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 The model 1 Law of large numbers (large population) 2 Large population and rare mutations 3 Deterministic limit Probabilistic limit Towards metastability 4 Loren Coquille Metastability with competition? Essen — June 26, 2014 2 / 20
The model Outline The model 1 Law of large numbers (large population) 2 Large population and rare mutations 3 Deterministic limit Probabilistic limit Towards metastability 4 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 = X i ( t ) Generator of the process X = ( X 0 ( t ) , . . . , X L ( t )) ∈ N L + 1 : L � Lf ( X ) = ( f ( X + e i ) − f ( X )) · b i ( 1 − ε ) X i clonal birth i = 0 L L c ij X j ) X i natural death � � + ( f ( X − e i ) − f ( X )) · ( d i + and competition i = 0 j = 1 L � � + ( f ( X + e j ) − f ( X )) · X i ε b i / 2 mutation i = 0 j ∼ i where e i = ( 0 , . . . , 1 , . . . , 0 ) . Loren Coquille Metastability with competition? Essen — June 26, 2014 4 / 20
Law of large numbers (large population) Outline The model 1 Law of large numbers (large population) 2 Large population and rare mutations 3 Deterministic limit Probabilistic limit Towards metastability 4 Loren Coquille Metastability with competition? Essen — June 26, 2014 5 / 20
Law of large numbers (large population) Law of large numbers [Fournier, Méléard, 2004] K N ) L + 1 with The rescaled process X K = 1 K ( X 0 ( t ) , . . . , X L ( t )) ∈ ( 1 generator L ( f ( X + e i � Lf ( X ) = K ) − f ( X )) · b i ( 1 − ε ) KX i i = 0 L L ( f ( X − e i c ij � � + K ) − f ( X )) · ( d i + K KX j ) KX i i = 0 j = 1 L ( f ( X + e j � � + K ) − f ( X )) · KX i ε b i / 2 , i = 0 j ∼ i 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 of differential equations : L dx ε b j � � i c ij x ε x ε x ε , dt = ( 1 − 2 ε ) b i − d i − i + ε i = 0 , . . . , L j j 2 j = 0 j ∼ i 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 of differential equations : L dx ε b j � � i c ij x ε x ε x ε , dt = ( 1 − 2 ε ) b i − d i − i + ε i = 0 , . . . , L j j 2 j = 0 j ∼ i Canonical vocabulary: Monomorphic equilibrium : ¯ x i = ( b i − d i ) / c ii Invasion fitness : f ij = b i − d i − c ij ¯ x j Loren Coquille Metastability with competition? Essen — June 26, 2014 7 / 20
Large population and rare mutations Outline The model 1 Law of large numbers (large population) 2 Large population and rare mutations 3 Deterministic limit Probabilistic limit Towards metastability 4 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 : x ε ( 0 ) = ¯ x 0 e 0 . � � � � � � � � If the invasion fitnesses f i 0 and f iL satisfy : � � � � � � � then the sequence of rescaled deterministic processes � � x ε 0 ( t log ( 1 /ε )) , . . . , x ε L ( t log ( 1 /ε )) t � 0 converges, as ε → 0 , towards the process � ¯ pour 0 � t � L / f L 0 x 0 e 0 x ( t ) = ¯ x L e L pour t > L / f L 0 Loren Coquille Metastability with competition? Essen — June 26, 2014 9 / 20
Large population and rare mutations Deterministic limit 7 6 5 4 3 2 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 <——————> L ∼ f L 0 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 )) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 log ( O ( ε )) � 20 log ( O ( ε 2 )) � 40 log ( O ( ε 3 )) � 60 log ( O ( ε 4 )) � 80 <——> <—–> T 1 T 2 Loren Coquille Metastability with competition? Essen — June 26, 2014 11 / 20
