The Dynamics of Two Harvesting Species with variable Effort Rate with the Optimum Harvest Policy

Transcription

1 Inernionl OPEN ACCESS Journl Of Modern Engineering Reserch (IJMER) The Dynmics of Two Hrvesing Species wih vrible Effor Re wih he Opimum Hrves Policy Brhmpl Singh; nd Professor Suni Gkkhr; Deprmen of Mhemics, J.V.JAIN College Shrnpur (UP) 47; Indi; Deprmen of Mhemics, Indin Insiue of Technology Roorkee, Roorkee 47667, Indi ABSTRACT: The dynmics of wo ecologiclly independen species which re being hrvesed wih vrible effor hve been discussed. The dynmics of effor is considered seprely. The locl nd globl dynmics of he sysem is sudied. The exisence of Hopf bifurcion wih respec o he ol cos of fishing is invesiged. The co-exisence of species in he form of sble equilibrium poin nd limi cycle is possible. The qusi-periodic behvior is lso possible for some choices of prmeers. An expression for Opimum populion level hs been obined. I. INTRODUCTION Exploiion of biologicl resources s prciced in fishery, nd foresry hs srong impc on dynmic evoluion of biologicl populion. The over exploiion of resources my led o exincion of species which dversely ffecs he ecosysem. However, resonble nd conrolled hrvesing is beneficil from economicl nd ecologicl poin of view. The reserch on hrvesing in predor-prey sysems hs been of ineres o economiss, ecologiss nd nurl resource mngemen for some ime now. The opiml mngemen of renewble resources hs been exensively sudied by mny uhors [,, 3, 7, 8, ]. The mhemicl specs of mngemen of renewble resources hve been discussed by []. He hd invesiged he opimum hrvesing of logisiclly growing species. The problem of combined hrvesing of wo ecologiclly independen species hs been sudied [, 3]. The effecs of hrvesing on he dynmics of inercing species hve been sudied Meseron-Gibbons [4], Chudhuri e.l. [6-9] wih consn hrvesing, he prey predor model is found o hve ineresing dynmicl behvior including sbiliy, Hopf bifurcion nd limi cycle [4, 5,, 5]. The muli species food web models hve found o hve rich dynmicl behvior [6, 8]. S Kumr e. l. [7] hve invesiged he hrvesing of predor species preding over wo preys. In his pper he dynmics of wo ecologiclly independen species which re being hrvesed hve been discussed when he dynmics of effor is considered seprely. II. THE MATHEMATICAL MODEL Consider wo independen biologicl species wih densiies Xnd Xwih logisic growh. The Mhemicl model of wo hrvesing prey species wih effor re is given by he following sysem of ordinry differenil equions: dx X A q XE r X X f( X, X, E) dt K B X B X dx X Aq EX r X X f( X, X, E) dt K B X B X de pq A X pqa X E h( X, X ) C Ek C Ef3( X, X ) dt B X B X The logisic growh is considered for he wo preys. The model does no consider ny direc compeiion beween he wo populions. The consns () K,r, A, nd B, re model prmeers ssuming i i i i only posiive vlues. The effor E is pplied o hrves boh he species nd C is ol cos of fishing. The IJMER ISSN: Vol. 5 Iss. 8 Augus 5 6

2 hrvesing is proporionl o he produc of effor E nd he fish populion densiy X i.the cch-biliy coefficiens q i re ssumed o be differen for he wo species. In he model, he hird equion considers he dynmics of effor E. The consns p nd p re he price of he per uni prey species. The ls equion of () implies h he re of increse of he effor is proporionl o he re of ne economic revenue. The consn k is he proporionliy consn. Le he consn M is he reference vlue of E. Inroduce he following dimensionless rnsformions: rt, y X K ( i,), x E M, w A q E r, w B K, w B K i i i 3 w r r, w A q M r, w kk M, w kk M ; The dimensionless nonliner sysem is obined s: dy wx y y y f( y, y, x) d w y w3 y dy w5x y ( y) w4 y f( y, y, x) d w y w3 y dx p w w y p w w y d w y w y x C xf3 y y 3 (, ) III. EXISTENCE OF EQUILIBRIUM POINTS Since y ; i,, he underlying non-liner model () is bounded nd hs unique soluion. i There re mos seven possible equilibrium poins of he nonliner hrvesing model:,,,,,,,,,,, E E E E, 3 E6 y, y, x. Theorem 3. The equilibrium poin 4 E y,, x, x ( y )( w y ) w ; y C ( w p w Cw ) 4 6 E, y, x, x w (- y )( w y ) w ; y C ( w p w - Cw ), is fesible only when C w p w ( w ) (3) Theorem 3. The equilibrium poin E 5, y, x is fesible only when C w p w ( w ) (4) The proofs of he wo heorems re srighforwrd s y ; i, i. Theorem 3.3 The posiive non-zero equilibrium E 6 of nonliner hrvesing model () exiss provided he following condiions re sisfied: C w p w w ; C w p w w (5) Proof: Proof is given in [6]; he equilibrium poins re x (- y )( w y w y ) w y 3 ( w w w )( w p w Cw ) Cw w ( ) ( ) Cw -( w - w w )( w p w - Cw ) w ( w p w - Cw ) w w ( w p w - Cw ) w5 w7 pw5 Cw3 ww 4 w pw 6 Cw y The posiive non-zero biologicl equilibrium E 6 y, y, x IJMER ISSN: Vol. 5 Iss. 8 Augus 5 7 () exiss provided he condiions (5) re sisfied.

