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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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