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Branching Random Walks: selection, survival and genealogies Damien Simon ´ e Ecole normale sup´rieure, Paris, France u Institut f¨r theoretische Physik, University of Cologne, Germany ´ in collaboration with E. Brunet and B. Derrida Eindhoven, March 25th 2009 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 1 / 17 Free branching random walks x branching rate β: A −→ A + A t (reproduction) diﬀusion: random walks (mutations or exploration) D. Simon () Branching Random Walks Eindhoven, March 25th 2009 2 / 17 Free branching random walks x branching rate β: A −→ A + A t (reproduction) diﬀusion: random walks (mutations or exploration) Linear expansion : √ velocity vc = 2 Dβ density(x , t) ∝ e βt e −x 2 /(2Dt) ∝ 2vc t D. Simon () Branching Random Walks Eindhoven, March 25th 2009 2 / 17 The questions 1 size distribution 2 position of the rightmost individual D. Simon () Branching Random Walks Eindhoven, March 25th 2009 3 / 17 The questions 1 size distribution Absorbing boundaries 2 position of the rightmost individual 3 inﬂuence of boundary conditions : phase transition typical extinction times D. Simon () Branching Random Walks Eindhoven, March 25th 2009 3 / 17 The questions 1 size distribution 2 position of the rightmost individual Size N = 4 3 inﬂuence of boundary t conditions : phase transition typical extinction times 4 constant size (or nearly) : t +1 selection velocity genealogies D. Simon () Branching Random Walks Eindhoven, March 25th 2009 3 / 17 Survival in presence of an absorbing wall t = 0 : one individual at distance x > 0 from the wall Speed v of the wall towards the right Qs (x , t) : proba. that the generated population is still alive at t D. Simon () Branching Random Walks Eindhoven, March 25th 2009 4 / 17 Survival in presence of an absorbing wall t = 0 : one individual at distance x > 0 from the wall Speed v of the wall towards the right Qs (x , t) : proba. that the generated population is still alive at t F-KPP Travelling wave equation (Fisher-Kolmogorov-Petrovski-Piscounov) : 2 2 ∂t Qs = ∂x Qs − v ∂x Qs + β(Qs − Qs ) Berestycki, Aronson & Weinberger D. Simon () Branching Random Walks Eindhoven, March 25th 2009 4 / 17 Survival in presence of an absorbing wall t = 0 : one individual at distance x > 0 from the wall Speed v of the wall towards the right Qs (x , t) : proba. that the generated population is still alive at t F-KPP Travelling wave equation (Fisher-Kolmogorov-Petrovski-Piscounov) : 2 2 ∂t Qs = ∂x Qs − v ∂x Qs + β(Qs − Qs ) Qs (x , t = 0) = 1, x >0 Qs (x = 0, t) = 0 Berestycki, Aronson & Weinberger D. Simon () Branching Random Walks Eindhoven, March 25th 2009 4 / 17 Slow dynamics of Qs (x , t) : a ﬁrst approach Qs (x , t) 1 Lt 0 x linearized KPP Wall D. Simon () Branching Random Walks Eindhoven, March 25th 2009 5 / 17 Slow dynamics of Qs (x , t) : a ﬁrst approach Qs (x , t) Scaling form for |v − vc | 1 and Qs 1: 1 Lt πx Qs (x , t) B sin e v (x −Lt )/2 Lt 0 x linearized KPP dynamics for Lt : Wall 2π 2 ∂t Lt = (v − vc ) + vc Lt 2 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 5 / 17 Slow dynamics of Qs (x , t) : a ﬁrst approach Qs (x , t) Scaling form for |v − vc | 1 and Qs 1: 1 Lt πx Qs (x , t) B sin e v (x −Lt )/2 Lt 0 x linearized KPP dynamics for Lt : Wall 2π 2 ∂t Lt = (v − vc ) + vc Lt 2 Long times t → ∞ : if v > vc , then Lt ∝ (v − vc )t if v = vc , then Lt ∝ t 1/3 if v < vc , then Lt → L and relaxation τ1 ∝ (vc − v )−3/2 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 5 / 17 Second approach : relation between Qs and an inﬁnite travelling wave Qv , v < vc ∗ Survival at long times : Qs (x ) = lim Qs (x , t) t→∞ solution Qv (x ) on R ∗ Qs (x ) given by the positive right part of Qv . same thing from the relaxation spectrum... D. Simon () Branching Random Walks Eindhoven, March 25th 2009 6 / 17 Second approach : relation between Qs and an inﬁnite travelling wave Qv , v < vc ∗ Survival at long times : Qs (x ) = lim Qs (x , t) t→∞ Qv (x ) 1 solution Qv (x ) on R ∗ Qs (x ) given by the positive right part of Qv . same thing from the relaxation spectrum... x D. Simon () Branching Random Walks Eindhoven, March 25th 2009 6 / 17 Second approach : relation between Qs and an inﬁnite travelling wave Qv , v < vc ∗ Survival at long times : Qs (x ) = lim Qs (x , t) t→∞ Qv (x ) 1 solution Qv (x ) on R ∗ Qs (x ) given by the positive right part of Qv . same thing from the relaxation spectrum... x x0 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 6 / 17 Second approach : relation between Qs and an inﬁnite travelling wave Qv , v < vc ∗ Survival at long times : Qs (x ) = lim Qs (x , t) t→∞ ∗ Qs (x ) Qv (x ) 1 solution Qv (x ) on R ∗ Qs (x ) given by the positive right part of Qv . same thing from the relaxation spectrum... x x0 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 6 / 17 Near the critical velocity Qs (x , t): probability that a population generated by an initial individual at x is still alive at t v > vc t→∞ −→ Qs (x , t) − − 0 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 7 / 17 Near the critical velocity Qs (x , t): probability that a population generated by an initial individual at x is still alive at t v > vc t→∞ −→ Qs (x , t) − − 0 v = vc 1/3 Qs (x , t) ∝ e −A t Kesten 78 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 7 / 17 Near the critical velocity v < vc Qs (x , t): probability that a population generated by an initial individual at x is t→∞ −→ ∗ Qs (x , t) − − Qs (x ) > 0 still alive at t When v → vc , v > vc t→∞ −1/2 Qs (x , t) − − 0 −→ ∗ Qs (x ) ∝ e −A(vc −v ) v = vc 1/3 Qs (x , t) ∝ e −A t Kesten 78 S. & Derrida, 07/08 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 7 / 17 Near the critical velocity v < vc Qs (x , t): probability that a population generated by an initial individual at x is t→∞ −→ ∗ Qs (x , t) − − Qs (x ) > 0 still alive at t When v → vc , v > vc t→∞ −1/2 Qs (x , t) − − 0 −→ ∗ Qs (x ) ∝ e −A(vc −v ) Relaxation times: v = vc τ1 ∝ (vc − v )−3/2 Qs (x , t) ∝ e −A t 1/3 τ2 , τ3 , . . . ∝ (vc − v )−1 Kesten 78 S. & Derrida, 07/08 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 7 / 17 Conditionned regime Size Observation m 1 Time t T D. Simon () Branching Random Walks Eindhoven, March 25th 2009 8 / 17 Conditionned regime Size Observation m 1 Time t T For v < vc : “quasi-stationary” cf. without space (GW) D. Simon () Branching Random Walks Eindhoven, March 25th 2009 8 / 17 Conditionned regime Size Observation m 1 Time t T For v < vc : “quasi-stationary” For v > vc cf. without space (GW) cf. random media D. Simon () Branching Random Walks Eindhoven, March 25th 2009 8 / 17 − “Quasi-stationary” regime for v → vc Nt Nt+t Gu,v ,t (x , t) = Ex u xi (t) v xj (t + t ) i=1 j=1 also satisﬁes the KPP equation !... Spatial extension : L ∝ (vc − v )−1/2 Eqs (ρ(x )) x L D. Simon () Branching Random Walks Eindhoven, March 25th 2009 9 / 17 − “Quasi-stationary” regime for v → vc Nt Nt+t Gu,v ,t (x , t) = Ex u xi (t) v xj (t + t ) i=1 j=1 also satisﬁes the KPP equation !... Spatial extension : L ∝ (vc − v )−1/2 Eqs (ρ(x )) Quasi-stationary size : 1 K 3 A 2 Eqs (N) ∼ 3 e γc L ∼ K (vc − v ) 2 exp L 2(vc − v ) x L D. Simon () Branching Random Walks Eindhoven, March 