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Feedback control for nonmonotone competition models in the chemostat

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(1)Feedback control for nonmonotone competition models in the chemostat Gonzalo Robledo, Jean-Luc Gouzé. To cite this version: Gonzalo Robledo, Jean-Luc Gouzé. Feedback control for nonmonotone competition models in the chemostat. RR-5033, INRIA. 2003. �inria-00071551�. HAL Id: inria-00071551 https://hal.inria.fr/inria-00071551 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Feedback control for nonmonotone competition models in the chemostat Gonzalo Robledo — Jean–Luc Gouzé. N° 5033 3 December 2003. ISSN 0249-6399. ISRN INRIA/RR--5033--FR+ENG. THÈME 4. apport de recherche.

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(202) (  7 '. u. 2. u1 D+ l. 2. −. D. l. 1. −. +. S. S. ¼V`‰bYMriO¯i ¬D_go$[]^V`_ba_f•e6_fVXa4]^h     K    [ja k  K œ ¬DOd]'Y{h k O$aM_f]iOƒWl  [fa k  K ]iJ{O‡e„_bV`a4]^hKVXa VU 01 <  hiY{o#J]iJ„[]  xf K [fa k  / W  / W pfU 0 ) K K K pfUl6   W  / mM ) A RTF  / qx  /  •–_br'[jZXZ rj / K 01i œ ) A XZF !O‡J„[£¢fOz]iJ{[j]  K pŠ _fr. :  p / ) V  K  /  ) V K  V“•  pf K 6 ¬DOd]'Y{hƒ{YMV`Z k ]iJ{O hmlghm]iO$N   hmY{ƒ{hu]^V“]^Yg]iVXaM‰   ƒ4l     VXa/hmlghm]iOnN =–Q6?›œ } a{[jZX_f‰b_fY{hiZXlf W’xO ƒ{YMV`Z k ]iJMO hilghu]^O$N  l K hiYMƒ{hm]iV`]iYg]^V`a{‰   ƒWl  '   )  ' VX- a/hmlghm]i) OnN - =–6Q ?›œ ¬DOd]Y{h k O$a{_j]iO»ƒ4l   [fa k  K '  ) ]i' JM- O B¨6œ ) =¡- ~ ?P’V`]iJ riOneMZ\[fodO k ƒ4l  [ja k  r^Onhie6Ono›Ÿ ]^V`¢bO$ZXlfœ } a{[fZ`_b‰f_bY{hmZXl:’O k OnaM_j]^O ƒWl   [fa k  K  ]iJMO B¨6Uœ ' =9) j?’V“]^J r^O$eMZ\[f) o$O k ƒWl  ) [fa k  'r^Onhie6Ono›]^V`¢bO$ZXlfœ K O‡’V`ZXZ,N&[j—bOz]iJMO [bhihiYMNPeg]iVX_fa„h˜•–_br'hmlghm]iOnNPh   [fa k   i A ( D lF IKJMO•–_fZXZX_’V`aM‰SVXaMOn¨4Y{[fZ`V`]iVXOnhJ{_fZ k i /T : v <  sV pŒ p /Z : v  s/50 ) K K : : /T v <  sVqpf p /Z v  K s/56 ) A ( R lF B¨Wh$œ   [ja k  K  J{[£¢fOw_faMOze„_4hmV`]iVX¢fOzhm_bZ`YM]iVX_fa  [ja k  K riOšhme6Onod]iVX¢fO$ZXlfœ B¨Whnœ  K  [ja k œ c_fr^O$_¢bO$r‡V`•Y: p :  ]^JMO$a   J{[£¢bO&_faMOe„_4hmV`]iVX¢fOhm_bZ`Yg]^V`_ba  K [fa k  riOšhme6) Onod]iVX¢fOnZ`lb- )     K [jr^OzVXa /%[ 01<Z [fa k     K [f) riOVXa VUl0u- %[, œ ) ) - . . . . . . . . . . . . . . >Ilt‘.

(203) 

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(205)  !". '<F. A ( X lF IKJMO•–_fZXZX_’V`aM‰SVXaMOn¨4Y{[fZ`V`]iVXOnhJ{_fZ k i. 

(206) :  /  2 ) y 2 -  p*  ) <#",& $$ #",& ++ •–_fr'[jZXZ  2 ) 052 -  j  0 =¡ QM? :  /  2 ) y 2 -  p*  )  <#",& $$ #"& ++  •–_br'[jZXZ  2 ) 052 -  jŽ6  A ( g lF IKJMO•–_fZXZX_’V`aM‰SVXaMOn¨4Y{[fZ`V`]iVXOnhJ{_fZ k i. 

