Models of structured populations in constant and fluctuating - - PowerPoint PPT Presentation

Models of structured populations in constant and fluctuating environments

Sepideh Mirrahimi

CNRS, IMT, Toulouse

Toulouse, June 16-19, 2014

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Darwinian evolution of a structured population density

• We study the

Darwinian evolution

• f a population

structured by phenotypical traits, under selection and mutation

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Example of evolution: creation of antibiotic resistance of bacteria under drug selection

Morbidostat : a selective pressure is applied continuously to the bacterial population. It automatically tunes drug concentration such that a constant growth rate is maintained.

Following evolution of bacterial antibiotic resistance in real time, Rosenthal et Elowitz, Nature Genetics 2012 3 / 26

The influence of fluctuating temperature on bacteria

Bacterial pathogen Serratia marcesens evolved in fluctuating temperature (daily variation between 24◦C and 38◦C, mean 31◦C),

• utperforms 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. 2013 4 / 26

Environment fluctuation can select for generalism ⇒ It can also increase organisms’ ability to invade novel environments.

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Environment fluctuation can select for generalism ⇒ It can also increase organisms’ ability to invade novel environments. However, generalism can still be costly in terms of reduced fitness in some ecological contexts: Evolution in fluctuating tempera- ture decreased bacterial virulence in D. melanogoster host:

Figure from: Fluctuation temperature leads to evolution of thermal generalism and preadaptation to novel environments, Ketola et al. 2013 5 / 26

A typical model

    

∂ ∂t nε − ε∆nε = nε

ε R

• x, Iε
• ,

nε(·, t = 0) = n0

ε(·),

Iε(t) =

• Rd η(x) nε(x, t) dx.
• x ∈ Rd:

phenotypical trait

• nε(x, t): density of trait x
• η(x): consumption rate
• Iε(t): total consumption
• R(x, Iε): growth rate
• ε: a small parameter

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A typical model

    

∂ ∂t nε − ε∆nε = nε

ε R

• x, Iε
• ,

nε(·, t = 0) = n0

ε(·),

Iε(t) =

• Rd η(x) nε(x, t) dx.
• x ∈ Rd:

phenotypical trait

• nε(x, t): density of trait x
• η(x): consumption rate
• Iε(t): total consumption
• R(x, Iε): growth rate
• ε: a small parameter

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The typical behavior of the solutions

A simple typical growth rate: R(x, I) = 1 − x2 2 − I Dynamics of the dominant trait

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Preliminary properties

• − C ≤ ∂R

∂I (x, I) ≤ −C −1 < 0

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Preliminary properties

• − C ≤ ∂R

∂I (x, I) ≤ −C −1 < 0

• maxx∈Rd R(x, IM) = 0,

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Preliminary properties

• − C ≤ ∂R

∂I (x, I) ≤ −C −1 < 0

• maxx∈Rd R(x, IM) = 0,
• Im ≤ Iε(0) ≤ IM.

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Preliminary properties

• − C ≤ ∂R

∂I (x, I) ≤ −C −1 < 0

• maxx∈Rd R(x, IM) = 0,
• Im ≤ Iε(0) ≤ IM.

= ⇒ Im < Iε(t) ≤ IM.

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Preliminary properties

• − C ≤ ∂R

∂I (x, I) ≤ −C −1 < 0

• maxx∈Rd R(x, IM) = 0,
• Im ≤ Iε(0) ≤ IM.

= ⇒ Im < Iε(t) ≤ IM. Moreover, after extraction of a subsequence, (Iε)ε converges a.e. to I(t).

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Some notations

• n(x, t): weak limit of nε(x, t) as ε vanishes
• We expect n to concentrate as Dirac masses
• Hopf-Cole transformation: nε(x, t) = exp

uε(x, t) ε

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The Hamilton-Jacobi limit

Theorem (Barles, SM, Perthame - 2009)

After extraction of a subsequence, uε converges locally uniformly to a continuous function u, a viscosity solution to           

∂ ∂t u = |∇u|2 + R(x, I(t))

max

x∈Rd u(x, t) = 0,

u(0, x) = u0(x).

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Convergence to a monomorphic population

nε(x, t) − − ⇀

ε→0 n(x, t),

supp n(x, t) ⊂ {u(x, t) = 0} ⊂ {R(x, I(t)) = 0}.

