HAL Id: hal-00453846
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Submitted on 5 Feb 2010
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Hybrid genetic algorithm for dual selection
F. Ros, Serge Guillaume, M. Pintore, J. Chrétien
To cite this version:
F. Ros, Serge Guillaume, M. Pintore, J. Chrétien. Hybrid genetic algorithm for dual selection. Pattern Analysis and Applications, Springer Verlag, 2008, 11 (2), p. 179 - p. 198. �10.1007/s10044-007-0089-3�.
�hal-00453846�
HYBRID GENETIC ALGORITHM FOR DUAL SELECTION
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Abstract
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Keywords* ' ' ' '' * 34 '
1. Introduction
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2. Related work and contribution
2.1.1 Feature selection
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2.1.2 Editing techniques and outlier detection
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"7.4 . <* *B= 11 ' & ".# . &-/ / 4 " '-/("(/&! '# . <*0 *F= .4 ("!/ !/'/.&
46 .'/# . <*0= &-"75- -75/ .7(3/! " 1 1/!# 4/ .5 8 &- (/(/& ' 5"! &-(#
!'- &/'&7!/# .4 4/# 5. 1! .' 1 /# - 6/ 11/ !/4 . &-/ #& ) 2/ !# &-/ 4 6/!# &2 "
5"! &-( ' 4/# 5. #1 '/ /%1 "!/4 # !/ & 6/ 2 #( <*,= .2 " &-/( !/ &"" & (/
'".#7( .5 # &-/2 !/;7 !/ '".# 4/! 3/ &7. .5" &-/ "' #/ !'- .4 /6" 7& ". !2 1 !&# "
&-/ 5"! &-( &-"75- &-/ 1- "#"1-2 " (/(/& ' 5"! &-(# # 8 2# &-/ # (/ & 11/ !#
&- & / '- 1 !& '7 ! 11 ' & ". !/;7 !/# &# "8. (/(/& ' 5"! &-( . <,)= "! /% (1 / (/(/& ' 5"! &-(# !/ 1!/#/.&/4 "! &-/ ! 6/ .5 /#( . !"3/( D E 7 4! & '
## 5.(/.& !"3/( D E . (7( ! 1- " "! .5 !"3/( D E .4 "! !"&/ .
&!7'&7!/ !/4 '& ". D E
2.1.4 Dual selection
-/ 1 !& '7 ! 4"73/ / &7!/ #/ /'& ". 1!"3/( #&74 /4 . &- # 1 1/! ' . 3/ 6/8/4 #
#/ !'- 1!"3/( 8-/!/ &-/ '- /.5/ # &" "3& . &-/ - 5-/#& ' ## ' & ". 1/! "( .'/ 8 &- &-/
3/#& 4 & #73#/& & ' . 3/ &-/. #//. # #1/' ' (7 & "3:/'& 6/ 1!"3/( H - / &! 4 & ".
( &-/( & ' 11!" '-/# " /! 6 ! /&2 " #" 7& ".# /6" 7& ". !2 5"! &-(# #//(
1 !& '7 ! 2 #7 & 3/ &" #" 6/ (7 & "3:/'& 6/ "1& ( 9 & ". 1!"3/(# 3/' 7#/ &-/2 4/
# (7 & ./"7# 2 8 &- #/& " 1"## 3/ #" 7& ".# . # .5/ !7. " &-/ 5"! &-( .7(3/! "
(7 & "3:/'& 6/ /6" 7& ". !2 5"! &-(# - 6/ 3//. #7''/## 7 2 !/1"!&/4 . &-/ &/! &7!/ "!
#/6/! 2/ !# -/ &/'-. ;7/# ' . 3/ ' ## /4 . ". !/&" .4 !/&" /'-. ;7/# <, =
". !/&" &/'-. ;7/# 4" ."& 4 !/'& 2 .'"!1"! &/ &-/ '".'/1& " !/&" 1& (7( .4 !/
5/./! 2 / ' /.& 37& 3/&&/! 4 1&/4 &" ( . 5.5 ". 2 /8 "3:/'& 6/# -/ ("#& 8/
>."8. !/ &-/ 8/ 5-& .5 11!" '-/# 8- '- ( & '"(3. .5 &-/ "3:/'& 6/# .&" # .5/
"3:/'& 6/ 32 55!/5 & ". -/ I (/&-"4 <, = 1!"1"#/4 32 '- /! '".# #&# .
