Polymorphic evolution sequence and evolutionary branching

HAL Id: inria-00345399

https://hal.inria.fr/inria-00345399v2

Submitted on 9 Mar 2016

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Distributed under a Creative Commons Attribution| 4.0 International License

Polymorphic evolution sequence and evolutionary branching

Nicolas Champagnat, Sylvie Méléard

To cite this version:

Nicolas Champagnat, Sylvie Méléard. Polymorphic evolution sequence and evolutionary branching.

Probability Theory and Related Fields, Springer Verlag, 2011, 151 (1-2), pp.45-94. �10.1007/s00440- 010-0292-9�. �inria-00345399v2�

∗

†

!

!

!

!

"

# $%

!

!" #$ %&'"" !"!

"#()

*"+ ,### -!" " "

#"". !$ "##!.#( "!"

# "&) /(#" "# #

$ #"#0 ,##"" ,1 "(

"( " # " # ,

∗

!"#$ %#

& ' ()*+ "*,-""."

†

/ 0 / #112 &)' ( )*+

/3"*-/ 0""

"" "") 2 ! !( #" !, #

1" # 3 4) 5 # &! " # "

! # "( #! " ,# ((" # "

# 6.)3 47) 8#+"# #

! " 6& 3 47) *##

" # 6)) # # "7 #, #

##,( #) 9# !!("

" " # ! " ) 2 #

, " ) 9# " " "

" " " "( ,# ) 9# #

) 5" # " " # !

# " ( #(#(# # "!"#( #)

8 "", , ( "(" " " !.) 3 :

))4 ,#" # "" #, ",##"

"!"#(, "" )

: # "# # " # "! ) :

" + #"! #" # 6;" <" 34

= 3 47) >" " # " " 6&

) 3 4 ?" >"# 34 ) 3 47) 5" "# 1")

8 # # # ( !

# ) 8 "( #, "! "#( "

!!6))"!,!70 ,""# "##

($"+("",##""!

"#( " " # )

* , # " ! # (!

"" "( ) @" #

, " ## " "+ ! "& "A

" # "#" ) 9#" # ""

(!#( #" %&'"" !) 9#!"#

, ! # , #

#") 8#""( #

"# "(!#"$"%&'""

! ,## # ! # . # "" (

# " " ") * # (". # " ! .

) 3 4 #! ! ;# ( 3B4 ,# # "" #

" #+, ") 8## " "(

" # 6 " "7 " A " 9" : :

$ 69::7) 9# " , # #

! "( # , ")

8 # " , "+ # +

! 6>/:7,( +""# "#

" " : )) : + " #,

: ) "" " # # "! "#() 8 :

),"# " %&'""!"(#

, ") * " : )B # # >/: & # "

"& A " # " #" X ^#"

" ,# ,(# , #"". #" # "

" #( #" %&'"" !) 8 : ,

+ ,#!# "( #"( #>/: "(

" # , " +) 8 # #! # >/: + !

#"".) + 6: B7 , ! # " " " #

" # ( # A # "

" " "!"#( " 3 4 "#

1) 9 # , " #, :B) # ,!" "! ("

# " # >/: ! " # "( 5/ &, #

C $D 6& %,347) E!:B) ,#"".

9#" B) # ,#"!"#("! .( "

#"F3 4#! #"

%&'"" !) 9# #" # ")

% " # # ) <" , " " ! #

# "( "" , !

) 2 " "# , " ! # "(

((# ("1" "1"$ "

#!"6;# ( )347G + !

#$,# " " (#,! #

"#" ) 9# "# "("#"!

"! ! & ) 34 " H

"<$,#"6?">"#347) 8#

"! "#( "! " ! !A) 2#"

"# , # #" , " "! # #

" ,( ! # "! " "

6?";# (347) 9#"$""!#

# "# ",) ## "#"

# ",# ,#"

#=3 4!# ) 9#!+ ,#!#""

1" " "6 "" # " "! "#(

,#" # " "#( ""7) 9# #, # I

"" (# (# #"!

"#()

% " #" # ,## " "# ( "

"( " # !" # $)

9#" ,,#! ,##

# "! (" "! +!!#" 3B 4)

9# " X R^l l≥1^{) E" !} x, y∈ X^,

" # ,( ( ""

λ(x)∈R+ ^{# " "# " #( "} x⁾

μ(x)∈R+ ^{# " C"D # " #( "} x⁾ r(x) :=λ(x)−μ(x) ^{#C"D (",# " "} x⁾

K ∈N "" (# . #"")

α(x,y)

