MuliCrf Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 INTRNATIONAL JOURNAL OF NGINRING SCINC AN TCHNOLOGY www.ijes-ng.com MuliCrf Limied. All righs reserved Bioeconomic modelling of pre predor ssem using differenil lgebric equions T.. r nd unl Chkrbor b eprmen of Mhemics Bengl ngineering nd Science Universi Shibpur Howrh-7 b eprmen of Mhemics MCV Insiue of ngineering 4 G.T.Rod N Liluh Howrh-74 -mils: kr7@gmil.com T.. r Corresponding uhor; kc_mckv@hoo.co.in unl Chkrbor Absrc We propose biologicl economic model bsed on pre-predor dnmics where he pre species re coninuousl hrvesed nd predion is considered wih pe II funcionl response. The dnmic behvior of he proposed biologicl economic prepredor model is discussed. Coninuous pe gesionl del of predors is incorpored nd is effec on he dnmicl behvior of he model ssem is nled. Through considering del s bifurcion prmeer he occurrence of Hopf bifurcion of he proposed model ssem wih posiive economic profi is shown in he neighborhood of he co-eising equilibrium poin. Finll some numericl simulions re given o verif he nlicl resuls nd he ssem is nled hrough grphicl illusrions. ewords: Bioeconomics differenil lgebric ssem ime del Hopf bifurcion. Inroducion The pplicion of mhemicl biolog hs n immense impc owrds he developmen of commonl used biologicl resources like fisher wildlife nd foresr. Recenl Scieniss nd reserchers give emphsis on he inercion beween mhemics nd biolog which iniie new reserch re. Inercions of mhemics nd biolog cn be divided ino hree cegories. The firs clss involves rouine pplicion of eising mhemicl echniques o biologicl problems. Such pplicions influence mhemics onl when he impornce o biologicl pplicions requires furher developmens. In oher cses however eising mhemicl mehods re insufficien bu i is possible o develop new mhemics wihin he convenionl frmeworks. In he finl clss some fundmenl issues in biolog pper o require new houghs quniivel or nlicll. Mos of our biologicl heories evolve rpidl; herefore i is necessr o develop some useful mhemicl models o describe he consequences of hese biologicl ssems. I is observed h hese newl developed mhemicl models re significnl influenced hrough he biologicl heories in he ps nd he consequen epnsion of hose heories in recen ime. For his purpose differenil lgebric equions cn be considered s n imporn ool for he nlsis of biologicl model. A generl pre predor model consiss of he inercions beween species herefore he model includes compeiion evoluion nd dispersion beween he species for he purpose of seeking resources o susin heir sruggle for heir own eisence. r nd Msud 6 represened he ge of muri hrough ime del which leds o ssems of rerded funcionl differenil equions. The considered pre-predor model wih Holling pe of predion nd hrvesing of predor species nd observed h when he ime del is smll boh he pre nd predor populions rech periodic oscillions round he equilibrium in finie ime hen converges o heir equilibrium vlues nd in non-del cse hrvesing effor hs n effec of sbiliing he equilibrium. Broer e l. 5 invesiged wo-dimensionl predor-pre model wih five prmeers dped from he Volerr-Lok ssem b non monoonic response funcion. The described vrious domins of srucurl sbili nd heir bifurcions. The effec of consn re hrvesing on he dnmics of predor-pre ssems hs been invesiged b i nd Tng 998 Merscough e l. 99 nd io nd Run 999 nd he obined ver rich nd ineresing dnmicl behviours. Feng 7 considered differenil equion ssem wih diffusion nd ime dels which models he dnmics of predor-pre inercions wihin hree biologicl species.