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Convergence to a monomorphic population

nε(x, t) − − ⇀

ε→0 n(x, t),

supp n(x, t) ⊂ {u(x, t) = 0} ⊂ {R(x, I(t)) = 0}. In particular, if R : R → R is strictly concave with respect to x, then nε(x, t) − − ⇀

ε→0 n(x, t) = ρ(t) δ(x − x(t)),

with R (x(t), I(t)) = 0 and ρ(t) =

I(t) η(x(t)).

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Canonical equation and long time behavior

Theorem (Lorz, SM, Perthame - 2013)

Under concavity and smoothness assumptions,

• nε(x, t) −

− ⇀

ε→0 n(x, t) = ρ(t) δ(x − x(t))

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Canonical equation and long time behavior

Theorem (Lorz, SM, Perthame - 2013)

Under concavity and smoothness assumptions,

• nε(x, t) −

− ⇀

ε→0 n(x, t) = ρ(t) δ(x − x(t))

• x(t) and ρ(t) are ’smooth’

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Canonical equation and long time behavior

Theorem (Lorz, SM, Perthame - 2013)

Under concavity and smoothness assumptions,

• nε(x, t) −

− ⇀

ε→0 n(x, t) = ρ(t) δ(x − x(t))

• x(t) and ρ(t) are ’smooth’
• R
• x(t), I(t)
• = 0, with I(t) = η (x(t)) ρ(t).

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Canonical equation and long time behavior

Theorem (Lorz, SM, Perthame - 2013)

Under concavity and smoothness assumptions,

• nε(x, t) −

− ⇀

ε→0 n(x, t) = ρ(t) δ(x − x(t))

• x(t) and ρ(t) are ’smooth’
• R
• x(t), I(t)
• = 0, with I(t) = η (x(t)) ρ(t).
• ˙

x(t) =

• −D2u
• x(t), t

−1 · ∇xR

• x(t), I(t)
• ,

x(0) = x0

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Canonical equation and long time behavior

Theorem (Lorz, SM, Perthame - 2013)

Under concavity and smoothness assumptions,

• nε(x, t) −

− ⇀

ε→0 n(x, t) = ρ(t) δ(x − x(t))

• x(t) and ρ(t) are ’smooth’
• R
• x(t), I(t)
• = 0, with I(t) = η (x(t)) ρ(t).
• ˙

x(t) =

• −D2u
• x(t), t

−1 · ∇xR

• x(t), I(t)
• ,

x(0) = x0

• I(t) −

→

t→∞ IM,

x(t) − →

t→∞ xM. with

0 = R(xM, IM) = max

x∈Rd R(x, IM).

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x1 x2 x3 xM

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What if the environment is fluctuating?

   ε ∂

∂t nε − ε2∆nε = nε R

• x, t

ε, Iε(t)

• ,

nε(·, t = 0) = n0

ε,

Iε(t) :=

• RN η(x)nε(x, t)dx,

with R 1-periodic in the second variable. Can we describe the limit as ε → 0?

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Some assumptions

• −2K1 ≤ D2

x R(x, s, I) ≤ −2K2

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Some assumptions

• −2K1 ≤ D2

x R(x, s, I) ≤ −2K2

• −K5 ≤ ∂

∂I R(x, s, I) ≤ −K6

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Some assumptions

• −2K1 ≤ D2

x R(x, s, I) ≤ −2K2

• −K5 ≤ ∂

∂I R(x, s, I) ≤ −K6

• max0≤s≤1, x∈RN R(x, s, IM) = 0

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Main result

Theorem ( SM, Perthame, Souganidis - 2014 )

There exist a fittest trait x and a total population size ρ such that, along subsequences ε → 0, nε(·, t)− ⇀ ̺(t)δ(· − x(t)) weakly in the sense of measures, Iε− ⇀ I := ̺η(x) in L∞(0, ∞) weak-⋆, and R(x, t ε, Iε(t))− ⇀ R (x, x(t)) weakly in t and strongly in x.

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Idea of the proof

• From the concavity property one can prove that, for some

x(t) ∈ RN, nε− ⇀ n, supp n = {x(t)}.

• The case with constant environment (R(x, I)):

There exists a unique constant I(t) such that R(x(t), I(t)) = 0. One can then prove that, as ε → 0, Iε(t) − → I(t), nε− ⇀ I(t) η(x(t))δ(x − x(t)).

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• Periodic environment (R(x, s, I)): more work to determine

the (weak) limit of Iε.

