Bifurcation Analysis of a Food Chain in a Chemostat with Distinct Removal Rates

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1 Interntionl Journl of Applied Science nd Engineering 5 : 7- Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte Srker Md Sohel Rn eprtent of Mthetic Univerit of hk hk- Bngldeh Abtrct: In thi pper we conider clicl food chin odel decribing predtor-pre interction in cheott The Micheli-Menten kinetic i ued the uptke for both predtor nd pre We oberve the dnicl behvior of the odel round ech of the equilibri nd point out the echnge of tbilit We ue Lpunov function in the tud of the globl tbilit of predtor-free equilibriu Uing reovl rte of pre the bifurction preter we prove tht the odel undergoe Hopf bifurction round interior equilibriu It h been found tht the dnicl behvior of the odel i ver enitive to the preter vlue With the id of nuericl iultion we nlze the odel eqution nd illutrte the ke point of nlticl finding nd deterine the effect of operting preter of the cheott on the dnic of the te Keword: Cheott; food chin; globl tbilit; Hopf bifurction; diiptive; ulc criteri Introduction The cheott i lbortor pprtu ued to tud generl propertie of popultion growth nd interction ong icro-orgni under nutrient liittion in controlled environent The continuou culture odel with Monod kinetic for nutrient uptke h received gret del of ttention ince it w firt introduced nd coplete theticl theor of thi odel h been developed There re n rticle devoted to the tud of the cheott both fro the eperientl nd the odeling point of view A detiled epoition of the theticl theor of the cheott i given in [] Moreover the cheott odel i the trting point for n vrition food chin food web etc tht ield ore relitic biologicl nd theticl proble [-8] The dnic of predtor pre nd ubtrte interction h becoe ubiquitou tool for tuding nuber of indutril field uch wte tretent biorector [9-] Predtion i direct interction which occur when individul fro one popultion derive their nourihent b cpturing nd ingeting individul fro nother popultion [-4] A iple food chin in cheott hd been tudied in [5-9] nd relted eperient re decribed in [-] The dnic of tri-trophic food chin odel tht incorporte either Micheli-Menten or generl onotone repone function for ll trophic level nd reovl rte for the pre nd predtor popultion re either equl to the whout rte of the cheott or ditinct hd been eined b n reercher [ ] It h been hown tht thee iple food chin with one predtor nd one pre ehibit tht the predtor feed ecluivel on either the pre or the pre nd the nutrient nd the pre conue the nutrient in the cheott Correponding uthor; e-il: rnthdu@gilco Received 9 eceber Revied 7 M 5 C 5 Chong Univerit of Technolog ISSN Accepted 9 Jul 5 Int J Appl Sci Eng 5 7

2 Srker Md Sohel Rn In [] the uthor retricted their ttention to the ce in which the pre i of logitic growth nd predtor hve Holling tpe II functionl repone nd etblihed the globl tbilit for the ce of etinction of top-predtor Severl tudie ued bifurction nli to find out if coeitence of ll trophic level i poible [5 6 5] In [6] the uthor crried out the nli of their odel nuericll b finding both locl nd globl bifurction of equilibri nd of liit ccle with repect to cheott control preter Li nd Kung [6] conidered iple food chin nd tudied globl tbilit of equilibri nd preented it dnic nuericll while in [4] the uthor tudied nlticll the globl tbilit of equilibri of the odel which i the etenion of thi iple food chin Srh [8] eined the dnicl behvior of tri-trophic food chin odel with globl tbilit of equilibriu point nd Hopf bifurction of olution In thi pper we re going to eine the dnic of food chin where the predtor feed ecluivel on the pre nd on the nutrient nd the pre conue the nutrient in the cheott The Micheli-Menten functionl repone i ued for both predtor nd pre In ddition we confine our interet to find criteri under which the odel predict tht the popultion will be ble to perit t ted tte in the culture veel for n indefinitel long period of tie Thi tud lo focu on globl tbilit of equilibriu point bifurction nli round interior equilibriu nd enitivit profile of tte vrible with repect odel preter Our reult in thi pper re etenion to thoe in [ ] Thi pper i orgnized follow In Section the food chin odel with Micheli-Menten functionl repone i decribed In ection oe eleentr propertie uch boundedne invrince of non-negtivit diiptivit nd the equilibri nd their tbilitie re invetigted Section 4 i devoted to dicu globl tbilit nli of equilibri In Section 5 we dicu Hopf bifurction of olution Section 6 del with enitivit nli nd nuericl iultion Finll hort dicuion i given in Section 7 The odel The food chin we nlze in thi pper conit of ubtrte pre nd predtor Let denote the nutrient concentrtion the concentrtion of the pre popultion nd the concentrtion of the predtor popultion t tie t Our odel i decribed b the following ordinr differentil eqution: d with ' dt In the te denote the input concentrtion of the nutrient i re the il re the growth ield contnt nd the input uptke i re the hlf turtion contnt i rte fro the feed pup nd the whout rte of the cheott chber nd re the reovl rte of pre nd predtor popultion repectivel All preter hve poitive vlue 8 Int J Appl Sci Eng 5