4 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 Oros 4 presened forml frmework for he nlsis of Hopf bifurcions in del differenil equions wih single ime del. He deermined closed-form liner lgebric equions nd clculed he criicli of bifurcions b norml forms. Co nd Freedmn 996 obined he crierion of persisence nd globl rcivi for predor-pre model wih ime-del due o gesion. Yfi e l. 7 considered model wih one del nd unique non rivil equilibrium. The sudied he dnmics of he model in erms of he locl sbili nd of he descripion of he Hopf bifurcion non rivil equilibrium. The proved h del ken s prmeer of bifurcion crosses some criicl vlues nd deermined he direcion of he Hopf bifurcion nd he sbili or insbili of he bifurcing brnch of periodic soluions. r sudied Guss-pe pre predor model wih selecive hrvesing nd inroduce ime del in he hrvesing erm. He concluded in generl del differenil equions ehibi much more compliced dnmics hn ordinr differenil equions since ime del could cuse sble equilibrium o become unsble nd cuse he populion o flucue. Celik 9 considered rio dependen predor-pre ssem wih ime del where he dnmics is logisic wih he crring cpci proporionl o pre populion. Broer nd Giko nled he complee globl quliive of quric ecologicl model priculrl he sudied he globl bifurcions of singulr poins nd limi ccles. Zhng nd Zhng 9 ssemicll sudied hbrid predor pre economic model which is formuled b differenil-difference-lgebric equions. The proved h his model ehibis wo bifurcion phenomen he inersmpling insns. Lr nd Mrine 9 considered discreeime conrol dnmicl model wih uncerinies represening bioeconomic ssem proposed hrough sochsic vibili pproch o mnge nurl resources in susinble w due o uncerinies dnmics nd conflicing objecives ecologicl socil nd economicl. An efficien lgorihm for individul-bsed sochsic simulion of biologicl populions in coninuous ime presened b Allen nd hm 9. I is observed h numerous number of reserch ricles of he populion dnmics proposed he inercion beween he species nd he sbili nlsis of he populion in presence of hrvesing effor bu quie few number of ricles considered he bioeconomic models o invesige he dnmicl behvior of he ecossem owrds he posiive economic profi. Agin for he long run susinbili of he ecossem i is necessr o compre he sic s well s dnmicl effecs of hrvesing hrough considering he economic perspecive of he model ssem. Thus o formule biologicl economic ssem from n economic poin of view nd invesige he relisic sic nd dnmicl behvior of he model ssem we need o use differenil lgebric equions. Afer going hrough he bove lierure surve we cn no find n biologicl economic model ssem using differenil lgebric equions where pre populion is hrvesed nd he dnmicl behvior of such model ssem is sudied hrough considering se feedbck conroller. In his pper our objecive is o emine he dnmicl behvior of biologicl economic pre predor model where pre populion is hrvesed using differenil lgebric nd bifurcion heor. The coninuous gesion del of predor populion is lso incorpored in he model. We hve divided he pper in wo prs in he firs pr we consider he model ssem wih ero economic profi nd singulri induced bifurcion is obined he inerior equilibrium of he model ssem. To reduce he singulri induced bifurcion se feedbck conroller is designed. Bu in he second pr we consider he model ssem wih posiive economic profi nd he occurrence of Hopf bifurcion is found he inerior equilibrium poin hrough considering del s bifurcion prmeer. I is lso proved h he ime del cn cuse sble equilibrium o become unsble.. The model nd is quliive properies In his secion we consider pre-predor model wih Holling pe of predion nd coninuousl hrvesing of pre species he ecologicl se up of which is s follows. I is ssumed h he predor is no hrvesed nd hence hrvesing does no ffec he growh of he predor populion direcl. However i is considered h he predors hve compeiion mong hemselves for heir survivl. Agin here eiss conflic beween predors nd hrvesers for common resource i.e. pre species. The growh of pre is ssumed o be logisic. Le us ssume nd re respecivel he sie of he pre nd predor populion ime. Thus he consequen model becomes d d d d r h β d where r is he inrinsic growh re of he pre is he environmenl crring cpci of pre is he miml relive increse of predion is Michelis-Menen consn h is he hrvesing ime d is he deh re of predor he predor consumes pre he re β we ssume < β < since he whole biomss of he pre is no rnsformed o he biomss of he predor. ensi dependen morli re describes eiher self limiion of consumers or he influence of predion. is he inrspecific coefficien of he predor populion. Self limiion cn occur if here is some oher fcor oher hn food which becomes limiing high populion densiies.