Lemma (Cell problem)

Let x = x(t) for some t ≥ 0. There exists a unique 1-periodic positive solution I(x, s) : [0, 1] → (0, IM) to   

d ds I(x, s) = I(x, s) R

• x, s, I(x, s)
• ,

I(x, 0) = I(x, 1). Then Iε(t)− ⇀ 1 I(x(t), s)ds, R(x, t ε, Iε(t))− ⇀ R (x, x(t)) := 1 R(x, s, I(x(t), s))ds.

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Weak convergence of Iε

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7 8 t I

Dynamics of the total population Iε(t) with R(x, s, I) = (2 + sin (2πs)) 2 − x2 I + .5 − .5, η(x) = 1, ε = 0.01. The Iε’s oscillate with period of order ε around a monotone curve I.

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The Hamilton-Jacobi limit

We use again the Hopf-Cole transformation nε = exp uε ε

• .

Theorem ( SM, Perthame, Souganidis - 2014)

Along subsequences ε → 0, uε → u locally uniformly in RN × [0, ∞), where u is a viscosity solution of          ut = R(x, x(t)) + |Dxu|2, max

x∈RN u(x, t) = 0 = u(x(t), t),

u(·, 0) = u0(·). Moreover, x satisfies the canonical equation ˙ x(t) =

• −D2

x u(x(t), t)

−1 · D1R(x(t), x(t)).

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In terms of adaptive dynamics...

• R(x, y): the effective fitness of a mutant x in a resident

population with a dominant trait y,

• D1R: the selection gradient, which represents the capability
• f invasion.

−D2

x u(x(t), t)

−1 is an indicator of the diversity around the dominant trait in the resident population.

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Fluctuations may increase the population size

We consider the following example R(x, s, I) = b(x) − D1(s)D2(I), η ≡ 1, with D1 1-periodic and D2 concave and increasing. The goal is to compare the long time limit of of the total population size for the model with fluctuations (I f

M) to the one

• btained from the model with the “averaged rate” (I c

M) :

Rav(x, I) = b(x) − D1,avD2(I) with fav = 1 f ds.

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Total population in constant and fluctuating environment

Let b(xM) = max

x

b(x).

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Total population in constant and fluctuating environment

Let b(xM) = max

x

b(x). According to the previous computations in the model with no fluctuation, the final total population I c

M is given by

b(xM) = D1,avD2(I c

M).

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Total population in constant and fluctuating environment

Let b(xM) = max

x

b(x). According to the previous computations in the model with no fluctuation, the final total population I c

M is given by

b(xM) = D1,avD2(I c

M).

Equivalently in the model with fluctuation we obtain, at the final time, I f

M =

1 I (xM, s) ds where, I is a periodic function which solves d ds I = I

• b(xM) − D1(s)D2(I)
• .

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Some computation for the fluctuating case

By integrating correspondent equations for suitable functions of I, we obtain b(xM) = 1 D1(s)D2(I(xM, s))ds 1 D2(I(xM, s))ds b(xM) = 1 D1(s)D2

2(I(xM, s))ds.

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Some computation for the fluctuating case

By integrating correspondent equations for suitable functions of I, we obtain b(xM) = 1 D1(s)D2(I(xM, s))ds 1 D2(I(xM, s))ds b(xM) = 1 D1(s)D2

2(I(xM, s))ds.

From the Cauchy-Schwarz inequality : b(xM)2 < D1,av b(xM) 1 D2(I(xM, s))ds

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Some computation for the fluctuating case

By integrating correspondent equations for suitable functions of I, we obtain b(xM) = 1 D1(s)D2(I(xM, s))ds 1 D2(I(xM, s))ds b(xM) = 1 D1(s)D2

2(I(xM, s))ds.

From the Cauchy-Schwarz inequality : b(xM)2 < D1,av b(xM) 1 D2(I(xM, s))ds and thus b(xM) < D1,av 1 D2(I(xM, s))ds.

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Comparison between the fluctuating and the averaged model

From the computations for both cases b(xM) = D1,avD2(I c

M) < D1,av

1 D2(I(xM, s))ds,

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Comparison between the fluctuating and the averaged model

From the computations for both cases b(xM) = D1,avD2(I c

M) < D1,av

1 D2(I(xM, s))ds, and thus D2(I c

M) <

1 D2(I(xM, s))ds.

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Comparison between the fluctuating and the averaged model

From the computations for both cases b(xM) = D1,avD2(I c

M) < D1,av

1 D2(I(xM, s))ds, and thus D2(I c

M) <

1 D2(I(xM, s))ds. From the concavity and monotonicity of D2 we obtain I c

M < I f M =

1 I (xM, s) ds. Therefore the population is more important in the fluctuating case !

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Thank you for your attention !

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