Large population and rare mutations Deterministic limit During time T 1 The process is close to the solution of the linear system 0 0 0 0 0 0 ε b 0 f 10 0 0 0 0 2 ε b 1 0 f 20 0 0 0 dy 2 dt = y ... ... 0 0 0 0 ε b L − 2 0 0 0 f L − 1 , 0 0 2 ε b L − 1 0 0 0 0 f L 0 2 with initial condition y ( 0 ) = (¯ x 0 , 0 , . . . , 0 ) . Loren Coquille Metastability with competition? Essen — June 26, 2014 12 / 20
Large population and rare mutations Deterministic limit During time T 1 The process is close to the solution of the linear system 0 0 0 0 0 0 ε b 0 f 10 0 0 0 0 2 ε b 1 0 f 20 0 0 0 dy 2 dt = y ... ... 0 0 0 0 ε b L − 2 0 0 0 f L − 1 , 0 0 2 ε b L − 1 0 0 0 0 f L 0 2 with initial condition y ( 0 ) = (¯ x 0 , 0 , . . . , 0 ) . Indeed, d dt ( x ε i − y i ) = f i 0 ( x ε i − y i ) + O ( ε x ε i + 1 ) + Error is negative until time t such that x L − 1 feels the presence of x L : T 1 = 2 − O ( ε L − 1 ) + O ( ε L + 1 e f L 0 t ) > 0 ⇒ log ( 1 /ε )( 1 + o ( 1 )) f L 0 Loren Coquille Metastability with competition? Essen — June 26, 2014 12 / 20
Large population and rare mutations Deterministic limit During time T 2 The process is close to the solution of the linear system 0 0 0 0 ε b 0 f 10 0 0 2 0 ... ... dy 0 0 dt = y ε b L − 2 0 0 f L − 2 , 0 2 ε b L f L − 1 , 0 0 2 0 f L 0 x 0 , O ( ε ) , . . . , O ( ε L − 2 )) . with initial condition y ( 0 ) = (¯ Loren Coquille Metastability with competition? Essen — June 26, 2014 13 / 20
Large population and rare mutations Deterministic limit During time T 2 The process is close to the solution of the linear system 0 0 0 0 ε b 0 f 10 0 0 2 0 ... ... dy 0 0 dt = y ε b L − 2 0 0 f L − 2 , 0 2 ε b L f L − 1 , 0 0 2 0 f L 0 x 0 , O ( ε ) , . . . , O ( ε L − 2 )) . with initial condition y ( 0 ) = (¯ Indeed, d dt ( x ε i − y i ) is negative until time t such that x L − 2 feels the presence of x L − 1 : T 2 = 2 log ( 1 /ε )( 1 + o ( 1 )) f L 0 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 : 0 0 0 0 0 ε b 0 f 10 0 0 0 2 ε b 1 0 f 20 0 0 0 2 ... ... 0 0 0 dy ε b k − 2 dt = 0 0 0 f k − 1 , 0 y 2 ε b k + 1 f k , 0 0 0 2 ... ... 0 0 0 ε b L 0 0 f L − 1 , 0 2 0 0 0 f L 0 L If L is even, the time to reach the swap is log ( 1 /ε )( 1 + o ( 1 )) . f L 0 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: dy 0 / dt = ( b 0 − d 0 − c 00 y 0 − c 0 L y L ) y 0 dy L / dt = ( b L − d L − c 0 L y 0 − c LL y L ) y L dy i / dt = 0 , ∀ 0 < i < L with initial condition y 0 ( 0 ) = ¯ x 0 y L ( 0 ) = η > 0 y i ( 0 ) = O ( ε min { 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: dy 0 / dt = ( b 0 − d 0 − c 00 y 0 − c 0 L y L ) y 0 dy L / dt = ( b L − d L − c 0 L y 0 − c LL y L ) y L dy i / dt = 0 , ∀ 0 < i < L with initial condition y 0 ( 0 ) = ¯ x 0 y L ( 0 ) = η > 0 y i ( 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 is O ( 1 ) . Loren Coquille Metastability with competition? Essen — June 26, 2014 15 / 20
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