Activator-Inhibitor Model of a Dynamical System: Application to an Oscillating Chemical Reaction System

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1 Actvtor-Inhtor Model of Dynmcl System: Applcton to n Osclltng Chemcl Recton System C.G. Chrrth*P P,Denn BsuP P * Deprtment of Appled Mthemtcs Unversty of Clcutt 9, A. P. C. Rod, Kolt-79 # Deprtment of Mthemtcs Scottsh Church College nd 3 Urquhrt Squre, Kolt-76 Astrct In ths pper we hve frst studed the role of ctvtor-nhtor technque n modellng generl dynmcl system nd then s pplcton nvestgted the crter of oscllton of non-lner chemcl recton system. Keywords: Actvtor-Inhtor Model, Informtonl Networ, Dynmcl System, Chemcl Recton System, Crter of Oscllton.. Introducton A comple system s composed of mny prts, elements or components whch re connected n more or less complcted fshon[,,3]. In delng wth comple systems we re confronted wth the chllenge of fndng unfyng prncples coverng ll systems rrespectve of the rnches of scence they elong to. In order to descre comple system t mcroscopc level, we need n enormous mount of dt or nformton[4]. A mcroscopc descrpton llows n enormous compresson of nformton whch re concerned not wth the ndvduls mcroscopc dt ut rther wth the glol propertes of the system estlshng vrous reltons mong the vrous mcroscopc qunttes. Nonequlrum thermodynmcs s the pproprte rnch of scence delng wth such reltons of comple system[5]. The method s nlogous to Rosen's mthemtcl representton of comple system[]. It s sed on Hggns ctvton nd nhton model of osclltng chemcl recton[6]. In the present pper we hve frst dscussed the mthemtcl technques of ctvton-nhton n the modellng of generl dynmcl system. We hve then nvestgted the role of ctvton-nhton technque n the modellng of n osclltng chemcl recton system.. Actvton nd Inhton n Dynmcl System 48

2 Let us consder generl dynmcl system descred y the system of dfferentl equtons f (,,..., n ), (,,..., n) (.) The functons f re ssumed to e contnuous nd to hve contnuous prtl dervtves n some open set Ω {, }. Followng Hggn's ctvton nd nhton model of dynmcl system we consder the followng oservtonl qunttes[] (,,..., V n ), (,,,..., n) (.) where V s the net rte t whch the sustnce (or rectnt) s eng produced s result of ntercton occurrng n the system. The qunttes, (,,,..., n) s Hggn noted hve nformtonl correlton. For nstnce, the qunttes defned ove hve three possltes[] ( ), ( ) <,( c) > (.3) If () holds, then the ncrese (or decrese) n the nstntneous vlue of wll cuse the rte of producton of e. V. to e ncresed (or decresed). In ths cse we shll cll e n ctvtor of. If to, then s self-ctvtor. If () holds, the reverse s true, n ncrese (or decrese) n the nstntneous vlue of wll cuse the rte of producton of to e decresed (or ncresed). In tht cse s sd to e n nhtor of. For, s then self-nhtor. f (c) holds, the rte of producton of ndependent of ltertons n the nstntneous vlues of. The quntty s the ntercton of wth ts flu V s nown s the self- s couplng nd the qunttes ( ), whch s the ntercton of wth the flu of (.e. V ) s nown s the crosscouplng. An mportnt prolem out the qunttes s to fnd out the rnges of vldty of the ctvtor nd nhtor whch re defned reltve to some ntl stte ( sttonry pont or crtcl pont). Snce y hypothess ll the prtls of the producton rtes re contnuous, t follows tht f s n ctvtor for reltve to some ntl stte, t s n ctvtor for n some entre 49

3 neghourhood of tht stte ; lewse f s n nhtor for reltve to some ntl stte, we cn thus, decompose the stte spce Ω of the system nto set of suregons, such tht, n ny su-regon, the sgn of ll frst prtl dervtves V the sme for ll ponts n the suregon[]. On the ss of ctvtor- nhtor ny dynmcl system (ochemcl, morphogenetc, ecologcl or neutrl etc) cn e converted nto n nformtonl networ of ctvtor nd nhtor whch seems more nturl thn the dynmcl one[]. The qunttes defned y[],,,,..., n (.4) lso hve the chrcters of nformtonl correltons. It s esy to show tht f > n stte, t mens tht enhnces or potenttes the effect of on. Under these consdertons we cll n gonst of. Lewse f < n stte, ttenutes the effect of on s nd hence we cll ntgonst of. We cn contnue the tertng process n ths wy to get successve networs,... to gve n nformtonl, descrpton of the dynmcl system (.). ply The system of networs { } sgnfcnt role n the study of stlty, nstlty nd perodcty of the dynmcl system. In the net secton we shll study n fndng out the role of the networ { } the crter of stlty, nstlty nd perodcty of chemcl recton system 3. Actvton nd Inhton n Chemcl Recton System Let us consder system of two chemcl speces descred y the netc equtons[7] f ( f (, ), (3.) where nd re the concentrtons of two rectnts dependng on tme, nd re eternlly gven fed n tme. Let us ssume tht the system s homogeneous such tht the chemcl speces re dstruted unformly throughout ts entre whole spce. The stedy stte (or sttonry pont) of the system s such tht f (, ), ( ) f (, ), (,) ) (3.) 4