#&! & 2 .5 &-/ 1"17 & ". . #/6/! #731"17 & ".# / '- - 6.5 4 /!/.& "3:/'& 6/# &"
( . 5/ # (/.& "./4 . <, = &- # (/&-"4 # '! &/! ". 3 #/4 11!" '- 8-/!/ / '- & (/
'-!"("#"(/ # '-"#/. "! !/1!"47'& ". 1"&/.& 2 4 /!/.& "3:/'& 6/ 8 4/' 4/ 8- '- (/(3/! " &-/ 1"17 & ". 8 3/ '"1 /4 .&" &-/ ./8 1"17 & ". /'-. ;7/# 8- '- 4 !/'& 2 .'"!1"! &/ &-/ '".'/1& " !/&" 1& (7( 8/!/ !#& 1!"1"#/4 32 " 43/!5 <,+= "(/
, 11!" '-/# 7#/ "! /% (1 / &-/ 4"( . .'/ ! .> / &-/ .7(3/! " '-!"("#"(/# 32 8- '- '-!"("#"(/ # 4"( . &/4 &" 4/&/!( ./ &-/ &./## 6 7/ 6/!# &2 .4 / & #( 1!/#/!6 & ".
!/ '/.&! . (7 & "3:/'& 6/ 1!"3/(# .4 &-/ ("#& !/'/.& .4 8 4/ 2 7#/4 11!" '-/# <,B=
.&/5! &/ &-/#/ #1/'&# # &-/ 1!"3/( " #/ /'& .5 # (7 & ./"7# / &7!/# .4 .#& .'/
1 &&/!.# !/ /##/.& 2 - .4 /4 32 &-/ 55!/5 & .5 11!" '- 8/ ( & &- # !/6/8 &" &-/ ("#&
' "#/ 2 !/ &/4 (/&-"4# .4 / 6/ &-/ !/ 4/! .&/!/#&/4 . &- # 5"3 !/ &" !/ 4 4/4 ' &/4 1 1/!# D // <,0 ,*= "! &7&"! #L#7!6/2# .4 <))= "! ("!/ #1/' ' 1 1/!E 1/!# !/ &/4
&" &-/ # (7 & ./"7# #/ /'& ". " / &7!/# .4 .#& .'/# !/ 6/!2 /8 . .7(3/! .4 7.;7/#& ". 32 &-/ ("#& ("7# !/ &-"#/ 32 > ><) = .4 K7.'-/6 <) = " &-/(
( &" 4/# 5. "1& ( ./ !/#& ' ## /!# & # 8"!&- (/.& ". .5&- & "7! "3:/'& 6/ # 4 /!/.&
# 8/ ( & .4 .5&-/ !5/#& 1 &&/!. #/& 8- '- 8 1!/4 '& &-/ 3 &2 " &-/ 4 & 3 #/ &"
4 #'! ( . &/ &-/ ' ##/# .4 ."& &-/ #( /#& 1 &&/!. #/& '".& . .5 '! & ' .#& .'/# "!
' ## ' & ". -/2 4" -"8/6/! 44!/## # ( ! 4"73/ #/ /'& ". 1!"3/( . &/!(# "
1!"3/( '"(1 /% &2 .4 !/ &-/!/ "!/ '".# 4/!/4 # &-/ ' "#/#& !/ /!/.'/# "! &- # 1 1/! H / - 6/ !/ 42 (/.& "./4 &-/ 8"!> " > > . # - # 4/ " 1/! "!( .5 #/ /'& ". 32