3 Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte It i convenient to introduce dienionle vrible In prticulr we define i i t t i i Then oitting the br to iplif the nottion the te becoe with Without lo of generlit we cn conider the te inted of nd we cn lw reinterpret our finding in ter of originl vrible Eleentr propertie eitence of equilibri nd their tbilitie Boundedne nd non-negtivit of olution In thi ection we hll how tht the te i diiptive b proving tht olution of te re non-negtive nd bounded Theore All olution of the te with initil vlue in lrge t R re non-negtive nd bounded nd for Proof Firt let i olution of Suppoe tht for ll t i not true Let t in{ t : t & } Then t t [ t But fro the firt eqution of we hve t Thi iplie there eit uch tht i increing on t t Therefore we hve t / contrdiction Thu for ll t Let now t in{ t : t } We firt ue tht t Then for t [ t ] Let A in tt Then for t [ t ] A which iplie tht t ep[ At ] contrdiction Therefore for ll t A iilr rguent how tht t i burd Thu the te with poitive initil condition t t produce poitive olution for t Furtherore if then Let Int J Appl Sci Eng 5 9

4 Srker Md Sohel Rn in in then in If u i olution of u in u with in t u then u inu e nd li u But u t in which en tht u o we conclude tht li t in Therefore olution of re bounded nd the te i diiptive Finll for t t o we hve e nd thu for lrge t Thi coplete the proof The equilibri: eitence nd locl tbilit We will find the following poible equilibri of te in the for E Etinction of ll popultion: E Survivl of popultion onl: E where Survivl of popultion onl: E where Survivl of popultion nd : E where nd re defined the unique olution of in nd the tif the eqution with nd The vlue nd repreent the brek-even concentrtion of nutrient It i e to ee tht if i i i then the correponding popultion tend to zero Thu in order to void the popultion vnihing we hll ue tht i i i To dicu the eitence of equilibri we tht equilibriu point will not eit if n one of it coponent i negtive The whout equilibriu point E lw eit The eitence condition for E i nd iilrl for E Finll the feibilit condition for the ied culture interior equilibriu point E ut be poitive or Note tht Q i decreing in with Q Q nd Q So if nd onl if or equivlentl In the net tep we will invetigte the locl tbilit of thee equilibriu point b finding the eigenvlue of the ocited Jcobin trice The Jcobin tri due to the lineriztion of bout n rbitrr equilibriu E R i given b Int J Appl Sci Eng 5

5 Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte J E efine the nuber R R R nd It i e to how tht the eigenvlue of E equivlentl R nd J The Jcobin tri t E i given b E R J will be negtive if R nd R or Henceforth we let M define the tri upper left hnd tri M Since trce M nd det M b Routh-Hurwitz criterion the eigenvlue of M hve negtive rel prt The third eigenvlue of J E i Therefore E i locll ptoticll tble LAS if nd onl if or equivlentl nd tht i R Siilr clcultion how tht E i LAS if nd onl if R When E eit the Jcobin tri due to lineriztion of bout E i given b the epreion Int J Appl Sci Eng 5