(207) :  /  2 ) y 2 -  p*  - <#",& $$ #"& ++ •–_br'[jZXZ  2 ) 052 -  jŽ0 =¡ f~ ? :  /  2 y 2  p*    <  ",& $  ",& +  •–_fr'[jZXZ  2 052  j  6 ) ) $ + IKJMOn_fr^O$N ' VXNSe{Z`VXOnh†]iJ{[j]  0C2 0C2  [fa k / K 0C2 K 0C2 K  [friOhi_fZXYg]^V`_ba{h'‰bZ`_bƒ{[jZXZ`l[fhilWNSeM]i_j]^V“Ÿ on[jZXZ`l‡hm]^[fƒMZ`O_j•6hmlghm]iO$N&h   [ja k  l) K r^-Onhie„Ošo›]^V`¢bO$ZXlfœ/) _fr^O$_- ¢fOnr 2  [ja k 2 K =>=  ? 0u@ ?D[jr^O k O$¦{aMO k ƒW,l i : :  / ) /  s 0 2  : - v / ) /  : ) /<    s 2 )  ) v - /; :   !:  :  : ) K K K  / ) V  s 0 2 K  : - v / ) V !-  : ) )/   K  s 6 K2  : ) v : - /    :  : :  : ) ) K K %l[bhihiYMNPeg]^V`_ba A RZF ’OJ{[£¢fOw]iJ„[] 2 xŠ2 [ja k 2 xf2 - œ ) ) ) - ) - U7 <,WT>

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(210)  : 

(211)  : d .X26].7- 129` A DlF J A XiF -426I.7- 19)` /    )  =>=  ?01@ ' ?P) 1g@.Y\' - 70)+9 A ( D F J A ( g lF N9

(212)  26 1f729`* .cY!"

(213)  =–6Q ?X3

(214)  .Y+i 2  \<_V] ` ^ŠK!] a  2 )  7   \<_V] ^` K!a 2 )  7  „2 K 0 ) ) 2 -K  \<_V] ` ^„K!] a  2 -  7   \]_V^` K!a 2 -  7  f2 - 0 =¡ j? K  \  ] Š ^ ]    \ ] ^. 7 7   _V` K a    _V` K!a   M 6 

(215) O)`9

(216) 7( LWL\

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(221) f. 0

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(223) V) 65 3

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(228) Ye>13

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(230) -   ' d61- 426 d], 7

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(242)    61-  26\1 - , 7

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(245) <; 7+- 1YO . . . . .   "d:cd. IKJMO‡hilWhm]iOnN == QM?V\hhm_bZ`¢bO k aWYMNPOnriV\o$[fZ`ZXl:Y{hiV`a{‰P[Po$_fYMeMZXO‡_j• Q†[fZ k [faMOYMeg]#[j—fOz•–YMa„o›]iVX_fa„h1i.  6 @ [fa k  W /   m   @Yy„QyŠ SyŠ yŠ ') '%Q†O$a{o$Of DV`]‡•–_fZXZX_’'h†]^J{[] %[    @   ,[    [ja k %[  @ œ&IKJMOPe{[fr^[fNPOd]iOnr^hz_f•hilWhm]iOnN ’O$r^O ¦{pWO k [] <    T:  [fa k) :  ? œ - c_fr^O$_¢bO$rš M’Ood_ba{hiV k Onr']^JMO•=[fNPV`ZXlc_j•B•–OnO k ƒ{[bo#—o$_fa4]ir^_fZ ) Z\[£’'Kh i / V2 y 2 m Y

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(248) „ j

(249)  VU 6 ) l05U - 6  ? ( [fa k B¨Whnœ =¡~ ? Ÿ=2j?B[friOOš¨4YMV`¢[fZ`Onab] ]^_ i.

(250) m  Š@G?

(251) y l - y3  f?

(252)  h 

(253) 83U 0 

(254) W f 

(255) y  - y3C?  @

(256)  h 

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(258) q [fa k  

(259)  riOšhme6Onod]iVX¢fOnZ`lP[ja k A ( RTF ) V\hK¢fOnriV`¦{O k VXa/[ShiYMƒ{hiOd]'_f•

(260) œ . . . . . . . >Ilt‘.

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(263)  !". 7. 7. 6. 6. 5. 5. 4. 4. x. x. 2. 2 3. 3. 2. 2. 1. 1. 0. 0. 2. 4. 6. 8. 10. 12. 0. 14. 0. 2. 4. 6. 8. 10. 12. 14. x. x1. 1. 