5 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 The funcionl form of hrves is generll considered using he phrse cch-per-uni-effor CPU hpohesis Clrk 99 o describe n ssumpion h cch per uni effor is proporionl o he sock level. Thus we consider h q where is he hrvesing effor used o hrves pre populion nd q is he cchbili co-efficien of pre populion. The Anrcic krill-whle communi is good emple of he presen model. rill is min source of food of whles nd he Anrcic krill populion is being incresingl hrvesed. On he oher hnd he mororium imposed b IWC on killing of whles coninues. Lrge cches from he lower rophic level krill cn hve serious implicions for producion boh he lower rophic level krill nd he higher rophic level whle. I is herefore necessr o regule hrvesing he lower rophic level. Le us eend our model b considering he following lgebric equion - c - s where c is he consn fishing cos per uni effor p is he consn price per uni biomss of lnded fish nd s is he ol economic ren obined from he fisher. Thus using & ssem becomes d r q d d β d d - c - s 4 Le us now consider his hrvesed pre predor ssem wih coninuous ime del due o gesion. Here he predor populion consumes he pre populion consn re β bu he reproducion of predors fer preding he pre populion is no insnneous hus i will be incorpored b some ime lg required for gesion of predors. Suppose he ime inervl beween he momens when n individul pre is killed nd he corresponding biomss is dded o he predor populion is considered s he ime del. Le us ke he enire ps hisor of pre biomss which is o be mesured b ep where < is considered s priculr ime in he ps nd represens he presen ime. Thus he pre biomss in predor's equion is replced b he following form ep d. 5 Under his ssumpion he finl ssem becomes d r q d d β d d d d - c -s. 6 The differenil lgebric ssem 6 cn be epressed in he following w
6 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 r q f s β f s f s d f s g s - c - s where. Le us now consider wo cses seprel wih ero economic profi nd wih posiive economic profi.. The model wih ero economic profi For s he ssem 6 becomes d r q d d β d d d d - c. 7 quilibrium poins: eisence nd sbili The following lemm represens ll possible non negive equilibrium poins of ssem 7. Lemm Ssem 7 hs wo equilibrium poins P nd P for n posiive se of prmeers. The hird boundr ~ equilibrium poin P ~ ~ eiss if nd onl if c < where ~ c ~ c ~ cr r. The inerior equilibrium cd d cβ cr poin P of he ssem 7 eiss if c β > cd d nd r >. When hese condiions c re sisfied nd re given b c cd d cβ c From ssem 7 we hve he following mri M f f g r r q c The chrcerisic polnomil of he mri M is given b g c μ μ μ r q cr β d cd d cβ. c c β.
7 where r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 cq β d r c r r d r cq c 4r cq β r c rβ β rβ β d r c cq cq β. c rβ rβ β β 4r r The sbili of he boundr equilibrium poins P nd P of ssem 7 is given in he following lemm. Lemm The boundr equilibrium poin P is unsble nd P is locll smpoicll sble if β < d. The eigen vlues of he chrcerisic polnomil he boundr equilibrium poin P re d r. Thus he boundr equilibrium poin P is unsble. Agin he eigen vlues of he chrcerisic polnomil he boundr equilibrium poin d d β P re r. I is clerl observed h he boundr equilibrium poin P is sble if β < d. Le us now sud he dnmic behvior of he differenil lgebric model ssem 7. The locl sbili of he boundr equilibrium poin ~ ~ ~ P nd he inerior equilibrium poin P cn be invesiged using he singulri induced bifurcion phenomen. Here we re ineresed o discuss he locl sbili of he model ssem 7 he inerior equilibrium poin P hrough bifurcion phenomen. For his purpose ol economic ren is ssumed o be he bifurcion prmeer i.e. μ s. Consequenl we hve he following heorem Theorem The differenil lgebric ssem 7 hs singulri induced bifurcion he inerior equilibrium poin P. When he bifurcion prmeer s increses hrough ero he sbili of he inerior equilibrium poin P chnges from sble o unsble. Proof. I is eviden h g c hs simple ero eigen vlue. Thus we cn define Δ s g c. i I follows from Lemm h rce f dj g g P. ii I cn be proved using Lemm h