3 I my furher be observed h condiions (3) nd (4) imply (5), h is if E 4 E5, y, x exiss hen E6 will lso exiss. However, he exisence of 6 E4 nd E5 provided he condiion (5) is sisfied. nd E is possible irrespecive of IV. STABILITY ANALYSIS The vriionl mrix bou he poin E is given by J w4 -C From he bove vriionl mrix, i is seen h here re wo unsble mnifolds long boh X,Y xis nd one sble mnifold long Z xis. Therefore he poin E is sddle poin, h is, very smll densiies of species he effor decreses nd ends o zero, while for smll effors he densiies of hrvesing species will sr incresing, The vriionl mrices bou he xil poin,, J - -/( w ) w w pw6 - C w 4 E nd E,, J -w - w /( w ) w5 pw 7 ( - C ) w3 nd re given by respecively. From he mrix J, i is seen h here exiss sble mnifold long X xis nd n unsble mnifolds long Z xis. Sble mnifold long Y xis exiss provided w p w 6 C( w ). Observe h his condiion violes he exisence of E y,, x 4. The poin E is sddle poin. Similrly, from he mrix J, i is seen h here exiss sble mnifold long Y xis nd n unsble mnifolds long X xis. Sble mnifold long Z xis exiss provided w p w 6 C( w ). This condiion excludes he exisence of equilibrium poin E 5, y, x The vriionl mrix bou he poin E3,, is given by.the poin E is sddle poin. w ( w w3 ) J3 w4 w5 ( w w3) w pw6 w5 pw C ( ww3) E3,, is sble provided he following condiion is sisfied: w pw 6 w5 pw7 C ( w w ) Thus, he equilibrium poin 3 (6) Theorem 4.: The equilibrium poin E y,, x 4 is loclly sympoiclly sble provided IJMER ISSN: Vol. 5 Iss. 8 Augus 5 8

4 ( w -) w y ( w - w ) w (7) 5 5 Proof. The vriionl mrix bou he poin E y,, x 4 3 J ww x y ( ) wy - wx 5 w4 ( wy ) x ww 6 p 3 ( wy ) ;, w w y x 3 ( wy ), 3 33 is given by, wy - ; ( ) 3 wy x w w p y w w w p w w w p ( wy ) The equilibrium poin E4 is loclly sble if he following condiions re sisfied: w x w (- y ) ; nd y ( w - w ) w ; 5 5 Subsiuion for x nd simplificion yields he sbiliy condiions s w w y w w w w w ( -) ( - ) The equilibrium E 4 is unsble when he condiion () is violed. Similrly, he sbiliy condiions for he equilibrium E 5 re sed in he heorem 4.. Is proof is omied. Theorem 4.: The equilibrium poin E 5, y, x is loclly sympoiclly sble provided ( w -) w y ( w - w ) w (8) The equilibrium E 5 is unsble when he condiion () is violed. The following heorem gives he condiions for he loclly sbiliy of he nonzero posiive equilibrium E y, y, x. poin 6 Theorem 4.3: The posiive non-zero biologicl fesible equilibrium E 6 y, y, x ; is loclly sympoiclly sble if he following condiions re sisfied: x w (- y ) ; (9) w w x w w (- y ) () w x y ( w w y w w y )(- y ) () 3 5 w w x y ( w w y w w y )(- y ) () Proof: Proof is given in [6]. Thus, he posiive non-zero biologicl fesible equilibrium E 6 is loclly sympoiclly sble if he condiions given by (9-) re sisfied. The following heorem gives he condiions for he globl sbiliy of he nonzero posiive equilibrium poin. Theorem 4.4 Le he locl sbiliy condiions given by (9-) hold. The posiive non-zero biologicl fesible E y, y, x is globl sble if he following condiion is sisfied: equilibrium 6 IJMER ISSN: Vol. 5 Iss. 8 Augus 5 9