! .4"( (7& & ". - ' (3.5 ' . #/!6/ "! #/ /'& .5 1!"&"&21/# D E .4 / &7!/#
#/1 ! &/ 2 "! # (7 & ./"7# 2 D E 6/. &- # #" 7& ". 11/ !# .'"(1 /&/ # &-/!/
# ." (/'- . #( &" 4! 6/ &-/ . # 9/ " &-/ 4 /!/.& #/&# &-/ !/#7 &# "3& ./4 !/
(1!/## 6/ !/ &/4 &" &# # (1 ' &2 K7.'-/6 <) = - # 1!"1"#/4 . & 8"!> "! &-/
# (7 & ./"7# /4 & .5" 1 &&/!.# .4 #/ /'& .5" / &7!/# 32 55!/5 & .5"3:/'& 6/# ? #/4 ".
# ( ! 1- "#"1-2 -/! 8"!>- # / 4 &" "&-/! #&74 /# #7'- # &-"#/ 32 Q " <) = .4 6/!2 !/'/.& 2 ." <)+= .4 -/. <)B=8- '- .'"!1"! &/# . .&/ 5/.& '!"##"6/! &"
(1!"6/ &-/ 1"17 & ". 4 6/!# &2 .4 11 2 ."&-/! (7 & "3:/'& 6/ 11!" '- . K7.'-/6
<) = &-/ 5" 8 # &" 4/# 5. "1& ( ./ !/#& ./ 5-3"! ' ## ' & ". 32 ( . ( 9.5&-/ # 9/#
" 3"&- 1!"&"&21/ .4 / &7!/ #/&# -/ 11 /4 #& .4 !4 8/ 4/# 5./4 &" 1/! "!(
/4 & ". .4 #/ /'& ". . # .5/ #&/1 .4 #-"8/4 &- & & '"7 4 '- /6/ 5""4 '"(1!"( #/
8 &- 3"&- - 5- 4 #'! ( . & ". .4 ("4/! &/ .7(3/! " / &7!/# -/ !/#7 &# 1!/#/.&/4 8/!/
3/&&/! 32 ! &- . &-/ "&-/! '"(3. & ".# " &/#&/4 11!" '-/# 8-/!/ #/ /'& ". .4 /4 & ".
8/!/ 11 /4 . 4 /!/.& #&/1# ''"!4 .5 &" &-/ &/! &7!/ !/#7 &# 6 ! "7# 1!"( # .5 11!" '-/# 3 #/4 ". # #//( &" 3/ -/ 1 7 . ( . 5.5# (7 & ./"7# / &7!/ .4 .#& .'/
1 &&/!.# -/ 4! 83 '>" # !/( .# &-/ 4 '7 &2 " #/&& .571 .4 4! 6.5&-/ 5"! &-(
&" "3& . 5""4 #" 7& ".# . !/ #". 3/ & (/ /#1/' 2 8 &- !5/ 4 & 3 #/ &-"75- &-/#/
#1/'&# !/ '!7' "! 1! '& & "./!# &-/!/ # ." ' / ! 57 4/ "! .2 (/.& ". . &-/ &/! &7!/
3"7& -"8 &" #/& &-/ 5/./& ' 1 ! (/&/!# ."! 3"7& &-/ & (/ !/;7 !/4 "! &-/ 5"! &-(# &"
'".6/!5/ &"8 !4 5""4 #" 7& ".#
. &-# 1 1/! 8/ 1!"1"#/ &" &!/ &&-/ &8" "4 1!"3/( " /4&.5 .#&.'/ 1 &&/!.# .4
#//'&.5 / &7!/# # #.5/"1&( 9 &".1!"3/( 32&-/7#/" #1/' '-23!49/45/./&' 5"!&-( //'&.5"1&( / &7!/# # 3#"7&/2./'/##!2 "!' ## ' &".17!1"#/#