K ∈R+ ^{# &"} " "(# ""!# ( "x ^{" #( "} y^{) 8 !"}

)

u_Kp(x) ^,# u_K, p(x) ∈ (0,1] ^{# "! # " "# "}

,#"x^{) :} u_K ^"")

m(x, h)dh ^{# , "} x+h∈ X^{"" ,#}

"x^{) 8 " #} {y−z:y, z ∈ X }⁾

* " ! t ≥0 ^" N_t ^{# # #(}

" X^{) % !} x₁, . . . , x_Nt ^{# " # )}

9# # t≥0 ^{" !}K ^{" ! #}

" X

ν_t^K= 1 K

Nt

i=1

δ_xi, ⁶⁾⁷

,#" δ_x ^{# " "} x^{) %} ν, f ^{# (" # "}

f ^{,#" #"}ν ^: (ν) ^")

9# ν_t^K,1 = ^N_K^{t " !} x ∈ X ^{# "} ν_t^K,1_{x}

t ^"x⁾

% M_F ^{# ( "} X ^{$ ,# # ,&}

(!

M^K = 1

K n i=1

δ_xi :n≥0, x₁, . . . , x_n∈ X

.

2 #( "x ^# ν_t^{K ("# #" ,#}

" λ(x) ^,#"

μ(x) +

α(x, y)ν_t^K(dy) =μ(x) + 1 K

Nt

i=1

α(x, x_i).

9# ""K^{#"(# #,( #+"}

# ) 2,"## " "(",#

"!1−u_Kp(x)^{,# "!}u_Kp(x)^#,",#"

y^#"( y=x+h^,#" h^"",#,m(x, h)dh⁾

8#" ," # "(ν_t^K, t≥0) M^{K"& ",#}

("" "! " φ ^"M^K R ^! L^Kφ(ν) =

X

φ

ν+δ_x

K

−φ(ν)

(1−u_Kp(x))λ(x)Kν(dx) +

X

R^l

φ

ν+δ_x+h K

−φ(ν)

u_Kp(x)λ(x)m(x, h)dhKν(dx)

+

X

φ

ν−δ_x

K

−φ(ν) μ(x) +

Xα(x, y)ν(dy)

Kν(dx). ⁶⁾⁷

E" ν ∈ M^{K # (" ,# "} Kν(dx) ⁶⁾⁷ "" "

# ) 9#""6"7"#"#,#

#"6"7"#"#,###""6"7

"# # ( " ) 9#! "!

# #" " # # # " #

")

% !627 #,( #"

627 λμ α ^{"" #" +}λ,¯ μ,¯ α <¯ +∞^{# #} λ(·)≤λ,¯ μ(·)≤μ¯ α(·,·)≤α.¯

627 r(x) =λ(x)−μ(x)>0 ^{" !}x∈ X^{#" +}α >0^## α ≤α(·,·)⁾

627 9#" + m¯ :R^l→ R+ ^{# #} m(x, h)≤m(h)¯ ^{" !}x ∈ X h∈R^l

¯

m(h)dh <∞⁾

E" + K^{"627 627 ( #} E(ν₀^K,1) <∞^#+

$ , " D(R+,M^K) ^,# ("" L^{K #}

" 3B4) 2 627 " # " + ( " (

+ )

9# (" "## " " # #"

! " ( " "! +) % " # #

"! "#( ,# + ,#" # "" #

" " " ""6J(#(" 34

& 3 47) 8 # #" ( " " #

"# " !" &") 9# "" "#,(0

X = [−2,2]; μ(x)≡0; p(x)≡p, λ(x) = exp(−x²/2σ²_b),

α(x, y) = ˜α(x−y) = exp(−(x−y)²/2σ_α²).

6)7

m(x, h)dh ^{# ,}N(0, σ²)^")) Y ^{6"K ,#"}σ²⁷

x+Y ∈ X⁾

9#(",#"λ(x)⁺ x= 0 ^{#" ,"}

# #α(x, y) ^{+ "} x=y ^,# |x−y|^{"() 8#}

&",I6α≡1^7,"",#+(",#

") <," "#" (

1" "#, E())) 8E() )67# !

" " " ( " " ! # "! "

,#" # , 6" "7 (" " "

1" " ) 9# # "! "#( # "

! ( 6 )() 3 47 ( ,# + " "

6 )() 3 47 " ,## # # #

( ! " 6 ,# ((" # "7)

45p= 0.1 K = 1000 σ= 0.01 σb = 0.9 σα= 1.0^"

465p= 0.1 K= 1000 σ= 0.01 σb= 0.9 σα= 0.7^"

E(" )0 & ' (

'( ')#(

K −1.0

8 # "" # ! "! "#( " "!