6 Srker Md Sohel Rn E J J ij The eigenvlue of J E tif the eqution where trce J J det JE J J J J J J J J JE JJJ JJJ JJJ JJJ The Routh-Hurwitz criterion tht E will be LAS if nd onl if nd We cn urize the bove reult in the following theore Theore If R nd R then onl E eit nd it i LAS If R nd R then E nd E eit E i untble nd E i LAS If R nd R then E nd E eit E i untble nd E i LAS If R nd R then E E nd E eit nd E nd E re untble E i LAS if nd nd therefore the te will be uniforl peritent 4 lobl nli In the previou ection we howed the eitence nd locl tbilit nli of ll equilibri In thi ection we hll preent the globl tbilit of the equilibri of te The proof for E i ver trightforwrd Mot iportntl we hll how tht if onl E nd E eit under reonble uption E i globll ptoticll tble The proof involve the contruction of Lpunov function nd the ppliction of the Lpunov-LSlle theore We hll ue ethod iilr to [] Theore If R nd R then E i the onl equilibriu point nd ll olution of converge to E Proof It i cler tht if R nd R then b Theore E i the onl equilibriu point nd LAS Now to prove tht E i globll ptoticll tble ue tht i 4 Int J Appl Sci Eng 5

7 olution of Since for lrge t nd Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte or R there i v uch tht v for t ufficientl lrge nd v in Since t i non-negtive thi how tht li It follow fro the third eqution of tht t ' where in which iplie tht li Then the t t firt eqution of ield tht li Hence E i globll ptoticll tble nd the theore i proved t If R nd R then E i locll ptoticll tble We will ppl the ulc criterion [6] to how tht E i globll ptoticll tble in the plne Let B Then in the olution plne of we hve B B ie doe not chnge ign nd i not identicll zero for Hence there re no periodic olution on the plne nd E will be globll ptoticll tble in the plne The following theore how tht E i globl ttrctor if it eit Theore 4 If R nd R then ll olution of tif li t t t Proof Let H t H Then li H nd H li H H Let d ln Then li nd for Let H F Since H nd H F tifie F for li F li F F for li F nd Int J Appl Sci Eng 5

8 Srker Md Sohel Rn 4 Int J Appl Sci Eng 5 efine Lpunov function in [] on the region R c c d d where c will be defined lter Then on nd iff nd The tie derivtive of long trjectorie of i given b 4 c c c F c H c To dicu the ign of we will invetigte ech ter of Let tifie in F F For F ince for nd for b definition of for nd iff For ince R we hve for n choice of c nd iff For if then for n choice of c nd iff Finll for 4 we hve c c c 4

9 Let c then 4 Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte nd 4 iff Therefore ech ter of i non-poitive Hence E i globl ttrctor b Llle invrince principle [ 7] Thi coplete the proof 5 Hopf bifurction nli In thi ection we hll dicu tht our odel undergoe Hopf bifurction b uing bifurction rel preter Clerl b theore there i no Hopf bifurction t E E nd E So we re going to vr in order to obtin the deired Hopf bifurction for round E Firt we recll tht the eigenvlue of J E tif the eqution nd it coponent re defined in 4 B the Routh Hurwitz criteri necer nd ufficient condition for ll the root of to hve negtive rel prt re H : nd : H Now in order to hve Hopf bifurction we ut violte either H or H Suppoe Clerl will hve two pure iginr root if nd onl if 5 for oe vlue of Since t there i n open intervl contining for oe for which uch tht for Thu for the chrcteritic eqution cnnot hve poitive rel root For we hve ee [ p8 ] 6 which h three root i i For i i the root re in generl of the for To ppl Hopf bifurction theore to ee [8] we need to verif the trnverlit condition i Re i 7 Subtituting i nd i into nd clculting the derivtive with repect to we obtin Int J Appl Sci Eng 5 5