. '& &-/1!/#/.'/" !!//6 .&"!!/47.4 .& / &7!/#( 2'". 7#/&-/ / !..5 5"!&-(
.4 / 4 &" 3 4 ' ## ' &". 1/! "!( .'/# "!/"6/! &-/!/ #." (357&2 &" #2 &- &
!/47'.5 &-/ "!5. &! ..5 32 !/("6.5 @3 4 .#&.'/#@ # >/2 &" .'!/ #/ &-/
' ## ' &".1/! "!( .'/# #&-/1!"3/(#" .#&.'/!/47'&". .4 / &7!/#//'&". !/
."& 8 2# .4/1/.4/.& 8/ 1!"1"#/ 8 2 &" - .4/ &-#4"73/ 1!"3/( &' # #.5/
1!"3/( "!( 9/4 # " "8#
*
/& ROSR G RT3/ &-/ #/& " / &7!/# 4/#'! 3 .5 "3:/'&# # . 4 (/.# ". 6/'&"!# .4 JOS9 G 91T 9:∈ 3/ &-/ 4 & #/& ##"' &/4 8 &- / '- &/( 9: : O G 1 # ' ## 3/
) !"( &-/ #/& OS G T 6/. ' ## /! &-/ "3:/'& 6/# " 4 & !/47'& ". .4 / &7!/
#/ /'& ". !/ &" .4 #73#/&# ⊆ R .4 ⊆ J #7'- &- & &-/ ' ## ' & ". ''7! '2" # ( %( .4 & &-/ # (/ & (/ "1& ( 9/ &-/ # 9/# " &-/ !/47'/4 #/&# &" - 6/ U U ( . ( .4 U U ( %( 8-/!/ U U 4/."&/# ' !4 . &2 -/ "!(7 & ". # &-/. &-/ " "8 .5
.4 .4 . &-/ '"(3 ./4 #1 '/ &" ( . 5/ &-!// 4 /!/.& "3:/'& 6/# . &-/ # (/
5"! &-( #7'- &- &
D E # ( %( D E
U U # ( %(
U U # ( . (
" #" 6/ &- # 1!"3 /( 8/ 1!"1"#/ -23! 4 - 6 .5 &-/ 4"73 / "3:/'& 6/ " !/47' .5 D/% (1 /#E .4 #/ /'& .5 D / &7!/#E 8- / !/ '- .5 &-/ - 5-/#& ' ## ' & ". #'"!/ ?2 4 1& .5&-/ '-!"("#"(/ #&!7'&7!/ # ' . .&/5! &/ / &7!/ #'-/(/ 3 / &" #/ /'& 1""
" / &7!/# !"( (7& 4 (/.# ". #1 '/ .4 & &-/ # (/ & (/ 1 &&/!. #/ /'& ". #'-/(/
-/ 1!/#/.&/4 . &- # 1 1/! .&/5! &/# 4/4 ' &/4 -/7! #& '# .4 (/'- . #(# "! &-/ 47
#/ /'& ". 1!"3 /( " &-/ 3/#& " "7! >."8 /45/ &-/!/ 11/ !# &" 3/ ." !/1"!&/4 (/&-"4 8- '- # (7& ./"7# 2&!/ &# &- # 4"73 / 1!"3 /( " .#& .'/# !/47'& ". .4 / &7!/# #/ /'& ".
&" '- /6/ &- # 1 !& '7 ! "3:/'& 6/ -/ !/ 4/! #-"74 "3#/!6/ &- & &-/ 11!" '- # &/'-. ' 2 ' "#/ &" &- & " (/&-"4# 4/6"&/4 &" 4/# 5. .5./ !/#& ./ 5-3"! ' ## /!# 37& &-/ -/7! #& '#
.4 (/'- . #(# .&!"47'/4 &" ( >/ &-/ (/&-"4 / ' /.& .4 1! '& ' !/ 4 /!/.& .