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 8. β β c d q q r g g f f p r P iii I cn lso be shown using Lemm h. Δ Δ Δ β β λ λ λ c d q q r g g g f f f p r p I is observed from i-iii h ll he condiions for singulri induced bifurcion Venksubrmnin e l. 995 re sisfied. Hence he differenil lgebric ssem 7 hs singulri induced bifurcion he inerior equilibrium poin P nd he bifurcion vlue is s. Agin i is noed h g g dj f rce M P. g f g g f f M p Δ Δ Δ λ λ λ Consequenl from Lemm we hve. > M M Hence i cn be concluded from Venksubrmnin e l. 995 h when s increses hrough ero one eigenvlue of he model ssem 7 moves from C o C long he rel is b diverging hrough. Consequenl he sbili of he model ssem 7 is influenced hrough his behvior i.e. he sbili of he ssem he inerior equilibrium poin P chnges from sble o unsble. In consequence o he bove heorem i is cler h he differenil lgebric model ssem 6 becomes unsble when he economic ineres of he hrvesing is considered o be posiive. If we consider economic perspecive of he fisher i is obvious h fisher gencies re ineresed owrds he posiive economic ren erned from he fisher. I is lso noed h n impulsive phenomenon cn occur hrough singulri induced bifurcion in pre predor ecossem which m led o he collpse of he susinble ecossem of he pre predor fisher. Therefore i is necessr o reduce he impulsive phenomenon from he pre predor ecossem o resume he susinbili of he ecossem nd sbilie he model ssem when posiive economic ineres is considered for fisher mngers. Thus o sbilie he model ssem 6 in cse of posiive economic ineres se feedbck conroller i 989 cn be designed of he form u w where u snds for ne feedbck gin. Le us inroduce he se feedbck conroller o he model ssem 6 nd rewrie model ssem s follows:
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 9. - c - s u d d d d d q r d d β 8 Consequenl we hve he following heorem: Theorem The differenil lgebric model ssem 8 is sble he inerior equilibrium poin P of he model ssem 7 if. m > β r r r r u Proof. For he differenil lgebric model ssem 8 we cn obin he following Jcobin he inerior equilibrium poin P of he model ssem 7. β u r J p Therefore he chrcerisic polnomil of he mri J is given w w w μ μ μ where u r w u r u r w. β u r w
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 According o he Rouh Hurwi crierion i cn be concluded h he model ssem 8 is sble he inerior equilibrium poin P of he model ssem 7 if he ne feedbck gin u sisfies he following condiion: u > m r r. β r r Hence i is possible o elimine singulri induced bifurcion which is responsible for impulsive phenomenon in susinble ecossem using suibl designed ne feedbck gin. Agin he economic ineres of fisher mngers cn lso be chieved using he se feedbck conroller funcion i.e. he sbili cn be resumed for he model ssem 8 when posiive economic ineres is considered. 4. The model wih posiive economic profi In his secion we consider he model ssem wih posiive economic profi i.e. s. Here we invesige he ssem behvior for wo sepre cses wih nd wihou ime del. 4. The model wihou ime del The model ssem 6 wihou ime del cn be wrien s d r q d d β d d - c - s. The inerior equilibrium poin of he ssem 9 is P 9 β d s where nd sisfies he following equion c where C 4 4 r C C C C C C cr r r C d β cr cr r r qs C cd d cβ cr cr r qs 4 cd cr qs. C From ssem 9 we hve he following mri r c N β.
