Towards metastability in a model of population dynamics with - - PowerPoint PPT Presentation

Towards metastability in a model of population dynamics with competition

Loren Coquille (HCM Bonn) Joint work with A. Bovier Essen — June 26, 2014

Loren Coquille Metastability with competition? Essen — June 26, 2014 1 / 20

Outline

1

The model

2

Law of large numbers (large population)

3

Large population and rare mutations Deterministic limit Probabilistic limit

4

Towards metastability

Loren Coquille Metastability with competition? Essen — June 26, 2014 2 / 20

The model

Outline

1

The model

2

Law of large numbers (large population)

3

Large population and rare mutations Deterministic limit Probabilistic limit

4

Towards metastability

Loren Coquille Metastability with competition? Essen — June 26, 2014 3 / 20

The model

The model

Trait space = {0, 1, . . . , L} Number of individuals of trait i = Xi(t) Generator of the process X = (X0(t), . . . , XL(t)) ∈ NL+1 : Lf (X) =

L

• i=0

(f (X+ei) − f (X)) · bi(1 − ε)Xi clonal birth +

L

• i=0

(f (X−ei) − f (X)) · (di +

L

• j=1

cijXj)Xi natural death and competition +

L

• i=0
• j∼i

(f (X+ej) − f (X)) · Xiεbi/2 mutation where ei = (0, . . . , 1, . . . , 0).

Loren Coquille Metastability with competition? Essen — June 26, 2014 4 / 20

Law of large numbers (large population)

Outline

1

The model

2

Law of large numbers (large population)

3

Large population and rare mutations Deterministic limit Probabilistic limit

4

Towards metastability

Loren Coquille Metastability with competition? Essen — June 26, 2014 5 / 20

Law of large numbers (large population)

Law of large numbers [Fournier, Méléard, 2004]

The rescaled process X K = 1

K (X0(t), . . . , XL(t)) ∈ ( 1 K N)L+1 with

generator Lf (X) =

L

• i=0

(f (X + ei K ) − f (X)) · bi(1 − ε)KXi +

L

• i=0

(f (X − ei K ) − f (X))·(di +

L

• j=1

cij K KXj)KXi +

L

• i=0
• j∼i

(f (X + ej K ) − f (X)) · KXiεbi/2, bounded parameters and convergence of the initial condition,

Loren Coquille Metastability with competition? Essen — June 26, 2014 6 / 20

Law of large numbers (large population)

Law of large numbers [Fournier, Méléard, 2004]

converges in law, as K → ∞, towards the solution of the non-linear system

• f differential equations :

dxε

i

dt =  (1 − 2ε)bi − di−

L

• j=0

cijxε

j

  xε

i + ε

 

j∼i

xε

j

bj 2   , i = 0, . . . , L

Loren Coquille Metastability with competition? Essen — June 26, 2014 7 / 20

Law of large numbers (large population)

Law of large numbers [Fournier, Méléard, 2004]

converges in law, as K → ∞, towards the solution of the non-linear system

• f differential equations :

dxε

i

dt =  (1 − 2ε)bi − di−

L

• j=0

cijxε

j

  xε

i + ε

 

j∼i

xε

j

bj 2   , i = 0, . . . , L Canonical vocabulary: Monomorphic equilibrium : ¯ xi = (bi − di)/cii Invasion fitness : fij = bi − di − cij¯ xj

Loren Coquille Metastability with competition? Essen — June 26, 2014 7 / 20

Large population and rare mutations

Outline

1

The model

2

Law of large numbers (large population)

3

Large population and rare mutations Deterministic limit Probabilistic limit

4

Towards metastability

Loren Coquille Metastability with competition? Essen — June 26, 2014 8 / 20

Large population and rare mutations Deterministic limit

Deterministic limit K → ∞ followed by ε → 0

Theorem Start with initial condition : xε(0) = ¯ x0e0. If the invasion fitnesses fi0 and fiL satisfy :

• then the sequence of rescaled deterministic processes
• xε

0(t log(1/ε)), . . . , xε L(t log(1/ε))

• t0

converges, as ε → 0, towards the process x(t) = ¯ x0e0 pour 0 t L/fL0 ¯ xLeL pour t > L/fL0

Loren Coquille Metastability with competition? Essen — June 26, 2014 9 / 20

Large population and rare mutations Deterministic limit

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1 2 3 4 5 6 7

<——————> ∼

L fL0 log(1/ε)

Loren Coquille Metastability with competition? Essen — June 26, 2014 10 / 20

Large population and rare mutations Deterministic limit

Idea of the proof : look at the logarithmic scale

log(O(1)) log(O(ε)) log(O(ε2)) log(O(ε3)) log(O(ε4))

0.2 0.4 0.6 0.8 1.0 1.2 1.4 80 60 40 20

<——> <—–>

T1 T2

Loren Coquille Metastability with competition? Essen — June 26, 2014 11 / 20

Large population and rare mutations Deterministic limit

During time T1

The process is close to the solution of the linear system dy dt =          

εb0 2

f10

εb1 2

f20 ... ...

εbL−2 2

fL−1,0

εbL−1 2

fL0           y with initial condition y(0) = (¯ x0, 0, . . . , 0).