#/'& ". 8/ &-/. 1!/#/.& &-/ -23! 4 .4 #1/' ' 2&-/ 4 /!/.& -/7! #& '# (1 /(/.&/4
&" #" 6/ &-/ 47 #/ /'& ". 1!"3 /(
3. The hybrid algorithm
-/ 8-" / 1!"'/47!/ # ( 4/ 71 " &8" 4 #& .'& #&/1# & # #7(( ! 9/4 . &-/ 4 5! ( " 5
-/ !#& "./ 8- '- ' . 3/ ' /4 1!/ ( . !2 1- #/ # 17!/ -/ 5" # &"
1!"("&/ 4 6/!# &2 8 &- . &-/ '-!"("#"(/ 1"17 & ". . "!4/! &" !/("6/ &-/ 7.7#/4 / &7!/# .4 &" 1!/1 !/ &-/ #/'".4 #&/1 ' /4 &-/ '".6/!5/.'/ 1- #/ -/. &-/ "3:/'& 6/ #
&" .4 #/& " 1"## 3 / #" 7& ".# .#&/ 4 " 4 6/!# &2 .&/!. (/'- . #(# !/ .&!"47'/4 &"
6"7! / & #( .4 #"(/ "' &7. .5 # '"(3 ./4 8 &- &-/ 47! .5&-/ '".6/!5/.'/ 1- #/
. &- # 1- #/ '"(17& .5!/#"7!'/# 4/4 ' &/4 &" "' &7. .5 !/ 1!"5!/## 6/ 2 .'!/ #/4
& #-"74 3/ ."&/4 &- & &-/ &! .# & ". 3/&8//. &-/ 1!/ ( . !2 .4 &-/ '".6/!5/.'/ 1- #/ # 7&"( & '
!/#/!6 .53"&- / & #( .4 4 6/!# &2'".#& &7&/# &-/ ( . '- /.5/ "!
-/ ( " "7! 1 !& & ". .5 1- #/ # !#& 2 &" /.'"7! 5/ 4 6/!# &2 .4 #/'".4 2 / & #(
&-!"75- &-/ '-" '/ " >."8. 5/./& ' 5"! &-(# -/ ( . 5/(/.& " 4 6/!# &2 .4 / & #( #
#" ( . 5/4 .# 4/ / '- 1- #/ H / - 6/ .'"!1"! &/4 &8" (/'- . #(# / . !'- 6/
1"17 & ". .4 3!/ >.5(/'- . #( . "!4/! &" 7&" 3 .'/ 4 6/!# &2 .4 / & #(
-/ !'- 6/ 1"17 & ". # 7#/4 # !/1"# &"!2" #" 7& ".# 1!"6 4/# . /%&! #"7!'/ " !/#7&#
.4 6"7!# ("!/ / & #(
'- & (/ # 5. " 1!/( &7!/ '".6/!5/.'/ # 4/&/'&/4 . &-/ '7!!/.& 1"17 & ". &-/ 3!/ >.5 (/'- . #( .&/5! &/4 8 &- &-/ ( . "3:/'& 6/ " 1!/6/.& .5 1!/( &7!/ '".6/!5/.'/
/.'"7! 5/# 4 6/!# ' & ". 32!/ #//4 .5#/ /'&/4 '-!"("#"(/#
-/ & (/ / &7!/ # . /##/.& '&"! "! &-/ 7#/ " #
23! 4 & ". 8 &- "' 11!" '-/# ' . ;7'>2 3/'"(/ 7.1! '& ' "#& " &-/ >."8.
(/(/& ' 11 ' & ".# 4/ 8 &- !/ & 6/ 2 #( #2#&/(# .7#/4 .4 8"!#/ / &7!/# !/
!/("6/4 & &-/ /.4 " &-/ !#& 1- #/ . "!4/! &" 6" 4 .//4 /## 2 -/ 62 ' '7 & ".#
7!&-/!("!/ "' 11!" '-/# !/ .'"!1"! &/4 . #7'- 8 2&- & '"(17& & ". '"#&#
!/ 6" 4/4 .4 &-/ !/ 3 &2 " &-/ !/#7&# /.- .'/4 # #-"8. . &-/ 4 5! ( #/6/!
