evolutionary dynamics of phenotypically structured...

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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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


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


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


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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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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Introduction

Table of contents

1 Introduction

2 Main results

3 Case studies

4 Adaptation in a shifting and fluctuating environment

5 / 29


Main results

Table of contents

1 Introduction

2 Main results

3 Case studies


5 / 29


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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Main results





),


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

).


uε,p(t, x) = u + εv + O(ε2).

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Main results





),


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

).


uε,p(t, x) = u + εv + O(ε2).6 / 29


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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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


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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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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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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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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Case studies

Table of contents

1 Introduction

2 Main results

3 Case studies


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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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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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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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Case studies

Example 1: when the fluctuations act on the optimal trait


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


Case studies

Example 1: numerical computations for ε small


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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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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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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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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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.

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Case studies

Example 1: numerical computations for ε large


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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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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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]

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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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Adaptation in a shifting and fluctuating environment

Table of contents

1 Introduction

2 Main results

3 Case studies


22 / 29



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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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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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 .

25 / 29



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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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.

27 / 29



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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R(e, x) = r − g(e)(x − θ)2, g :=1T

∫ T

0g(e(s))ds.


√g .


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

28 / 29





R(e, x) = r − g(e)(x − θ)2, g :=1T

∫ T

0g(e(s))ds.


√g .


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

28 / 29



Thank you for your attention !

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