10 Srker Md Sohel Rn 6 Int J Appl Sci Eng 5 N K L M L K 8 where b b K 6 b L b b b M b b N Since N L M K we hve Re i L K N L M K i nd Hence there i Hopf bifurction t We cn now forulte the following: Theore 5 Suppoe H hold Then te ehibit Hopf bifurction leding to fil of periodic olution tht bifurcte fro E for uitble vlue of in the neighborhood of 6 Senitivit nli nd Nuericl iultion Mn dnic odel of biologicl procee cn be written under the following generl for: ; t t X g t Y X X t t X f dt t dx 9 where X i vector of tte vrible d d vector of preter Y vector of output nd t the independent vrible Let X Z be the enitivit of tte vrible X with repect to preter then the enitivit tri eqution cn be epreed in copct forul: Z f Z X f dt dz Eqution i lo clled firt forwrd enitivit eqution Siultneou integrtion of eqution of te 9 nd provide vlue of enitivit function [9] with repect to tie According to 9 the enitivit of the output i Y wrt the preter i evluted Y g Z X g Y

11 Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte However thee bolute enitivit function re not norlized nd the re not ueful for copring the effect of different input fctor for wht reltive enitivit function hould be ued Reltive enitivitie re idel for copring preter becue the re dienionle norlized function Hence L -nor of the reltive enitivit of the function Y i to vrition in the preter j i given b: en nor j Yi Y i j 6 Nuericl iultion In thi ection our i i to preent nuericl iultion to illutrte the ke reult of theoreticl finding epecill bifurction round interior equilibriu nd enitivit of tte vrible wrt preter We left the iultion work for other equilibri thee re iple The figure hve been contructed b proper choice of the kinetic preter o tht ll the intereting behvior of the te re oberved We chooe the bic preter of the odel to be The initil condition 67 i ued to generte olution curve nd trjectorie in ll figure Our iultion work Fig ugget tht E i globl ttrctor if it i locll ptoticll tble A certin preter incree or decree further w E loe it tbilit nd ocilltor olution pper which i to be the reult of Hopf bifurction For howing the dnic of the te chnge the preter et 5555 given fied preter nd vried preter rel bifurction preter Fig how ce in which nd te poee periodic olution nd reult Hopf bifurction round E Fig indicte tht perturbing while chnging to nd keeping other preter in Fig fied led to bifurction Thi ee to detro the periodic olution nd poibl led to the globl tbilit of E where Fig 4 while chnging to 9 nd keeping other preter in Fig fied led to the intbilit of E Therefore vring the vlue of nd ffect the dnic of in ver urpriing nd ignificnt w Furtherore t ech tge we hve preented the norlized enitivit profile b Eq to ee the ffect of preter on the dnic of the odel Highet vlue indicte the ot enitivit of the preter All of the coputtion nd viuliztion hve been perfored in MATLAB R7 Net we ee how the preter nd ffect the dnic of if i fied Aue tht i i re lrge enough o tht R nd R then ll popultion will be whed out E i tble in the cheott A i grdull decreed eventull there i bifurction when R nd R In thi ce E loe it tbilit nd the new bifurcted ted tte E i ptoticll tble A i grdull decreed the net bifurction occur when R nd R hold In thi ce E loe it tbilit nd new ted tte Int J Appl Sci Eng 5 7

concentrtion concentrtion Srker Md Sohel Rn E pper 8 5 6 4 5 5 5 5 5 5 4 tie d d b Figure : The olution curve pproch poitive equilibriu E b L - nor of the norlized enitivit of the tte vrible wrt

12 concentrtion concentrtion Srker Md Sohel Rn E pper tie d d b Figure : The olution curve pproch poitive equilibriu E b L - nor of the norlized enitivit of the tte vrible wrt preter tie b d d c d Figure : The olution pper to pproch periodic olution nd Hopf bifurction occur round E b A plot of trjectorie in three dienionl view c Projection of trjectorie onto the plne d L - nor of the norlized enitivit of the tte vrible wrt preter 8 Int J Appl Sci Eng 5

concentrtion Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte 8 9 6 8 4 7 6 5 4 tie 4 8 6 4 b 5 7 6 5 4 6 5 4 4 5 6 7 d d c d Figure : The olution ocillte but eventull pproche poitive

