2.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Example 2: numerical computation for ε large
R(t, x) = r − g(e(t))x2, g(e) = e, e(t) = 1.5 + cos(2πt),
r = 2, ε = 1.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (t)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Mom
ents
of th
e d
istr
ibution o
f th
e p
opula
tion
approx
2
2approx
Comparison between the analytical and the numerical approximations ofthe moments of the population phenotypic distribution.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Examples with two maximum points for RAn example with symmetric maxima:
R(t, x) =
{1− (x − 0.1)2(x − 0.9)2(1− x) if t ∈ [0, 1/2],
1− (x − 0.1)2(x − 0.9)2x if t ∈ [1/2, 1]
21 / 29
Evolutionary dynamics of phenotypically structured populations in time-varying environments
Case studies
Examples with two maximum points for RAn example with non symmetric maxima:
R(t, x) =
{1− (x + 1)2(x − 1)4(1− x) if t ∈ [0, 1/2],
1− (x + 1)2(x − 1)4x if t ∈ [1/2, 1]
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Table of contents
1 Introduction
2 Main results
3 Case studies
4 Adaptation in a shifting and fluctuating environment
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
A shifting and fluctuating environmentAn environmental change with linear trend but with fluctuations:
∂∂t n − σ
∂2
∂x2n = n
(R(e(t), x − ct)− κρ
),
ρ(t) =∫R n(t, x) dx .
e(t): periodic
−ct : a change of the optimal trait with linear trend
A change of variable n(t, x) = n(t, x + ct):∂∂t n − c ∂
∂x n − σ∂2
∂x2n = n
(R(e(t), x)− κρ
),
ρ(t) =∫R n(t, x) dx .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
A shifting and fluctuating environmentAn environmental change with linear trend but with fluctuations:
∂∂t n − σ
∂2
∂x2n = n
(R(e(t), x − ct)− κρ
),
ρ(t) =∫R n(t, x) dx .
e(t): periodic
−ct : a change of the optimal trait with linear trend
A change of variable n(t, x) = n(t, x + ct):∂∂t n − c ∂
∂x n − σ∂2
∂x2n = n
(R(e(t), x)− κρ
),
ρ(t) =∫R n(t, x) dx .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
AssumptionsNotation:
R(x) =1T
∫ T
0R(e(t), x)dt.
Assumptions:e is T -periodic with respect to t.
R is bounded and takes negative values for large |x |.
There exists a unique xm ∈ R such that
maxx∈Rd
R(x) = R(xm) > 0.
There exists a unique x < xm such that
R(x) +c2
4σ= R(xm) > 0.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
AssumptionsNotation:
R(x) =1T
∫ T
0R(e(t), x)dt.
Assumptions:e is T -periodic with respect to t.
R is bounded and takes negative values for large |x |.
There exists a unique xm ∈ R such that
maxx∈Rd
R(x) = R(xm) > 0.
There exists a unique x < xm such that
R(x) +c2
4σ= R(xm) > 0.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Critical speed for survivalLet λσ be the principal eigenvalue for{
∂tp − σ∂xxp − R(e, x)p = λσp, (t, x) ∈ [0,+∞)× R,p(t, x) = p(t + T , x), p > 0.
Define
cσ =
{2√−σλσ if λσ < 0,
0 otherwise.
Proposition (Figueroa Iglesias and M., Preprint )
(i) If c ≥ cσ, then ρ(t)→ 0 as t →∞.(ii) If c < cσ, then n(t, ·) converges to a the unique positive andperiodic solution to
∂∂t np − c ∂
∂x np − σ∂2
∂x2np = np
(R(e, x)− κρp
),
ρp(t) =∫R np(t, x) dx .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Critical speed for survivalLet λσ be the principal eigenvalue for{
∂tp − σ∂xxp − R(e, x)p = λσp, (t, x) ∈ [0,+∞)× R,p(t, x) = p(t + T , x), p > 0.
Define
cσ =
{2√−σλσ if λσ < 0,
0 otherwise.