("!#σ_b σ_α^,##" " " !#,##"

"( ,##(#(",# " #" "() 8 E())67 σ_α > σ_b ^#"

"!"#( ,#" E())67 σ_α < σ_b ^{"! "#(}

") *" # #" # # " "

" " ( " # "( ) 8 # ,

" A , 1" ""0 #(#(",# " 6"

7 6" " "# " # 7) 8σ_α ^(##

, # " ,"#)

8 "" !. # # "! "#( , "(( "

#" ( ! # 0 "( 6K →+∞⁷

"" 6u_K → 0^{7 6}h ^{" !}εh ^,# ε → 0⁷⁾

9# # ( #" # "# ," " 3 4) 9#

# " , 6"( "" 7 , (

"( # " # !

6>/: 9#" ) : )B)7) 9# "( "" "+(#

! E())67 67!# E())67 67 " !)

# # "! p ^{# "! " E() )) 9#}

" ,#!σ ^{#" &# A # )}

: ,# σ = 0.01 p = 0.0001 ^{, #, A " #}

") 9#(#" #"(,#K → ∞u_K→0^#

,() E"!# "" ",(

6" !7"!6"70 # "

#H, ,(") :!#

"( ," !

, ### ") 9#

"! #"!#" #

, " !, " "" , # ) 9#

! ""( # ! "# #

" "( # )

9# >/: ! "! "#(0 : B) , (

"""!"#(#>/:6B) 7,# "

"#("" 69#"B) 7,## "" #"""

# + )

5#" " "( E())) 8E())67,"#

" " σ_α ^{" "! "#( " 1" ) 9#}

#, # "! "#( ! " " "! "! ,

"!" " " ") ##(#" ""!

"#( """ #""#( #>/: : )B)

, ,#" " #, C"#D +#

E() )67) 8 E() )67 # . A σ ^{! " ,#}

E() )67) 9# #, # # " #

"# # " "(#! ! B ,# " E() )67) 9#

, #! A 9#" B)B : B) ,#" , " # #

"# >/:,#"t/σ^{2 "(,#}σ →0^#

""! 1" $ &, # ! " )

2 # " #" "( #, , ! # ,

" " ! x₁, . . . , x_d ^{, , ) * " # !}

# " ,# 6u_K = 0 ^" K ≥1⁷⁾

: + x₁, . . . , x_d ∈ X ^{# "} i ∈ {1, . . . , d} ν₀^K,1_{xi} ^#

45 p = 0.0001 K = 1000 σ = 0.08 σb = 0.9 σα= 1.0^"

465 p = 0.0001 K = 1000 σ = 0.08 σb = 0.9 σα= 0.7^"

E(" )0 & ' (

'( ')#(

K −1.0

"" "( " n_i(0) ∈ R+^{) 9# "}

3 9#)B)4 ,# K → +∞ ^{# "} (ν_t^K,_{x1}, . . . ,ν_t^K,_{xd}) ^"(

" " # :&"# (! # # d

%&'"" !LV(d,x) ^,# (n1(0), . . . , n_d(0)) ^,)

# x = (x1, . . . , x_d) ∈ X^d$ LV(d,x)

%&'

n˙(t) =F^x(n(t)), t≥0, ^6)B7

n(t) = (n1(t), . . . , n_d(t))^$

F_i^x(n) :=n_iG^x_i(n) G^x_i(n) :=r(x_i)−^d

j=1

α(x_i, x_j)n_j. ⁶⁾⁷

9# $" LV(d,x) ^{" ( ! # " #! "} (P_i)_1≤i≤d ^,#"

P_i ^{# $ #"} n_i = 0 ^" G^x_i(n) = 0^{) % " # ,(}

+(")

45 p = 0.0001 K = 1000 σ = 0.08 σb = 0.9 σα= 0.4^"

465 p = 0.0001 K = 1000 σ = 0.04 σb = 0.9 σα= 0.7^"

E(" )0 & ' (

'( ')#(

K −1.0

#d≥0^$x₁, . . . , x_d ^"LV(d,x) ^!

! n¯(x)∈(0,∞)^{d $}

" ( ) " F^x n¯(x) ^{*+ , -}

$" i∈ {1, . . . , d}^$

G^x_i(¯n(x)) = 0 DF^x(¯n(x)) = ((−α(x_i, x_j)¯n_i(x)))_1≤i,j≤d. ⁶⁾⁷

5 x₁, . . . , x_d ∈ X ^{+ # "} y ∈ X ^{# "}

(" ! (",# ") 9# (",# " ( ! #

!

f(y;x) =f(y;x1, . . . , x_d) =r(y)−^d

j=1

α(y, x_j)¯n_j(x). ⁶⁾⁷

9# ! # $" %&'"" ! (" ! # ( #

)

. x₁, . . . , x_d∈ X ^{), /}

67 # i∈ {1, . . . , d}^$ f(x_i;x1, . . . , x_d) = 0^,