13 concentrtion Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte tie b d d c d Figure : The olution ocillte but eventull pproche poitive equilibriu E b A plot of trjectorie in three dienionl view c Projection of trjectorie onto the plne d L - nor of the norlized enitivit of the tte vrible wrt preter 7 icuion nd concluion In thi pper we conidered food chin with one pre nd one predtor in the cheott where the pre conue the nutrient nd the predtor conue the pre nd the nutrient We ued tht the functionl repone function re in Micheli-Menten for nd the reovl rte re different We perfored detiled coputtionl nli of thi odel The dnic behvior of thi odel depend on the nuber R R R nd R We etblihed tht te h olution which re eventull bounded in the future We lo etblihed ufficient condition for the eitence nd locl tbilit of the equilibri b uing Routh-Hurwitz criterion for te We found tht ll the popultion cnnot perit if the reovl rte of the pre i reltivel lrge Thi hppen when E i globl ttrctor of te We contructed Lpunov function on the be of [] to how tht E i globll ptoticll tble The globl ptotic tbilit of E iplie tht nutrient well the pre popultion cnnot upport the predtor nd conequentl the predtor will be whed out in the cheott regrdle of the initil denit level of pre nd predtor Net when E eit then ll the Int J Appl Sci Eng 5 9

concentrtion Srker Md Sohel Rn 8 6 8 4 6 4 4 tie 8 6 4 b 5 7 6 5 5 4 4 4 5 6 7 8 d d c d Figure 4: The olution ocillte nd ee to pproch periodic olution b A plot of trjectorie in three dienionl view c

14 concentrtion Srker Md Sohel Rn tie b d d c d Figure 4: The olution ocillte nd ee to pproch periodic olution b A plot of trjectorie in three dienionl view c Projection of trjectorie onto the plne d L - nor of the norlized enitivit of the tte vrible wrt preter pecie pre nd predtor coeit in the ene tht the te i uniforl peritent nd the conervtion principle i circuvented In thi ce witch of the tbilit of E occur We then ue the reovl rte of pre bifurction rel preter We found tht Hopf bifurction occur under certin condition t the interior equilibriu point E leding to fil of periodic olution bifurcte for E Finll both nlticll nd nuericll iultion how tht in certin region of the preter pce the food chin te h rich dnic including periodic nd ptotic behvior nd the odel enitivel depend on the preter vlue The reult of the conidered te in thi tud re etenion to the te of [ ] nd i ueful in the further tud of the coeitence of copeting popultion in the cheott Since ot of the food chin odel in cheott incorporte Micheli-Menten-tpe tpe II repone function with contnt ield coefficient ore detiled nli for thi te with other tpe repone function or ubtrte inhibition with vrible ield coefficien focu on bifurction nli liit ccle nd enitivit profile of tte vrible with repect odel preter will be provided in ner future Int J Appl Sci Eng 5