5 ( w - w w ) 4 w ( w y w y ) - w ( w y w y ) - w w ( w w p - w C) w5 ( ww 6 p - wc) Proof: Proof is given in [6]; (3) V. THE OPTIMUM HARVEST POLICY The economic ren (ne revenue) ny ime is given by pw w6 y pw5w7 y p( x, y, y) x C w y w3 y The presen vlue J of coninuous ime srem of revenues is given by he expression - pw w6 y pw5 w7 y - J p( x, y, y) e d x( C) e d w y w y (4) Consider he inegrnd of he presen vlue p w w y p w w y G( y, y, x, ) x C e w y w3 y y( y) y pw 6 w4 y( y) - y pw5 - Cxe. Here is he insnneous nnul re of discoun. Using clssicl Euler necessry condiions o mximize he posiive nonzero equilibrium such h G d G G d G G d G ; ;. y d y y d y x d x On solving he firs equion, we ge G d G y pw6e ; or y d y (5) y e & p w 6 Similrly oher condiions yield 4 5 IJMER ISSN: Vol. 5 Iss. 8 Augus 5 w ( y ) e & p w (6) Ce (7) Cse (): Le e for, hen y. (8) w4( y) (9) C () Combining () nd (3) gives y w y ( w )/. () 4 4 Cse (): Le e for bu hen y c. () w ( y ) c (3) 4

6 C (4) Elimining, c, c using (6) nd (7) we ge y w y ( w ). (5) 4 4 Hence he opimum line () nd (5) re sme in he boh cses. Therefore he opimum equilibrium will be he inersecion of line (5) nd firs isocline of (). VI. EXISTENCE OF HOPF BIFURCATION The chrcerisic equion of he bove vriionl mrix bou E 6 is obined s [6]. By he Rouh- Hurwiz crierion, he posiive nonzero equilibrium poin is loclly sympoiclly sble under he condiions (9-). The Hopf bifurcion occurs C C where he vlue of -becomes zero. Subsiuing ino (6), we ge, 3 ( )( ). This gives purely imginry roos nd one rel roo: ;,3 i (6) Trnsversliy condiion: - Le he chrcerisic equion be such h i conins pir of purely imginry ' ' roos i nd one rel roo, sy c : ( - )(( - )( - c or 3 ( c ' ) ). ( c Compring he coefficiens of (3) nd (6) we ge ' ' ' ( ) ( ) ' ) c (8) Differeniing (3) wih respec o bifurcion prmeer C, seingc C, nd rescling h ' C.we ge [8], ( ) (7) ( ' ) C C C C ( ) C C (9) Thus he rnsversliy condiion is sisfied. So here exiss fmily of periodic soluions bifurcing from E 6 in he neighborhood of C h is, C ( C, C ). VII. NUMERICAL SIMULATION As he soluion of he sysem is bounded, he long ime behvior of he soluion is obined s limi cycle, limi poin rcor, qusi-periodic. For globl dynmic behvior, numericl simulions of he underlying non-liner sysem re crried ou. Consider he biologicl fesible se of prmeers s w., w., w.6, w., w.5, w.3, w.7, p.5, p., (3) The sign chnge from posiive o negive for he expression - is observed s he vlues of prmeer C re vried. The exisence of hopf bifurcion is observed in he neighborhood of C.4. The vriionl mrix he hopf bifurcion poin C.4 is given by J IJMER ISSN: Vol. 5 Iss. 8 Augus 5