25th 2009 9 / 17 − “Quasi-stationary” regime for v → vc Nt Nt+t Gu,v ,t (x , t) = Ex u xi (t) v xj (t + t ) i=1 j=1 also satisﬁes the KPP equation !... Spatial extension : L ∝ (vc − v )−1/2 Eqs (ρ(x )) Quasi-stationary size : 1 K 3 A 2 Eqs (N) ∼ 3 e γc L ∼ K (vc − v ) 2 exp L 2(vc − v ) x Conjecture: x = N/Eqs (N) has distrib. e −x L (reminder: Galton-Watson xe −x ) D. Simon () Branching Random Walks Eindhoven, March 25th 2009 9 / 17 Limiting the size t t +1 but limited resources −→ constant size N D. Simon () Branching Random Walks Eindhoven, March 25th 2009 10 / 17 Limiting the size t t +1 but limited resources −→ constant size N Uniform choice t t +1 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 10 / 17 Limiting the size t t +1 but limited resources −→ constant size N Uniform choice Choice of the best N ind. t t t +1 t +1 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 10 / 17 Position of the population Choice of the N best ind. Uniform choice A 3A ln ln N E (vN ) vc − 2 − 2 ln N ln3 N E (vN ) = 0 1 DN ∝ 3 ln N DN 1 Brunet-Derrida 07 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 11 / 17 Position of the population Choice of the N best ind. Uniform choice A 3A ln ln N E (vN ) vc − 2 − 2 ln N ln3 N E (vN ) = 0 1 DN ∝ 3 ln N DN 1 Brunet-Derrida 07 to be compared with Wall at v + Conditioning 1 3 A 2 Eqs (N) ∝ (vc − v ) exp 2 2(vc − v ) S.-Derrida 08 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 11 / 17 Position of the population Choice of the N best ind. Uniform choice A 3A ln ln N E (vN ) vc − 2 − 2 ln N ln3 N E (vN ) = 0 1 DN ∝ 3 ln N DN 1 Brunet-Derrida 07 Questions : to be compared with Stochastic KPP vs. Wall at v + Conditioning KPP with boundaries 1 3 A 2 Eqs (N) ∝ (vc − v ) exp 2 universality/robustness ? 2(vc − v ) other similarities ? S.-Derrida 08 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 11 / 17 Genealogies and coalescence times T T3 T2 D. Simon () Branching Random Walks Eindhoven, March 25th 2009 12 / 17 Genealogies and coalescence times T T3 T2 from t < 0 towards t > 0 : branching-diﬀusion from t > 0 towards t < 0 : coalescence D. Simon () Branching Random Walks Eindhoven, March 25th 2009 12 / 17 A wider class of models... Ingredients : evolutive advantage x better reproduction for large x heredity of x mutations Models : D. Simon () Branching Random Walks Eindhoven, March 25th 2009 13 / 17 A wider class of models... Ingredients : evolutive advantage x better reproduction for large x heredity of x Constant size N t mutations Models : branching random walks with t +1 selection D. Simon () Branching Random Walks Eindhoven, March 25th 2009 13 / 17 A wider class of models... E3 E4 Ingredients : 3 4 evolutive advantage x E = mini (Ei + i ) better reproduction for large x 1 2 heredity of x E1 E2 mutations x ↔ energy Models : disorder ↔ mutations i branching random walks with selection directed polymers, polynuclear growth (PNG) D. Simon () Branching Random Walks Eindhoven, March 25th 2009 13 / 17 Inﬂuence of the selection N individuals at time t0 Partition of the population at t0 + ∆t according the ancestor at t0 Distribution of the fractions xi = Ni /N after ∆t ? D. Simon () Branching Random Walks Eindhoven, March 25th 2009 14 / 17 Inﬂuence of the selection N individuals at time t0 Partition of the population at t0 + ∆t according the ancestor at t0 Distribution of the fractions xi = Ni /N after ∆t ? Selection =⇒ some xi are dominant... This hierarchy can be seen in the trees D. Simon () Branching Random Walks Eindhoven, March 25th 2009 14 / 17 Ratios E (Tn ) /E (T2 ) when N → ∞ T T 2 T3 Neutral