15 Bifurction Anli of Food Chin in Cheott with itinct Reovl Rte Acknowledgeent The uthor would like to thnk the referee for their vluble coent Reference [] Sith H L nd Wltn P 995 The Theor of Cheott Cbridge Univerit Pre Cbridge UK [] Alhuzi K nd Ajbr A 5 nic of Predtor-Pre Interction in Continuou Culture Engineering in Life Science 5 : 9-47 [] Chiu C-H nd Hu S-B 998 Etinction of top-predtor in three-level food-chin odel Journl of Mheticl Biolog 7 4: 7-8 [4] El-Owid H M nd Monie A A On food chin in cheott with ditinct reovl rte Applied Mthetic E-Note : 8-9 [5] El-Sheikh M M A nd Mhrouf S A A 5 Stbilit nd bifurction of iple food chin in cheott with reovl rte Cho Soliton nd Frctl 4: [6] Li B nd Kung Y Siple food chin in cheott with ditinct reovl rte Journl of Mtheticl Anli nd Appliction 4 : 75-9 [7] Nrin F nd Rn S M S Three pecie food web in cheott Interntionl Journl of Applied Science nd Engineering 9 4: - [8] Al-Sheikh S A 8 The nic of Tri-Trophic Food Chin in the Cheott Interntionl Journl of Pure nd Applied Mthetic 47 : - [9] Alqhtni R T nd Nelon M I nd Worth A L Anli of cheott odel with vrible ield coefficient: Contoi kinetic ANZIAM Journl EMAC 5: C55-C7 [] Alqhtni R T Nelon M I nd Worth A L A fundentl nli of continuou flow biorector odel governed b Contoi kinetic I Reccle round the whole rector ccde Cheicl Engineering Journl 8: 99 7 [] W M S Mohd I B Mt M nd Slleh Z Mtheticl odel of three pecie food chin interction with ied functionl repone Interntionl Journl of Modern Phic: Conference Serie ol 9: 4 4 [] Wrno Sunro M S Slleh Z nd Mt M Mtheticl odel of three pecie food chin with Holling Tpe-III functionl repone Interntionl Journl of Pure nd Applied Mthetic 89 5: [] Boonrngin S nd Bunwong K Hopf bifurction nd dnicl behvior of tge tructured predtor hring pre Interntionl Journl of Mtheticl Model nd Method in Applied Science 8 6: 89-9 [4] Updh R K nd Rw S N Cople dnic of three pecie food-chin odel with Holling tpe I functionl repone Nonliner Anli: Modelling nd Control 6 : 5 74 [5] Butler J Hu S B nd Wltn P 98 Coeitence of copeting predtor in cheott Journl of Mtheticl Biolog 7 : -5 [6] Boer M P Kooi B W nd Koojin S A L M 998 Food chin dnic in the cheott Mtheticl Biocience 5 : 4-6 [7] Freedn H I nd Wltn P 977 Mtheticl nli of oe three-pecie food chin odel Mtheticl Biocience -4: Int J Appl Sci Eng 5

16 Srker Md Sohel Rn [8] Hting A nd Powell T 99 Cho in three-pecie food chin Ecolog 7 : [9] Klebnoff A nd Hting A 994 Cho in three pecie food chin Journl of Mtheticl Biolog 5: [] rke J F nd Tuchi H M 976 Predtion of Echerichi coli b Colpod tenii Applied nd Environentl Microbiolog 6: [] Jot J L rke J F Tuchi H M nd Fredrickon A 97 Microbil food chin nd food web Journl of Theoreticl Biolog 4 : [] Jot J L rke J F Fredrickon A nd Tuchi H M 97 Interction of Tetrhen prifori Echerichi coli Azotobcter vinelndii nd glucoe in inil ediu Journl of Bcteriolog : [] Tuchi H M rke S F Jot J L nd Fredrickon A 97 Predtor-pre interction of ictoteliu dicoideu nd Echerichi coli in continuou culture Journl of Bcteriolog : 47-5 [4] Ali E Aif M nd Ajbr A Stud of chotic behvior in predtor pre interction in cheott Ecologicl Modelling 59: 5 [5] Freedn H I nd Run S 99 Hopf bifurction in three-pecie food chin odel with group defene Mtheticl Biocience : 7-87 [6] Mrin A M Rubén O nd Rodríguez J A A ulc function for qudrtic te Theoreticl Mthetic & Appliction : [7] Wolkowicz S K nd Lu Z 99 lobl dnic of theticl odel of copetition in the cheott: enerl repone function nd differentil deth rte SIAM Journl of Applied Mthetic 5 : - [8] Mrden J E nd Mckrcken M 976 The Hopf Bifurction nd it Appliction Springer-erlg New York [9] Wlter E nd Pronzto L 997 Identifiction of Pretric Model fro Eperientl t Springer London Int J Appl Sci Eng 5

Transfer Functions. Chapter 5. Transfer Functions. Derivation of a Transfer Function. Transfer Functions

Transfer Functions. Chapter 5. Transfer Functions. Derivation of a Transfer Function. Transfer Functions 5/4/6 PM : Trnfer Function Chpter 5 Trnfer Function Defined G() = Y()/U() preent normlized model of proce, i.e., cn be ued with n input. Y() nd U() re both written in devition vrible form. The form of