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(265) GHJILKMNPOQSRT UVIKMNPOWXZY7R[5F*\+BB^]/_`2a4bF7_E F%c8\(de]fB^[hgF7E5Bji F7Ulk 7. 7. 6. 6. 5. 5. 4. 4. X2. X2 3. 3. 2. 2. 1. 0. 1. 0. 2. 4. 6. 8. X1. 10. 12. 14. 0. 0. 5. 10. 15. X. 1. mnopq rtseuv*wyx{z | }'~€‚ƒ„ w†x{z| }~€‡$ˆ‰5Š r‚‹4Šr7Œq r%Ž:s;‘n^’“ƒ Œ‰•”r%q nj– r7„—p‰•˜%Œ€r%™$n’(‰5r7ƒ ˜šrn’ '˜›Šnr%”re„ œp•r%q n˜7ž'’5n“pž^Ÿ‰ njŒ'ƒ ’@ /r%q r˜%q q njre„¡Œ'p$‰¢nƒ“£¥¤J‹h¦¢¤¡§Lp ’(nƒo©¨©ª©«J¬Z­4ƒ „©‰ Šr%®

(266) q rJ¯q r7’5r%ƒ€‰5re„ nƒ‚‰5Šr‘mno'’7°J±ˆ²•ƒ „Lseu°2³r‘„n’ ˜šp ’ ’4—q njr%´ ®*‰ Šr7’5rµq5re’(pžj‰ ’7°. ¶·¶¹¸Ÿº »½¼½¾½¾.

(267) b . 0

(268) (  7. ¼{_fr:e{[fr^[fNPOd]iOnr^h

(269)  Ul6  [ja k

(270) Ul6   x’O/o$[ja hiO$O/’V`]iJ ]iJMO/J{O$ZXež_j•zo$_fNPeMYg]^O$r&]^J{[] [bhihiYMNPeg]^V`_ba{h A ( XiF Ÿ A ( gF [jr^O‡¢fOnriV`¦{O k [fa k IKJMO$_briOnN 'on[jacƒ6O[jeMe{Z`VXO k [fa k ¼V`‰„œ hiJM_’'hK]^J{[]  [ 052 [) 0C2 -[  V\h‰fZX_fƒ„[jZXZ`l:[bhmlWNPeg]i_f]iV\o$[fZ`ZXl:hu]#[jƒMZXOfœ ¼{_frKe{[jr#[jNPO$]iO$r#h

(271) Ul6  [fa k

(272) k3U 6  @  W’Ozon[jahiO$Oz]^J{[]'[fh^hiYMNPeg]iVX_fa A ( XiF V\h¢fOnriV`¦{O k ƒMYg] a{_j] A ( giF œ ‰ ’V`a{‰&]i_ %[ j„%[ 0u%[  V`]†•–_bZ`ZX_’'h]^J{[]w]iJMOSo$[bhmOO ==V`V ?'_f•BIKJMO$_briOnN 'on[ja/ƒ6OeMeMZXV`O k [fa k ¼VX‰{dœ FPhiJM_’'hx]iJ{[j] /%[ 0C2[ ) 052i[ -  VXhK‰bZ`_bƒ{[jZXZXl:[fhil4NPeg]^_j]^VXon[jZXZ`lhm]^[fƒMZXOfœ ¼{_fr†e„[jr#[jNPOd]^O$r#h

(273) „#Ul6 )  [ja - k

(274) „#Ul6   M’Oon[ja™hiO$O‡]^J{[][fh^hmYMNPeg]^V`_ba{h A ( XZF Ÿ A ( giF [jr^O a{_j]¢bO$r^V“¦„O k [ja k IKJMO$_briOnN 'o$[ja{aM_j]Sƒ„O[jeMeMZXVXO k [fa k ¼V`‰„C œ 'n»hmJM_’'h ]iJ{[j] ,[ 052i[ 0C2i[  VXhaM_j] ) hm]^[fƒMZXOzƒ{Yg]]iJMOnriO‡V\h[SZ`VXNPV“]†o$lgodZXO [ja k ]iJMO‡Y{aMV“•–_briN e„Onr^hiVXhm]iOna{odO‡V\h[fo#J{V`On¢fO k œ A a9]iJMO¼VX‰{k œ 'n/’OhiO$O]iJ{[j]Y{hiVXaM‰

(275) [fhS[/ƒMV`•–YMr#o$[]^V`_ba e{[fr^[fNSO$]iOnrn •–_frP[»o$riV`]iV\o$[fZ˜¢[jZXYMO_f•

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