Towards metastability in a model of population dynamics with competition

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Towards metastability in a model of population dynamics with competition

Loren Coquille (HCM Bonn) Joint work with A. Bovier

Essen — June 26, 2014

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Outline

1 The model

2 Law of large numbers (large population)

3 Large population and rare mutations Deterministic limit

Probabilistic limit

4 Towards metastability

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The model

Outline

1 The model

Probabilistic limit

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The model

Trait space ={0,1, . . . ,L}

Number of individuals of trait i =X_i(t)

Generator of the processX = (X₀(t), . . . ,X_L(t))∈N^L+1 : Lf(X) =

L

X

i=0

(f(X+e_i)−f(X))·b_i(1−ε)X_i clonal birth

+

L

X

i=0

(f(X−e_i)−f(X))·(d_i +

L

X

j=1

c_ijX_j)X_i natural death and competition

+

L

X

i=0

X

j∼i

(f(X+e_j)−f(X))·X_iεb_i/2 mutation where e_i = (0, . . . ,1, . . . ,0).

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Law of large numbers (large population)

Outline

1 The model

Probabilistic limit

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Law of large numbers

[Fournier, Méléard, 2004]

The rescaled process X^K = _K¹(X₀(t), . . . ,X_L(t))∈(_K¹N)^L+1 with generator

Lf(X) =

L

X

i=0

(f(X +ei

K)−f(X))·b_i(1−ε)KX_i +

L

X

i=0

(f(X −e_i

K)−f(X))·(d_i +

L

X

j=1

c_ij

KKX_j)KX_i +

L

X

i=0

X

j∼i

(f(X+ e_j

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

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Law of large numbers

converges in law, asK → ∞, towards the solution of the non-linear system of differential equations :

dx_i^ε dt =



(1−2ε)b_i −d_i−

L

X

j=0

c_ijx_j^ε



x_i^ε+ε



 X

j∼i

x_j^εb_j 2



, i =0, . . . ,L

Canonical vocabulary:

Monomorphic equilibrium : x¯_i = (b_i −d_i)/c_ii Invasion fitness : f_ij =b_i−d_i−c_ijx¯_j

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Law of large numbers

converges in law, asK → ∞, towards the solution of the non-linear system of differential equations :

dx_i^ε dt =



(1−2ε)b_i −d_i−

L

X

j=0

c_ijx_j^ε



x_i^ε+ε



 X

j∼i

x_j^εb_j 2



, i =0, . . . ,L

Canonical vocabulary:

Monomorphic equilibrium : x¯_i = (b_i −d_i)/c_ii Invasion fitness : f_ij =b_i−d_i−c_ijx¯_j

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Large population and rare mutations

Outline

1 The model

Probabilistic limit

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Large population and rare mutations Deterministic limit

Deterministic limit K → ∞ followed by ε → 0

Theorem

Start with initial condition : x^ε(0) = ¯x₀e₀.

If the invasion fitnesses f_i0 andf_iL satisfy : ^æ ^æ

æ æ

æ

à à

ì ì ì ì ìà

then the sequence of rescaled deterministic processes

x₀^ε(t log(1/ε)), . . . ,x_L^ε(t log(1/ε))

t>0

converges, as ε→0, towards the process x(t) =

¯x₀e₀ pour06t6L/f_L0

¯

x_Le_L pour t>L/f_L0

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

1 2 3 4 5 6 7

<——————>

∼ _f^L

L0 log(1/ε)

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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

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During time T

1

The process is close to the solution of the linear system

dy dt =







0 0 0 0 0 0

εb0

2 f10 0 0 0 0

0 ^εb₂¹ f₂₀ 0 0 0

0 0 . .. ... 0 0

0 0 0 ^εb^L−2₂ f_L−1,0 0

0 0 0 0 ^εb^L−1₂ f_L0





 y

with initial condition y(0) = (¯x₀,0, . . . ,0).

Indeed, d

dt(x_i^ε−y_i) =f_i₀(x_i^ε−y_i) +O(εx_i+1^ε ) +Error is negative until time t such that x_L−1 feels the presence of x_L:

−O(ε^L−1) +O(ε^L+1e^f^L0^t)>0 ⇒ T₁= 2

f_L0 log(1/ε)(1+o(1))

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During time T

1

dy dt =







0 0 0 0 0 0

εb0

2 f10 0 0 0 0

0 ^εb₂¹ f₂₀ 0 0 0

0 0 . .. ... 0 0

0 0 0 ^εb^L−2₂ f_L−1,0 0

0 0 0 0 ^εb^L−1₂ f_L0





 y

with initial condition y(0) = (¯x₀,0, . . . ,0). Indeed, d

dt(x_i^ε−y_i) =f_i₀(x_i^ε−y_i) +O(εx_i+1^ε ) +Error is negative until time t such that x_L−1 feels the presence of x_L:

−O(ε^L−1) +O(ε^L+1e^f^L0^t)>0 ⇒ T₁= 2

f_L0 log(1/ε)(1+o(1))

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During time T

2

dy dt =







0 0 0 0

εb0

2 f₁₀ 0 0

0 . .. ... 0

0 0 ^εb^L−2₂ f_L−2,0

0

0 f_L−1,0 ^εb₂^L

0 f_L0





 y

with initial condition y(0) = (¯x₀,O(ε), . . . ,O(ε^L−2)).

Indeed, _dt^d(x_i^ε−y_i) is negative until time t such that xL−2 feels the presence of x_L−1:

T₂= 2

f_L0 log(1/ε)(1+o(1))

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During time T

2

dy dt =







0 0 0 0

εb0

2 f₁₀ 0 0

0 . .. ... 0

0 0 ^εb^L−2₂ f_L−2,0

0

0 f_L−1,0 ^εb₂^L

0 f_L0





 y

with initial condition y(0) = (¯x₀,O(ε), . . . ,O(ε^L−2)).

Indeed, _dt^d(x_i^ε−y_i) is negative until time t such that xL−2 feels the presence of x_L−1:

T₂= 2

f_L0 log(1/ε)(1+o(1))

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Step by step towards the swap

If L>5, we continue to compare the system with :

dy dt =







0 0 0 0 0

εb0

2 f₁₀ 0 0 0

0 ^εb₂¹ f₂₀ 0 0

0 0 . .. . .. 0

0 0 0 ^εb^k−2₂ f_k−1,0

0

f_k,0 ^εb^k+1₂ 0 0

0 . .. . .. 0

0 0 f_L−1,0 ^εb₂^L

0 0 0 f_L0





 y

If Lis even, the time to reach the swap is L

f_L0log(1/ε)(1+o(1)).

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The swap : Lotka-Volterra system with 2 traits

The system is now close to the solution of:







dy₀/dt = (b₀−d₀−c₀₀y₀−c_0Ly_L)y₀ dy_L/dt = (b_L−d_L−c_0Ly₀−c_LLy_L)y_L dy_i/dt =0, ∀0<i <L

with initial condition







y₀(0) = ¯x₀ y_L(0) =η >0

yi(0) =O(ε^min{L−i,i}), ∀0<i <L

unique stable equilibrium(0, . . . ,0,x¯_L)

time to enter an η−neighborhood of this equilibrium isO(1).

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The swap : Lotka-Volterra system with 2 traits

The system is now close to the solution of:







dy₀/dt = (b₀−d₀−c₀₀y₀−c_0Ly_L)y₀ dy_L/dt = (b_L−d_L−c_0Ly₀−c_LLy_L)y_L dy_i/dt =0, ∀0<i <L

with initial condition







y₀(0) = ¯x₀ y_L(0) =η >0

yi(0) =O(ε^min{L−i,i}), ∀0<i <L unique stable equilibrium(0, . . . ,0,x¯_L)

time to enter an η−neighborhood of this equilibrium isO(1).

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Step by step towards equilibrium

We continue to compare the system with the linear two blocs system:

dy dt =







f0L 0 0 0 0

εb₀

2 f1L 0 0 0

0 ^εb₂¹ f2L 0 0

0 0 . .. . .. 0

0 0 0 ^εb^L−k−2₂ fL−k−1,L

0

fL−k,L

εb_L−k+1

2 0 0

0 . .. . .. 0

0 0 fL−1,L εb_L

2

0 0 0 0





 y

The system reaches equilibrium after a time L

f_0L log(1/ε)(1+o(1)).

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Large population and rare mutations Probabilistic limit

Probabilistic limit (K , ε) → (∞, 0) with ε

^L

K 1

Theorem

Assume same hypothesis on the fitness landscape. Consider the process (X_t^K)_t>0 with initial condition X₀^K = ^N_K⁰^Ke₀ such that ^N_K⁰^K ^law→ ¯x₀ >0as

(K, ε)→(∞,0) with ε^LK 1 Then we have

K→∞lim X_t^K_log(1/ε)^(d)=

¯x₀e₀ for06t 6L/f_L0

¯x_Le_L for t >L/f_L0 under the total variation norm.

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Towards metastability

Outline

1 The model

Probabilistic limit

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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 !!!

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Thank you !