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 Thus he chrcerisic polnomil of he mri N P is given b μ b μ b r where b c β r b. c r We find h b > nd b > if >. c Hence he inerior equilibrium poin P of ssem 9 is smpoicll sble if r >. c In priculr if we consider inr-specific coefficiens of he predor populion is ero i.e. hen he inerior equilibrium of he model ssem 9 becomes P d where β d d r dr dr q ds sβ dq ds sβ r nd d β d β d β cd d cβ d β cd d cβ sβ ds. cd d cβ I is noed h for he eisence of he inerior equilibrium poin P i is necessr β > d cd d > cβ nd dr ps d β r >. β d β d cd d cβ In his priculr cse he chrcerisic polnomil of he mri N P is reduced o μ p μ p r where p c p > β r We find h p > if >. c. r Hence he inerior equilibrium poin P of ssem 9 is smpoicll sble if >. c
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 r d c d β d β d β Agin i is observed h for s s p nd hence he roos of he q c d β chrcerisic equion become purel imginr nd he re conjuge o ech oher. Also we hve d ds dq c d β [ rcen ]. P s s d c d β β Hence b he Hopf bifurcion heorem Hssrd e l. 98 he ssem 9 eners ino Hopf pe smll mpliude periodic soluion s s in bsence of ner he posiive inerior equilibrium poin P. 4. The model wih ime del In his secion we consider he model ssem 6. I is eviden h he coordines of he inerior equilibrium poin P ˆ ˆ ˆ ˆ of model ssem 6 is s follows: ˆ ˆ nd ˆ ˆ where ˆ sisfing he equion hus ˆ cn be evlued from equion. From he model ssem 6 we hve he following mri ˆˆ ˆ R ˆ r ˆ ˆ ˆ c ˆ ˆ ˆ ˆ β. ˆ Thus he chrcerisic polnomil of he mri R is given b μ d μ d μ d where ˆ ˆˆ ˆ ˆ d r ˆ ˆ ˆ c rˆ ˆ ˆ r ˆˆ ˆˆ ˆ ˆˆ d ˆ c ˆ ˆ c ˆˆβ r ˆˆ ˆˆ ˆ ˆˆ d. ˆ ˆ ˆ ˆ c r ˆ ˆ I is noed h d > nd d > if >. ˆ c ˆ Le us ssume A dd d. Then A ˆ ˆˆ ˆ
where r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 r ˆ 4 ˆ ˆ ˆ p q ˆ ˆ rˆ ˆ r ˆ ˆ ˆ ˆ c c ˆ ˆ r ˆˆ ˆˆ ˆˆ ˆ c ˆ 4 ˆ ˆ ˆ p q ˆ rˆ r ˆ r ˆ c ˆ c ˆ ˆ r ˆˆ ˆˆ ˆˆ ˆ ˆ c ˆ ˆ ˆˆ ˆ ˆ r ˆ ˆ ˆ c. ˆ ˆ ˆ ˆ ˆ 4 ˆ ˆ c ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ c ˆ 4 β ˆˆ ˆ ˆ Now we hve he following heorem which ensures he locl sbili of he inerior equilibrium poin P ˆ ˆ ˆ ˆ of he model ssem 6. Theorem If P ˆ ˆ ˆ ˆ r ˆ ˆ eiss wih > nd ˆ c > hen P ˆ ˆ ˆ ˆ is locll smpoicll sble. ˆ r ˆ ˆ Proof. The condiion > implies h d ˆ c > nd d >. Finll > implies ˆ h A d d d. Hence b Rouh Hurwi crierion he heorem follows. > Bifurcion nlsis Pre-predor models wih consn prmeers re ofen found o pproch sed se in which he species coeis in equilibrium. Bu if prmeers used in he model re chnged oher pes of dnmicl behvior m occur nd he criicl prmeer vlues which such rnsiions hppen re clled bifurcion poins. Now we nle he bifurcion of he model ssem 6 ssuming s he bifurcion prmeer. Theorem 4 ˆ If P ˆ ˆ ˆ ˆ r ˆ ˆ eiss wih > nd < hen simple Hopf bifurcion occurs he posiive ˆ c ˆ ˆ β unique vlue. Proof. The chrcerisic equion of he model ssem 6 P ˆ ˆ ˆ ˆ is given b μ d μ d μ d The equion hs wo purel imginr roos if nd onl if d d d for unique vlue of s which we hve Hopf bifurcion. Thus in he neighborhood of he chrcerisic equion cn' hve rel roos. For we hve μ d μ d. This equion hs wo purel imginr roos nd rel roo s μ i d μ i d nd μ d. The roos re of he following form μ p iq μ p iq nd μ d. To ppl Hopf bifurcion heorem s sed in Liu's crierion Liu994 we need o verif he rnsversli condiion
4 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 dp d. Subsiuing μ p iq in he equion nd differeniing he resuling equion w.r.. nd seing p nd q d we ge he rnsversli condiion s dp d d dd d d d d d d Thus from he epressions of d d nd d we find ˆ ˆ ˆ ˆˆ ˆ β v v ˆ ˆ ˆ dp ˆ > if < d ˆ ˆ v v ˆˆ vˆ ˆ ˆ β r ˆ ˆ where v. ˆ c ˆ. Thus i cn be concluded h he inerior equilibrium poin P ˆ ˆ ˆ ˆ is locll smpoicll sble for <. Furhermore ccording o he Liu's crierion simple Hopf bifurcion occurs equilibrium poin P ˆ ˆ ˆ ˆ pproches o periodic soluion. Hence he heorem follows. 