Selection Unif. choice, Voter Branching Directed (no advantage) walks polymers Dim. d 1 ≥ 2 & MF MF MF d =1 d =2 E (T3 ) E (T2 ) E (T4 ) E (T2 ) D. Simon () Branching Random Walks Eindhoven, March 25th 2009 15 / 17 Ratios E (Tn ) /E (T2 ) when N → ∞ T T 2 T3 Neutral Selection Unif. choice, Voter Branching Directed (no advantage) walks polymers Dim. d 1 ≥ 2 & MF MF MF d =1 d =2 E (T3 ) 4 5 = 1.25 E (T2 ) 3 4 E (T4 ) 3 25 1.39 E (T2 ) 2 18 Kingman, Cox, Brunet et al. 07 Limic-Sturm D. Simon () Branching Random Walks Eindhoven, March 25th 2009 15 / 17 Ratios E (Tn ) /E (T2 ) when N → ∞ T analytical T 2 T3 Neutral Selection Unif. choice, Voter Branching Directed (no advantage) walks polymers Dim. d 1 ≥ 2 & MF MF MF d =1 d =2 E (T3 ) 7 4 5 = 1.25 E (T2 ) 5 3 4 E (T4 ) 8 3 25 1.39 E (T2 ) 5 2 18 Kingman, Cox, Brunet, Derrida, S. 08 Brunet et al. 07 Limic-Sturm D. Simon () Branching Random Walks Eindhoven, March 25th 2009 15 / 17 Ratios E (Tn ) /E (T2 ) when N → ∞ T analytical T 2 T3 numerical Neutral Selection Unif. choice, Voter Branching Directed (no advantage) walks polymers Dim. d 1 ≥ 2 & MF MF MF d =1 d =2 E (T3 ) 7 4 5 = 1.25 1.25 1.36 1.29 E (T2 ) 5 3 4 E (T4 ) 8 3 25 1.39 1.39 1.53 1.43 E (T2 ) 5 2 18 Kingman, Cox, Brunet, Derrida, S. 08 Brunet et al. 07 Limic-Sturm D. Simon () Branching Random Walks Eindhoven, March 25th 2009 15 / 17 Ratios E (Tn ) /E (T2 ) when N → ∞ Neutral Selection Unif. choice, Voter Branching Directed (no advantage) walks polymers Dim. d 1 ≥ 2 & MF MF MF d =1 d =2 E (T3 ) 7 4 5 = 1.25 1.25 1.36 1.29 E (T2 ) 5 3 4 E (T4 ) 8 3 25 1.39 1.39 1.53 1.43 E (T2 ) 5 2 18 Brunet, Derrida, S. 08 Neutral and Mean Field : Kingman Coalescent Selection and Mean Field : Bolthausen-Sznitman Coalescent (spin glasses, etc.) intermediate cases ? D. Simon () Branching Random Walks Eindhoven, March 25th 2009 15 / 17 Conclusion branching random walks inﬂuence of the boundary conditions on the survival relation with the propagation of non-linear travelling waves impact of the selection universality/robustness in the tree structure (BRW, directed polymers) D. Simon () Branching Random Walks Eindhoven, March 25th 2009 16 / 17 Conclusion branching random walks inﬂuence of the boundary conditions on the survival relation with the propagation of non-linear travelling waves impact of the selection universality/robustness in the tree structure (BRW, directed polymers) Questions transition between the regimes neutral/selection (intermediate between “best N” and “uniform choice”) ? what about other models and other classes, besides Kingman and Bolthausen-Sznitman in presence of a ﬁtness coordinate ? D. Simon () Branching Random Walks Eindhoven, March 25th 2009 16 / 17 Some references Branching random walks with absorbing boundaries B. Derrida & D. S., Survival probability of a branching random walk in presence of an absorbing wall, EPL, 78 (2007) D. S. & B. Derrida, Quasi-stationary regime of a branching random walk in presence of an absorbing wall, J. Stat. Phys., 131 (2008). Coalescence times and selection ´ E. Brunet, B. Derrida, A.H. Mueller, S. Munier, Noisy traveling waves : eﬀect of selection on genealogies, Europhys. Lett., 76 (2006), 1. ´ E. Brunet, B. Derrida, D. S., Universal tree structures in directed polymers a and models of evolving populations, Phys. Rev. E, 78 (2008), ` paraˆ ıtre. D. S. & B. Derrida, Evolution of the most recent common ancestor of a population with no selection, J. Stat. Mech. (2006) P05002. D. Simon () Branching Random Walks Eindhoven, March 25th 2009 17 / 17

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