7 The eigenvlues of he mrix Jre obined s.989, i, i. From he eigenvlues he Hopf bifurcion is eviden is eviden in he neighborhood of C.4. The vlue of he expression - is.3 C =.56. Fig. () shows he phse spce rjecories converging o he poin (.4973,.36954,.598) for he d se (3) C.56, sring wih wo differen iniil condiions. In oher words, he nonrivil equilibrium poin (.4973,.36954,.598) is sble giving he persisence of he sysem for he given se of prmeers. Fig (b) shows he ime series for he d se (3) C.56. Figure () shows he phse spce rjecories nd heir ime series for he d se (3) C.4. Figure (b) shows he long ime behvior in phse spce rjecories nd heir ime series for he sme d se (3) C.4. The soluion is qusi periodic. The vlue of he expression - is -.57, C.4 () (b) Fig. : For he d se (34); () 3D behvior (b) ime series Fig. (). IJMER ISSN: Vol. 5 Iss. 8 Augus 5

8 Fig. (b). Qusiperiodic behvior However, for C., here exiss limi cycle for he sme se of d. As is eviden from he Fig 3. The vlue of he expression - is -.77, C.. Fig. 3. Limi cycle VIII. CONCLUSIONS In his model, sepre dynmics of hrvesing effor is considered. The locl nd globl persisence of he hrvesed preys hs been nlyzed. The opimum populion level is invesiged. The exisence of hopf bifurcion wih respec o ol cos of fishing is observed. REFERENCES []. Asep K.Suprion, Hugh P.Possinghm, Opiml hrvesing for predor-prey mepopulion, Bullein of Mhemicl Biology 6:49-65 (998). []. B. Dubey, Peeyush Chndr nd Prwl Sinh, A model for fishery resource wih reserve re, Nonliner Anlysis: Rel World Applicions 4( 4): (3). [3]. Berg Hugo A.vn den, Yuri N. Kiselev, S.A.L.M.Kooijmn, Orlov Michel V., Opiml llocion beween nurien upke nd growh in microbil richome, Journl of Mhemicl Biology 37:8-48 (998). [4]. Bruer, F., Soudck, A.C., Sbiliy regions nd rnsiion phenomen for hrvesed predor-prey sysems, J.Mh. Biol.7: (979). [5]. Bruer, F., Soudck, A.C., Sbiliy regions in predor-prey sysems wih consn- re prey hrvesing, J.Mh. Biol.8:55-7 (979). IJMER ISSN: Vol. 5 Iss. 8 Augus 5 3

9 [6]. Brhmpl Singh; The Globl Sbiliy of Two Hrvesing Species wih vrible Effor Re; Inernionl Journl of Educion nd Science Reserch Review (ISSN , impc fcor.4) pge no-6-3; Volume-, Issue-, April- 4. [7]. Chudhuri, K.S., A bioeconomic model of hrvesing of mulispecies fishery, Ecol.Model.3:67-79(986). [8]. Chudhuri, K.S., Dynmic opimizion of combined hrvesing of wo-species fishery, Ecol.Model.4:7-5(988). [9]. Chudhuri, K.S., Sh Ry, S., Bionomic exploiion of Lok-Volerr prey-predor sysem, Bull.Cl.Mh.Soc.83:75-86 (99). []. Chudhuri, K.S., Sh Ry, S., On he combined hrvesing of prey -predor sysem. J. Biol. sys. 4 (3): (996). []. Colin W.Clrk, Mhemicl bioeconomics: The opiml mngemen of renewble resources, John Wiley & Sons, USA (976). []. Di, G., Tng, M., Coexisence region nd globl dynmics of hrvesed predor-prey sysem, SIAM J.Appl, Mh.58:93- (998). [3]. Meng Fn, Ke Wng, Opiml hrvesing policy for single populion wih periodic coefficiens, Mhemicl Bioscience 5:65-77 (998). [4]. Meseron-Gibbons, M., On he opiml policy for he combined hrvesing of independen species, N.Res.model. :7-3 (987). [5]. Meseron-Gibbons, M., On he opiml policy for he combined hrvesing of predor nd prey, N.Res.model.3:63-9 (988). [6]. Nguyen Phong chu, Desbilising effec of periodic hrves on populion dynmics, Ecologicl Modeling 7:-9 (). [7]. S. Gkkhr, R. K. Nji, On food web consising of specilis nd generlis predor, Journl of biologicl Sysems (4): (3). [8]. S.Kumr, S.K.Srivsv, P.Chingkhm, Hopf bifurcion nd sbiliy nlysis in hrvesed one-predor-woprey model, Applied Mhemics nd Compuion 9:7-8 (). [9]. Y.Tkeuchi, Globl Dynmicl Properies of Lok-Voler Sysems, World Scienific (996). IJMER ISSN: Vol. 5 Iss. 8 Augus 5 4