slow dynamics is described by means of a general nonYnegative matrix. For the ...... Some of the authors are partially supported by Proyecto de Investigación.
Acta Biotheoretica 49: 261–276, 2001. c 2001 Kluwer Academic Publishers. Printed in the Netherlands.
/,1($5',6&5(7(3238/$7,2102'(/6:,7+ 7:27,0(6&$/(6,1)$67&+$1*,1* (19,5210(176,$872120286&$6( ÈQJHO%ODVFR/XLV6DQ]3LHUUH$XJHUDQG 5DIDHO%UDYRGHOD3DUUD
'HSDUWDPHQWR GH 0DWHPiWLFDV 8QLYHUVLGDG GH $OFDOi $OFDOi GH +HQDUHV 0DGULG 6SDLQ(PDLODQJHOEODVFR#XDKHVFRUUHVSRQGLQJDXWKRU 'HSDUWDPHQWR GH 0DWHPiWLFDV (76, ,QGXVWULDOHV 8QLYHUVLGDG 3ROLWpFQLFD GH 0DGULG-RVp*XWLpUUH]$EDVFDO0DGULG6SDLQ 805 &156 8QLYHUVLWp &ODXGH %HUQDUG /\RQ %RXO 1RYHPEUH 9LOOHXUEDQQH&HGH[)UDQFH
$%675$&7 ,QWKLVZRUNZHFRQVLGHUDVWUXFWXUHGSRSXODWLRQZLWKJURXSVDQGVXEJURXSVRILQGLYLGXDOV 7KHLQWUDJURXSG\QDPLFVLVDVVXPHGWREHIDVWLQFRPSDULVRQZLWKWKHLQWHUJURXSG\QDPLFV :HVWXG\OLQHDUGLVFUHWHPRGHOVZKHUHWKHVORZG\QDPLFVLVUHSUHVHQWHGE\DVLQJOHPDWUL[DQG WKH IDVW G\QDPLFV LV GHVFULEHG E\ PHDQV RI WKH ILUVW N WHUPV RI D FRQYHUJLQJ VHTXHQFH RI GLIIHUHQWPDWULFHV7KHQXPEHUNFDQEHLQWHUSUHWHGDVWKHUDWLREHWZHHQWKHWZRWLPHVFDOHV 7KH DLP RI WKLV ZRUN LV WR H[WHQG DJJUHJDWLRQ WHFKQLTXHV WR WKH FDVH RI IDVW FKDQJLQJ HQYLURQPHQWV7KHPDLQLGHDRIDJJUHJDWLRQLVWREXLOGXSDQHZV\VWHPZLWKORZHUGLPHQVLRQ WKDWVXPPDUL]HVWKHLQIRUPDWLRQFRQFHUQLQJWKHIDVWSURFHVV7KLV³DJJUHJDWHG´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
%/$6&2(7$/
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iQFKH]HWDO %UDYRGHOD3DUUDDQG6i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
/,1($5',6&5(7(3238/$7,2102'(/6
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
) ∈51
q, N q T
xt = ( xt1,1 ,..., xt1, N1 , xt2,1 ,..., xt2, N 2 ,..., xtq,1 ,..., xt
ZKHUH1 1«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∈51ZKLFKZHFRQVLGHUDVGLYLGHGLQWREORFNV0LMLMT:HWKHQKDYH
M 11 M 12 M 1q 21 M M 22 M 2 q M= q1 q2 M qq M M ZKHUH HDFK EORFN 0LM >0LMPO@KDV GLPHQVLRQV 1L×1M DQG FKDUDFWHUL]HV WKH UDWHV RI WUDQVIHUHQFH RI LQGLYLGXDOV IURP WKH VXEJURXSV RI JURXS M WR WKRVH RI JURXS L 0RUH VSHFLILFDOO\IRUHDFKP 1L DQG HDFK O 1M 0 LMPOUHSUHVHQWVWKHUDWHRI WUDQVIHUHQFH RI LQGLYLGXDOV GXH WR WKH VORZ SURFHVV IURP VXEJURXS O RI JURXS M WR
%/$6&2(7$/
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Â3NÂÂÂ3 $ :HDVVXPHWKDWWKHIDVWG\QDPLFVLVLQWHUQDOIRUHDFKJURXS 7KLV DVVXPSWLRQ PHDQV WKDW WKH IDVW G\QDPLFV GRHV QRW SURGXFH WUDQVIHUHQFH RI LQGLYLGXDOV DPRQJ GLIIHUHQW JURXSV &RQVHTXHQWO\ HYHU\ PDWUL[ 3 O O «N LV D EORFNGLDJRQDOPDWUL[ZKRVHEORFNVZHFDOO3OLL «T 3O GLDJ^3O«3OT` % :HDVVXPHWKDWWKHIDVWG\QDPLFVFRQVHUYHVWKHQXPEHURILQGLYLGXDOVRIHDFK JURXS 7KLVK\SRWKHVLVLVLQFOXGHGLQWKHPRGHOE\DVVXPLQJHDFKEORFN3OLWREHDFROXPQ VWRFKDVWLFPDWUL[LHDPDWUL[ZKRVHFROXPQVDUHSUREDELOLW\YHFWRUV 7KHUHIRUHWKHSURSRVHGPRGHOZLOOFRQVLVWRIWKHIROORZLQJGLVFUHWHV\VWHPWKDWZH ZLOOFDOO³PLFURV\VWHP´RU³RULJLQDOV\VWHP´ [W 03N«Â3[W ,Q D SUHYLRXV ZRUN 6iQFKH] HW DO DJJUHJDWLRQ WHFKQLTXHV KDYH EHHQ DSSOLHGWRWKHIROORZLQJPRGHO [W 03N[W ZKHUHWKHPDWULFHVFRUUHVSRQGLQJWRWKHIDVWG\QDPLFVDUHFRQVWDQW,QWKHQH[WVHFWLRQ ZHFRQVWUXFWDUHGXFHGV\VWHPWKHVRFDOOHGDJJUHJDWHGV\VWHP