4 (3.3) Epndng the functons f (, ),(,) out the sttonry pont (, ) we hve the system of equtons f (, ) f ( ) f ( ) neglectng hgher order terms of ( ) ( )( ) hve the system of lner equtons, where (3.4) δ, we (, ) δ ( ) ( ) f (,) (3.5) δ ( ) ( ) (3.6) s the vlue of the nformtonl correlton functon t the sttonry pont (, ). ( ) re the elements of the Jcon mtr J t the sttonry stte., J f ( ) f f f (3.7) ( The term V > mples tht s self-ctvtor of. V < mples tht s n nhtor of. V > mples tht s n ctvtor of. V < mples tht s self-nhtor of. The system (3.) whose Jcon mtr hs the sgn mtr of the form Sgn (J ) (3.8) s n ctvtor-nhtor system[8]. Let us now turn to the stlty nlyss of the system (3.) on the ss of the system ) 4

5 of lner equtons (3.5). The egenvlue equton s λ or λ βλ γ wth egenvlues γ, where β ± λ β 4γ β Trce( J ) γ det( J ) ( ) > (3.) (3.) (3.) We eep the concentrton fed nd chnge the concentrton. β f (3.9) c. If c < then 4 4 β > mplyng tht sttonry pont (3.) s unstle focus. Agn f c > 4 then β < mplyng tht the sttonry pont s stle focus. If, β. e. c 4 furcton pont t β or c 4. Ths stuton, we hve corresponds to the trnston of the system from the stle focus ( β < ) to the unstle focus crossng through the furcton pont β or. 4 Hopf-furcton theorem then predcts the estence of stle lmt cycle provded γ > [8]. Let C e closed ort enclosng the sttonry pont s the only sngulr pont. In order tht the closed ort e the lmt cycle t s necessry tht γ > or > or V V V V > (3.3) The prtls n the ove l.h.s. epresson re evluted t the sttonry pont. In vew of the contnuty of the prtls we hve seen tht, f these prtls hve defnte chrcter t the sttonry pont, ths chrcter must e retned throughout the entre neghourhood of the sttonry pont, 4

6 nd thus the nequlty (3.3) must hold throughout ths entre regon[]. We now suppose tht the closed curve C les entrely wthn ths regon of constnt chrcter. So lso the recton V V (3.4) vld for the sttonry pont s vld throughout the entre regon n the neghourhood of the sttonry pont. We V hve thus see tht the self-couplngs V nd must e of opposte sgn. From ths nd (3.3) t follows tht V V (3.5) mplyng tht the cross couplng terms V V nd re of opposte sgn. From the ove nlyss of self-couplng nd cross-couplng we see tht nd < V < V (3.6) V V < < (3.7) where the elements,(,,) re defned n the neghourhood of the sttonry pont. The sgned mtr correspondng to the Jcon mtr (3.7) s gven y Sgn (J ) (3.8) whch chrcterses n ctvtor nhtor system[8]. 4. Concluson In the present pper we hve studed two prolems. We hve frst studed the role of the technque of ctvtor-nhtor n the modellng of generl dynmcl system. As pplcton of the ctvtor-nhtor technque we hve studed non-lner chemcl recton system nd determned the condtons or crter of osclltng ehvour of the system. The ctvtornhtor networ ply sgnfcnt role n the study of stlty, nstlty nd the perodcty of the chemcl recton system. References: [] Rosen, R.: Dynmcl System Theory n Bology, Wley-Interscence Pulsher, New-Yor(97) 43

7 [] Rosen, R.(ed): In "Theoretcl Bology nd Complety", Acdemc Press, New- Yor(985) [3] Rosen, R.: Bull.Mth.Bol.,4,477(978) [4] Hen, H.: Synergetcs: An Introducton, Sprnger-Verlg, Berln(978) [5] Serr, R. et l: Introducton to the Physcs of Comple Systems, Pergmon Press, Oford(986) [6] Hggns, J.: J.Industrl nd Engneerng Chem.,59,8(967) [7] Murry, J.D.: Mthemtcl Bology, Sprnger-Verlg, Berln(989) [8] Allen, Lnd, J.S.: Introducton to Mthemtcl Bology, Person Prentce Hll, London(7) 44