Proposition (Figueroa Iglesias and M., Preprint )
(i) If c ≥ cσ, then ρ(t)→ 0 as t →∞.(ii) If c < cσ, then n(t, ·) converges to a the unique positive andperiodic solution to
∂∂t np − c ∂
∂x np − σ∂2
∂x2np = np
(R(e, x)− κρp
),
ρp(t) =∫R np(t, x) dx .
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
An asymptotic expansion for the critical speedRegime of small mutations:
σ = ε2, cε = cσ/ε.
Theorem (Figueroa Iglesias and M., Preprint )
cε = 2√
R(xm)− ε√−Rxx(xm) + o(ε).
We then define the renormalized speed
c = c/ε.
and study, for c < cε, ∂tnε,p − εc∂xnε,p − ε2∂xxnε,p = nε,p[R(e(t), x)− ρε,p(t)],
ρε,p(t) =
∫Rnε,p(t, x)dx , nε,p(0, x) = nε,p(T , x).
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
An asymptotic expansion for the critical speedRegime of small mutations:
σ = ε2, cε = cσ/ε.
Theorem (Figueroa Iglesias and M., Preprint )
cε = 2√
R(xm)− ε√−Rxx(xm) + o(ε).
We then define the renormalized speed
c = c/ε.
and study, for c < cε, ∂tnε,p − εc∂xnε,p − ε2∂xxnε,p = nε,p[R(e(t), x)− ρε,p(t)],
ρε,p(t) =
∫Rnε,p(t, x)dx , nε,p(0, x) = nε,p(T , x).
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
The population follows the optimum with a constant lagTheorem (Figueroa Iglesias and M., Preprint )
Assume that c < c∗ε − τ for τ > 0 and all ε ≤ ε0. Then, as ε→ 0,
nε,p(t, x)−−⇀%(t)δ(x − x).
with %(t) the unique periodic solution of
d%
dt= % [R(e, x)− %] .
In the original problem before the translation
nε,p(t, x)−−⇀%(t)δ(x − x − ct).
Recall: x the unique point such that R(x) + c2
4 = R(xm) andx < xm.
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Fluctuations acting on the pressure of the selection may helpthe population to follow the environmental change
A growth rate, with fluctuations on the pressure of the selection:
R(e, x) = r − g(e)(x − θ)2, g :=1T
∫ T
0g(e(s))ds.
Critical speed of survival:Periodic g : cε,p ≈ 2r − ε
√g .
Constant g = g(e0): cε,c ≈ 2r − ε√g(e0).
if g < g(e0) ⇒ c∗ε,c < c∗ε,p
⇒ the oscillations help the population to follow the environmentalchangeNote: opposite condition to have a more performant population inabsence of linear change ⇒ what is beneficial in a (in average)constant environment is disadvantageous in a changing environment
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Fluctuations acting on the pressure of the selection may helpthe population to follow the environmental change
A growth rate, with fluctuations on the pressure of the selection:
R(e, x) = r − g(e)(x − θ)2, g :=1T
∫ T
0g(e(s))ds.
Critical speed of survival:Periodic g : cε,p ≈ 2r − ε
√g .
Constant g = g(e0): cε,c ≈ 2r − ε√g(e0).
if g < g(e0) ⇒ c∗ε,c < c∗ε,p
⇒ the oscillations help the population to follow the environmentalchange
Note: opposite condition to have a more performant population inabsence of linear change ⇒ what is beneficial in a (in average)constant environment is disadvantageous in a changing environment
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Fluctuations acting on the pressure of the selection may helpthe population to follow the environmental change
A growth rate, with fluctuations on the pressure of the selection:
R(e, x) = r − g(e)(x − θ)2, g :=1T
∫ T
0g(e(s))ds.
Critical speed of survival:Periodic g : cε,p ≈ 2r − ε
√g .
Constant g = g(e0): cε,c ≈ 2r − ε√g(e0).
if g < g(e0) ⇒ c∗ε,c < c∗ε,p
⇒ the oscillations help the population to follow the environmentalchangeNote: opposite condition to have a more performant population inabsence of linear change ⇒ what is beneficial in a (in average)constant environment is disadvantageous in a changing environment
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Evolutionary dynamics of phenotypically structured populations in time-varying environments
Adaptation in a shifting and fluctuating environment
Thank you for your attention !
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