5. Numericl simulions nd discussion nd for > he inerior In his secion we ssign numericl vlues o he prmeers of he model ssem 6 nd compue some simulions using hose vlues. For he purpose of simulion eperimens we minl use he sofwre MATLAB 7. nd MATHMATICA 5.. This secion cn be clssified ino wo cegories. Firs cegor consiss of he resuls where he ol economic profi is considered o be ero. In he second cegor numericl simulions re represened wih posiive economic profi. 5. Simulion when ol economic profi is ero In order o ensure he nlicl resul of heorem numericll le us ssign he following numericl vlues o he prmeers of he model ssem 6;.95 r β.75 q.5 d..5 p 5 c. I is noed h when s he inerior equilibrium poin of he model ssem 6 is P ˆ ˆ ˆ ˆ P..6..99. Agin i is observed h when s -.he eigen vlues of he chrcerisic polnomil of he model ssem 6 re 77.759.5. 587 nd he eigen vlues become 8.746-.5-.48 when s.. Therefore i is cler from he bove resul h when s increses hrough ero wo eigen vlues of he chrcerisic polnomil of he model ssem 6 remin sme bu one eigenvlue of he model ssem 6 moves from C o C long he rel is b diverging hrough. Hence he sbili of he model ssem 6 he inerior equilibrium poin P ˆ ˆ ˆ ˆ chnges from sble o unsble. To sbilie he model ssem 6 in cse of posiive economic ineres le us consider se feedbck conroller of he form w u.99 consequenl we hve go he differenil lgebric model ssem 8 s follows
5 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 d.95.5 d d.75..5 d d d 5.5 - -s u.99. I is possible o evlue he numericl vlue of ne feedbck gin from heorem. For he bove model ssem we hve go u > m.96465.87 4.65. Considering u 5 we find he inerior equilibrium poin of he model ssem 8 s.64.6547.64.9998 when s nd he inerior equilibrium poin of he model ssem 8 becomes.794.879.794.994 when s.. I is eviden from Figure& h he differenil lgebric model ssem 8 is clerl sble when s increses hrough ero i.e. singulri induced bifurcion phenomenon is elimined from he differenil lgebric model ssem 6 he inerior equilibrium poin when ne economic profi increses hrough ero..6.4 u 5 nd s Pre biomss Predor biomss.. &.8.6.4. 5 5 5 5 4 45 5 Figure. Vriion of pre nd predor biomss wih he incresing ime when u 5 nd s.
6 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4.6.4 u 5 nd s. Pre biomss Predor biomss.. &.8.6.4. 5 5 5 5 4 45 5 Figure. Vriion of pre nd predor biomss wih he incresing ime when u 5 nd s.. 5. Simulion when ol economic profi is posiive In order o ensure he eisence of Hopf bifurcion le us consider he prmeers of he model ssem 6 s.6 r. β.8 q.5 6 d.5. p c s. Then he criicl vlue of he bifurcion prmeer 7.456. If we consider he vlue of 7. hen i is observed from he figure& h P ˆ ˆ ˆ ˆ is locll smpoicll sble nd he populions nd converge o heir sed ses in finie ime. Now if we grdull increse he vlue of keeping oher prmeers fied hen b heorem we hve go criicl vlue 7.456 such h P ˆ ˆ ˆ ˆ loses is sbili s psses hrough. Figure&4 clerl show he resul. I is lso noed h if we consider he vlue of 7.8 hen i is eviden from figure5&6 h he posiive equilibrium P ˆ ˆ ˆ ˆ is unsble nd here is periodic orbi ner P ˆ ˆ ˆ ˆ.
7 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 45 4 7. Pre biomss Predor biomss 5 & 5 5 5 4 5 6 7 8 9 Figure. Vriion of pre nd predor biomss wih he incresing ime when 7. <. 8 7 6 5 4 7. 9 8 5 5 5 5 4 45 5 55 Figure 4. Phse spce rjecories of pre nd predor biomss beginning wih differen iniil levels when 7. <.
8 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 45 4 7.456 Pre biomss Predor biomss 5 & 5 5 5 4 5 6 7 8 9 Figure 5. Vriion of pre nd predor biomss wih he incresing ime when 7.456. 8 7 6 5 4 7.456 9 8 5 5 5 5 4 45 5 55 Figure 6. Phse spce rjecories of pre nd predor biomss beginning wih differen iniil levels when 7.456.