Loren Coquille Metastability with competition? Essen — June 26, 2014 12 / 20

Large population and rare mutations Deterministic limit

During time T1

The process is close to the solution of the linear system dy dt =          

εb0 2

f10

εb1 2

f20 ... ...

εbL−2 2

fL−1,0

εbL−1 2

fL0           y with initial condition y(0) = (¯ x0, 0, . . . , 0). Indeed, d dt (xε

i − yi) = fi0(xε i − yi) + O(εxε i+1) + Error

is negative until time t such that xL−1 feels the presence of xL: −O(εL−1) + O(εL+1efL0t) > 0 ⇒ T1 = 2 fL0 log(1/ε)(1 + o(1))

Loren Coquille Metastability with competition? Essen — June 26, 2014 12 / 20

Large population and rare mutations Deterministic limit

During time T2

The process is close to the solution of the linear system dy dt =          

εb0 2

f10 ... ...

εbL−2 2

fL−2,0 fL−1,0

εbL 2

fL0           y with initial condition y(0) = (¯ x0, O(ε), . . . , O(εL−2)).

Loren Coquille Metastability with competition? Essen — June 26, 2014 13 / 20

Large population and rare mutations Deterministic limit

During time T2

The process is close to the solution of the linear system dy dt =          

εb0 2

f10 ... ...

εbL−2 2

fL−2,0 fL−1,0

εbL 2

fL0           y with initial condition y(0) = (¯ x0, O(ε), . . . , O(εL−2)). Indeed, d

dt (xε i − yi) is negative until time t such that xL−2 feels the

presence of xL−1: T2 = 2 fL0 log(1/ε)(1 + o(1))

Loren Coquille Metastability with competition? Essen — June 26, 2014 13 / 20

Large population and rare mutations Deterministic limit

Step by step towards the swap

If L 5, we continue to compare the system with : dy dt =                 

εb0 2

f10

εb1 2

f20 ... ...

εbk−2 2

fk−1,0 fk,0

εbk+1 2

... ... fL−1,0

εbL 2

fL0                  y If L is even, the time to reach the swap is L fL0 log(1/ε)(1 + o(1)) .

Loren Coquille Metastability with competition? Essen — June 26, 2014 14 / 20

Large population and rare mutations Deterministic limit

The swap : Lotka-Volterra system with 2 traits

The system is now close to the solution of:    dy0/dt = (b0 − d0 − c00y0 − c0LyL)y0 dyL/dt = (bL − dL − c0Ly0 − cLLyL)yL dyi/dt = 0, ∀0 < i < L with initial condition    y0(0) = ¯ x0 yL(0) = η > 0 yi(0) = O(εmin{L−i,i}), ∀0 < i < L

Loren Coquille Metastability with competition? Essen — June 26, 2014 15 / 20

Large population and rare mutations Deterministic limit

The swap : Lotka-Volterra system with 2 traits

The system is now close to the solution of:    dy0/dt = (b0 − d0 − c00y0 − c0LyL)y0 dyL/dt = (bL − dL − c0Ly0 − cLLyL)yL dyi/dt = 0, ∀0 < i < L with initial condition    y0(0) = ¯ x0 yL(0) = η > 0 yi(0) = O(εmin{L−i,i}), ∀0 < i < L unique stable equilibrium (0, . . . , 0, ¯ xL) time to enter an η− neighborhood of this equilibrium is O(1).

Loren Coquille Metastability with competition? Essen — June 26, 2014 15 / 20

Large population and rare mutations Deterministic limit

Step by step towards equilibrium

We continue to compare the system with the linear two blocs system: dy dt =              

f0L

εb0 2

f1L

εb1 2

f2L ... ...

εbL−k−2 2

fL−k−1,L fL−k,L

εbL−k+1 2

... ... fL−1,L

εbL 2

              y The system reaches equilibrium after a time L f0L log(1/ε)(1 + o(1)) .

Loren Coquille Metastability with competition? Essen — June 26, 2014 16 / 20

Large population and rare mutations Probabilistic limit

Probabilistic limit (K, ε) → (∞, 0) with εLK ≫ 1

Theorem Assume same hypothesis on the fitness landscape. Consider the process (X K

t )t0 with initial condition X K 0 = NK K e0 such that NK K law

→ ¯ x0 > 0 as (K, ε) → (∞, 0) with εLK ≫ 1 Then we have lim

K→∞ X K t log(1/ε) (d)

= ¯ x0e0 for 0 t L/fL0 ¯ xLeL for t > L/fL0 under the total variation norm.

Loren Coquille Metastability with competition? Essen — June 26, 2014 17 / 20

Towards metastability

Outline

1

The model

2

Law of large numbers (large population)

3

Large population and rare mutations Deterministic limit Probabilistic limit

4

Towards metastability

Loren Coquille Metastability with competition? Essen — June 26, 2014 18 / 20

Towards metastability

Towards metastability

Natural questions : study the limit (K, ε) → (∞, 0) on a non-trivial scale get a random time of swap exponentially distributed ? what are the strategies used by the system to swap at a given time ? There is no literature about metastability in models with competition. !!! non-reversibility !!!

Loren Coquille Metastability with competition? Essen — June 26, 2014 19 / 20

Towards metastability

Thank you !

Loren Coquille Metastability with competition? Essen — June 26, 2014 20 / 20