@&! '>#A !/ .'"!1"! &/4 &" !/47'/ 3"&- &-/ .7(3/! " #" 7& ".# &" 8- '- "' #/ !'- # 11 /4 .4 &-/ .7(3/! " .#1/'&/4 '-!"("#"(/ '"(1"./.&#
-/ !#& #73#/'& ". 5"/# .&" &-/ 4/& # 8- / &-/ #/'".4 # 4/4 ' &/4 &" &-/ -23! 4 '"(1"./.&
3.1 The Genetic algorithm
Chromosome:# &-/ "1& ( 9 & ". 1!"'/47!/ 4/ # 8 &- &8" 4 #& .'& #1 '/# &-/ / &7!/ #1 '/ .4 &-/ 1 &&/!.
#1 '/ 3"&- !/ ( . 5/4 32 &-/ '-!"("#"(/ !/1!/#/.&# &-/ 8-" / #" 7& ". & # /.'"4/4 # #&! .5" 3 &# 8-"#/ /.5&- # N1 3/ .5&-/ .7(3/! " 6 3 / / &7!/# .4 1
&-/ .7(3/! " 1 &&/!.# . &-/ &! . .5#/& . '-!"("#"(/ "! &-/ / &7!/ "! 1 &&/!.
#& .4# "! &# #/ /'& ". 8- / ) (/ .# & # ."& & >/. .&" ''"7.&
# &-/ .7(3/! " / &7!/# # >/ 2&" 3/ #( /! &- . &-/ .7(3/! " 1 &&/!.# . "!4/! &" #1//4 71 &-/ 1!"'/47!/ .4 &" (1!"6/ &-/ /%1 "! & ". 1"8/! " &-/ 5"! &-( &-/ &8" #1 '/# !/
( . 5/4 .4/1/.4/.& 2 & / '- &/! & ". 32 &-/ 5/./& ' "1/! &"!# #7'- # '!"##"6/! .4 (7& & ". - # (/ .# &-/ 8-" / '-!"("#"(/ # &-/ 7. ". " &8" 4 #& .'& #73'-!"("#"(/#
&-/ !#& "./ &" /.'"4/ &-/ / &7!/ #1 '/ .4 &-/ #/'".4 "./ &-/ 1 &&/!. #1 '/ . / '-
#73'-!"("#"(/ ' ## ' "./ 1" .& '!"##"6/! # 11 /4 & 1!"'/##/# .# 4/ '"(1 ! 3 / / 4# - # 8 24" .5 # >/ 2&" 2/ 4 3/&&/! /%1 "! & ". .4 !/#7&#
H / #71/! (1"#/ #"(/ !/#&! '& ".# "! '-!"("#"(/ &" !/1!/#/.& 6 4 #" 7& ". -/ !#&
"./ # "36 "7# &-/ .7(3/! " #/ /'&/4 / &7!/# # ."& 9/!" U U≥ &-/!8 #/ ." .17& #1 '/
8"74 3/ 4/ ./4 -/ "&-/! '".4 & ". (# & /.#7! .5&- & &-/ ' ##/# !/ ( . 5/4 32&-/
#2#&/( 8- &/6/! &-/ ! ' !4 . &2 -/ .7(3/! " 1!"&"&21/# " 56/. ' ## - # &" 3/
5!/ &/! &- . 4/ ./4 1!"1"!& ". !/;!/1 H &-"7& &- # >.4 " '".#&! .& 3 4 !/#7&# 8 &-
#( ' ##/# '"74 3/ '"(1/.# &/4 325""4 !/#7&# 8 &- !5/! "./#
. &-/ ' #/ " !5/ 6/'&"! # 9/# #7'- # &-"#/ "7.4 . &-/ '-/("(/&! ' / 4 &-/ ! .4"(
#" 7& ". # ."& 6/!2 11!"1! &/ # ."& ". 2&-/ & (/ & >/. 32&-/ 1!"'/## .'!/ #/# 37& &-/
1/! "!( .'/ " 4/# 5./4 ' ## /!# 8 ."& 3/ 57 ! .&//4 -/. &-/ . & '-!"("#"(/# !/
."& 5/./! &/4 . '"(1 /&/ 2 ! .4"( 8 2 &-/ .7(3/! " '& 6/ 3 &# # ( &/4 "! 3"&-
#1 '/# -/ .&/!6 # !/ < 1 1= .4 < ( .D E=D&21 ' 6 7/# !/ O) O) , .4
O )E
Fitness function:
-/ '-" '/ " &-/ &./## 7.'& ". # " 1! (/ (1"!& .'/ . 4/# 5. -/ "./ 8/ 1!"1"#/
& >/# .&" ''"7.& &-/ &-!// '".&! 4 '&"!2 "3:/'& 6/# ( %( 9/ &-/ ' ## ' & ". !/#7&#
( %( #/ &-/ .7(3/! " 1!"&"&21/# .4 ( . ( 9/ &-/ .7(3/! " / &7!/# & # " '"7!#/
4/ ./4 "! 6 4 '-!"("#"(/# ". 2 3/ .5&-/ #/ /'&/4 ' ## /! &# . 2& ' /%1!/## ".