9 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 45 4 7.8 Pre biomss Predor biomss 5 & 5 5 5 4 5 6 7 8 9 Figure 7. Vriion of pre nd predor biomss wih he incresing ime when 7.8 >. 8 7 6 5 4 7.8 9 8 5 5 5 5 4 45 5 55 Figure 8. Phse spce rjecories of pre nd predor biomss beginning wih differen iniil levels when 7.8 >. The foresid Hopf bifurcion cn lso be illusred if we consider noher se of numericl vlues o he prmeers of he model ssem 6.
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 Le us consider he following se of prmeers:.8 r.5 β.8 q.5 4 d.. p.5 c s. For his se of prmeers he criicl vlue of he bifurcion prmeer.678. I is clerl observed h simple Hopf bifurcion occurs nd for > he inerior equilibrium poin P ˆ ˆ ˆ ˆ pproches o periodic soluion. 55 5 45 Pre biomss Predor biomss 4 5 & 5 5 5 4 5 6 7 8 9 Figure 9. Vriion of pre nd predor biomss wih he incresing ime when <. 8 7 6 5 4 9 8 4 5 6 Figure. Phse spce rjecories of pre nd predor biomss beginning wih differen iniil levels when <.
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 55 5 45.678 Pre biomss Predor biomss 4 5 & 5 5 5 4 5 6 7 8 9 Figure. Vriion of pre nd predor biomss wih he incresing ime when.678. 8 7 6 5 4.678 9 8 4 5 6 7 Figure. Phse spce rjecories of pre nd predor biomss beginning wih differen iniil levels when.678.
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 55 5 45.5 Pre biomss Predor biomss 4 5 & 5 5 5 4 5 6 7 8 9 Figure. Vriion of pre nd predor biomss wih he incresing ime when.5 >. 8 7 6 5 4.5 9 8 4 5 6 7 Figure 4. Phse spce rjecories of pre nd predor biomss beginning wih differen iniil levels when.5 >. 6. Concluding remrks The pper nles he dnmicl behvior of pre predor model using differenil-lgebric ssems heor. In generl del differenil equions ehibi much more compliced dnmics hn ordinr differenil equions hus we hve sudied he
r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 effecs of coninuous ime-del on he dnmics of pre predor ssem. I is found h singulri induced bifurcion kes plce when ne economic revenue of he fisher is considered o be posiive. In consequence o he foresid bifurcion n impulsive phenomenon occurs nd he ssem becomes unsble. The mos imporn relisic feure of he pper is he se feedbck conroller which is designed o sbilie he model ssem when posiive economic ren is ken ino considerion. Numericl simulions re used o show h se feedbck conroller cn be designed o resume he sbili of model ssem in cse of posiive economic profi. In he second pr of he pper we hve discussed he behvior of he model ssem wih posiive economic profi here we hve divided our discussion in wo prs wih nd wihou ime del. In cse of wihou ime del i is observed h hough he model ssem is sble bu i is possible o ge criicl vlue of ol economic profi so h he model ssem becomes unsble when ol economic profi psses hrough he criicl vlue nd he model ssem eners ino Hopf pe smll mpliude periodic soluion. I is noed h coninuous ime del lso pls n imporn role o he dnmics of he model ssem. I is eviden from he obined resuls h he ime del cn cuse sble equilibrium o become unsble nd even simple Hopf bifurcion occurs when he ime del psses hrough is criicl vlue. The enire sud of he pper is minl bsed on he deerminisic frmework. On he oher hnd i will be more relisic if i is possible o consider he model ssem in he sochsic environmen due o some ecologicl flucuions nd oher fcors. Thus fuure reserch problem would be considered in sochsic environmen. Agin o chieve he commercil purpose of he fisher i is lso possible o deermine opiml hrvesing sregies using gme heor. Acknowledgemen Reserch of T.. r is