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` WHQGV ZKHQ N→ ∞ WR D FHUWDLQ QHFHVVDULO\ VWRFKDVWLF SULPLWLYHPDWUL[3L +HQFHIRUWKZHVXSSRVHWKHWKUHHDVVXPSWLRQVRQIDVWG\QDPLFV$%DQG& KROG
/,1($5',6&5(7(3238/$7,2102'(/6
7RFRQVWUXFWWKHDJJUHJDWHGV\VWHPZHQHHGWRPDNHXVHRIWKHIROORZLQJUHVXOW 3URSRVLWLRQ /HW^3N`EHDVHTXHQFHRIFROXPQVWRFKDVWLFPDWULFHVWKDWFRQYHUJHVWRDSULPLWLYH PDWUL[3/HWYEHWKHULJKWSUREDELOLW\QRUPHG HLJHQYHFWRURI3DVVRFLDWHGZLWKWKH HLJHQYDOXH7KHQ lim Pk ⋅ ... ⋅ P1 = v 7 k →∞
ZKHUH LV WKH FROXPQ YHFWRU RI WKH DSSURSULDWH GLPHQVLRQ ZLWK DOO LWV HQWULHV HTXDO WR 3URRI6HH$SSHQGL[ )URPDVVXPSWLRQV$% DQG & ZH FDQ DSSO\ 3URSRVLWLRQ WR WKH EORFNV RI WKH IDVWG\QDPLFV+HQFHLWIROORZVWKDWIRUHDFKL T
lim Pki ⋅ ... ⋅ P1i = v i 7
k →∞
ZKHUH Y L LV WKH ULJKW SUREDELOLW\ QRUPHG HLJHQYHFWRU RI WKH PDWUL[ 3 L FLWHG LQ $VVXPSWLRQ& 1RZ LI ZH GHILQH P i = v i 7 P = diag( P 1 ,..., P q ) 8 GLDJ^«` DQG 9 GLDJ^Y«YT`WKHQZHKDYH
P = lim Pk ⋅ ... ⋅ P1 = VU T k →∞
+HQFHWKHRULJLQDOV\VWHP FDQEHFRQVLGHUHGDVDSHUWXUEDWLRQRIWKHIROORZLQJ
Xt +1 = MPXt
7KLV QHZ V\VWHP FDOOHG WKH ³DX[LOLDU\ V\VWHP´ FDQ EH LQWHUSUHWHG DV WKH PLFURV\VWHPZKHQZHVXEVWLWXWHWKHIDVWSURFHVVFRUUHVSRQGLQJWRHDFKLQWHUYDO,W E\ WKH HTXLOLEULXP IUHTXHQFLHV RI WKH IDVW SURFHVV LQ ,W ,Q RWKHU ZRUGV ZH DUH OHWWLQJ N→∞ LQ WKH H[SUHVVLRQ RI WKH PLFURV\VWHP VXSSRVLQJ WKDW ,W LV ORQJ HQRXJK ZLWK UHVSHFW WR WKH SURMHFWLRQ LQWHUYDO RI WKH IDVW SURFHVV IRU WKH IDVW G\QDPLFV WR UHDFK HTXLOLEULXP IUHTXHQFLHV 1RWH WKDW ZH XVH FDSLWDO OHWWHUV ZKHQ UHIHUULQJ WR YDULDEOHV DVVRFLDWHGZLWKWKHDX[LOLDU\V\VWHP :HZLOOVHHWKDWWKHDJJUHJDWHGV\VWHPFDQEHSHUIHFWO\DJJUHJDWHG,QRUGHUWRGR VRZHGHILQHWKHJOREDOYDULDEOHVIRUHDFKWLPHWLQWKHIROORZLQJZD\
yt = ( yt1 ,..., ytq )
ZKHUH yti = Xti,1 + ... + Xti, N i LH\WLUHSUHVHQWVWKHWRWDOQXPEHURILQGLYLGXDOVLQJURXS LDWWLPHWDVVXPLQJWKDWWKHIDVWG\QDPLFVUHDFKHVLWVHTXLOLEULXPIUHTXHQFLHVLQHDFK WLPHXQLW 1RWHWKDWWKHVHJOREDOYDULDEOHVFDQEHREWDLQHGIURPWKHYDULDEOHVRIWKHDX[LOLDU\ V\VWHPDVIROORZV \W 87;W 7 7KHUHIRUHLIZHSUHPXOWLSO\HTXDWLRQ E\PDWUL[8 ZHREWDLQ
U T Xt +1 = U T MPXt = U T MVU T Xt 7KLVLVWKHDJJUHJDWHGV\VWHPZKLFKFDQEHH[SUHVVHGLQWKHIROORZLQJZD\
%/$6&2(7$/
yt +1 = Myt
ZKHUH M = U T MV :H KDYH UHGXFHG WKH GLPHQVLRQ RI WKH PRGHO IURP 1 1«1T WR T VLQFH LQ WKH DJJUHJDWHG V\VWHP HDFK JURXS LV UHSUHVHQWHG E\ RQHVLQJOHYDULDEOH ,Q QH[W VHFWLRQ ZH ZLOO VKRZ KRZ WKLV QHZ V\VWHP SURYLGHV LQIRUPDWLRQ RQ WKH DV\PSWRWLFEHKDYLRURIWKHSRSXODWLRQUHSUHVHQWHGE\WKHPLFURV\VWHP
5(/$7,216+,36%(7:((17+($**5(*$7('$1'7+( 25,*,1$/6