&" ( %( 9/ # # " "8#
= ) E
D * ' '
. λ* λ-
D E
Dλ λ1E '"(1/.# &/ / '- "&-/! .4 &-/2 ' . 3/ #//. # 1/. &2 &/!(# "! D E -/2 !/
!/#1/'& 6/ 2( %( "! ( . (7( " / &7!/# .4 ( %(7( " 1 &&/!.# .&!"47'/4 . &-/
'-!"("#"(/ -/2 !/ 4/# 5./4 . # ( ! 8 2λ1 D#// 5 E 4/1/.4# ". &8" 1 ! (/&/!#
V1 .4∆13/&8//. ) .4 V14/ ./# &-/ "8/#& 6 7/ "!λ1 .4∆1 # &-!/#-" 4
+
≤
<
=
/ #/
3 1 E D1
W 1 E D1 E
DV V X
1 1
1 1
1
1 D E
8-/!/ 1 # &-/ .7(3/! " 1 &&/!.# . &-/ '7!!/.& '-!"("#"(/ 1 .4 31 !/ ' '7 &/4 #" &- &
Dλ1E # 8-/. 1 L1 O .4 Dλ1E # V1 8-/. 1 L1 # ∆1 -/ 6 7/# " V1 .4 ∆1 - 6/ &" 3/
'-"#/. ' !/ 7 2 # &-/2 !/ !/1!/#/.& & 6/ " &-/ (1"!& .'/ 4/4 ' &/4 &" &-/ 1 &&/!. #/&
21 ' 6 7/# !/∆1O ) B .4 V1O ) ,B
. &-/ # (/ 8 2λ D#// 5 E 4/1/.4# ". &-!// 1 ! (/&/!# V ∆ ∆ V 4/ ./# &-/ "8/#&
6 7/ "!λ ∆ .4∆ D∆ P∆ E !/ &8" &-!/#-" 4#
+
≥
≤
=
/ #/
3 E D
W Y
W
X D+E
8-/!/ # &-/ .7(3/! " / &7!/# . &-/ '7!!/.& '-!"("#"(/ .4 3 !/ ' '7 &/4 #" &- &
DλE # 8-/. O∆ .4 DλE # V 8-/. &-/ .7(3/! " / &7!/# #∆ 21 ' 6 7/# !/∆ O ∆ O B .4 V O ) /%1!/## .5&- &λ 4/'!/ #/# ./ ! 2 ''"!4 .5&" .4 3
Population evolution:
"#& (/&-"4# #7'- # /&/!( . #& !"84 .5D E /#&! '&/4 "7!. (/.& / /'& ". D E .4 "&-/!# !/ '".& .7"7# 2 "">.5 "! 3 .'/ 3/&8//. / & #( .4 4 6/!# &2 . &-/ '7!!/.&
1"17 & ". H / 1!"1"#/ &" 7#/ &8" 4 #& .'& 1"17 & ".# 8 &- 4 /!/.& /6" 7& ". !7/# .4 ."
4 !/'& .&/! '& ". -/ !#& "./ # ' /4 &-/ '7!!/.& 1"17 & ". 1"1' &# /6" 7& ". # ( . 5/4 7# .5 ' ## ' 5/./& ' #'-/(/# D/ & #( E -/ #/'".4 "./ # ' /4 &-/
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