suppored b he Council of Scienific nd Indusril Reserch C S I R Indi Grn no. 56/ 8 / MR-II ded 7..8 References Allen G.A. nd hm C. 9. An efficien mehod for sochsic simulion of biologicl populions in coninuous ime BioSsems Vol. 98 No. pp. 7-4. Broer H.W. Nudo V. Roussrie R. Sleh. 5. Bifurcions of predor-pre model wih non monoonic response funcion C. R. Acd. Sci. Pris Ser. I Vol. 4 pp. 6-64. Broer H. W. nd Giko V. A.. Globl quliive nlsis of quric ecologicl model Nonliner Anlsis Vol. 7 No. pp. 68-64. Celik C. 9. Hopf bifurcion of rio-dependen predor-pre ssem wih ime del Chos Solions nd Frcls Vol. 4 pp. 474-484. Co Y. nd Freedmn H. I. 996. Globl rcivi in ime-deled predor-pre ssems J. Ausrl. Mh. Soc. Ser B Vol. 8 pp. 49-6. Clrk C. W. 99. Mhemicl bioeconomics: he opiml mngemen of renewble resources nd ed. John Wile nd Sons New York. i L. 989. Singulr conrol ssem Springer New York. i G. nd Tng M. 998. Coeisence region nd globl dnmics of hrvesed predor- pre ssem SIAM J. Appl. Mh. Vol. 58 No. pp. 9-. Hssrd B.. rinoff N.. nd Wn Y.H. 98. Theor nd pplicion of Hopf bifurcion Cmbridge universi press Cmbridge. Feng W. 7. nmics in -species predor-pre models wih ime dels iscree nd Coninuous nmicl Ssems Supplemen pp. 64-7. r T... Selecive hrvesing in pre-predor fisher wih ime del Mh. Compu. Model Vol. 8 No. ¾ pp. 449-458. r T.. nd Msud H. 6. Conrollbili of hrvesed pre-predor ssem wih ime del Journl of Biologicl Ssems Vol. 4 No. pp. 4-54. Lr M.. nd Mrine V. 9. Muli-crieri dnmic decision under uncerin: A sochsic vibili nlsis nd n pplicion o susinble fisher mngemen Mhemicl Biosciences Vol. 7 No. pp. 8-4. Liu W. M. 994. Crierion of Hopf bifurcions wihou using eigenvlues J. Mh. Anl. Appl. Vol. 8 No. pp. 5-56. Merscough M.R. Gr B.F. Hogrh W.L. nd Norbur J. 99. An nlsis of n ordinr differenil equion model for wo-species predor-pre ssem wih hrvesing nd socking J. Mh. Biol. Vol. No. 4 pp. 89-4. Oros G. 4. Hopf bifurcion clculions in deled ssems Periodic Polechnic Ser. Mech. ng. Vol. 48 No. pp. 89-. Venksubrmnin V. Schler H. nd Zborsk J. 995. Locl bifurcions nd fesibili regions in differenil-lgebric ssems I Trns Auom Conrol Vol. 4 No. pp. 99-. io. nd Run S. 999. Bogdnov-Tkens bifurcions in predor-pre ssems wih consn re hrvesing Fields Insiue Communicions Vol. pp. 49-56.
4 r nd Chkrbor / Inernionl Journl of ngineering Science nd Technolog Vol. No. pp. -4 Yfi R. Adnni F.. nd Aloui H.T. 7. Sbili of limi ccle in predor-pre model wih modified Leslie-Gower nd Holling-pe II schemes wih ime del Applied Mhemicl Sciences Vol. No. pp. 9 -. Zhng. nd Zhng Q.L 9. Bifurcion nlsis nd conrol of clss of hbrid biologicl economic models Nonliner Anlsis: Hbrid Ssems Vol. No. 4 pp. 578-587. Biogrphicl noes r. T.. r is n Associe Professor he eprmen of Mhemics Bengl ngineering nd Science Universi Shibpur in Indi. His reserch ineress include nmicl ssems sbili nd bifurcion heor populion dnmics mhemicl modeling in ecolog nd epidemiolog mngemen nd conservion of fisheries bioeconomic modeling of renewble resources. He wroe round 5 cdemic ppers on hose opics. He lso supervised severl sudens of mser nd docor degree. unl Chkrbor is Lecurer he eprmen of Mhemics MCV Insiue of ngineering Howrh in Indi. He is currenl doing his Ph.. under he guidnce of r. T.. r in he eprmen of Mhemics Bengl ngineering nd Science Universi Shibpur Indi. His reserch opic is Bio-economic modelling nd developmen of soluion echniques for he mngemen nd conservion of fisheries. He hs obined his pos grdue degree in Mhemics from he Universi of Burdwn in 6. Received Ocober 9 Acceped ecember 9 Finl ccepnce in revised form ecember 9