A Mathematical study of Two Species Amensalism Model With a Cover for the first Species by Homotopy Analysis Method

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Available online a www.pelagiaeseachlibay.co Advances in Applied Science Reseach,, 3 (3):8-86 A Maheaical sudy of Two Species Aensalis Model Wih a Cove fo he fis Species by Hooopy Analysis Mehod B.

SLC Insiue of Engineeing & Technology, Hyd 3 Depaen of Basic sciences & Huaniies.

1 Available online a Advances in Applied Science Reseach,, 3 (3):8-86 A Maheaical sudy of Two Species Aensalis Model Wih a Cove fo he fis Species by Hooopy Analysis Mehod B. Sia Rababu, K. L. Naayan and Shahanaz Bahul 3 ISSN: CODEN (USA): AASRFC Depaen of Basic sciences & Huaniies.AVN Insiue of Engineeing & Technology, Hyd Depaen of Basic sciences & Huaniies.SLC Insiue of Engineeing & Technology, Hyd 3 Depaen of Basic sciences & Huaniies. JNT Univesiy, Kukapally, Hyd _ ABSTRACT The pesen pape is devoed o an analyical invesigaion of a wo species aensalis odel wih a paial cove fo he fis species o poec i fo he second species. The seies soluion of he non-linea syse was appoxiaed by he Hooopy analysis ehod (HAM) and he soluions ae suppoed by nueical exaples. Key Wods: Aensalis, zeo ode defoaion, ebedding paaee, linea opeao, Non linea opeao and HAM. _ INTRODUCTION Sybioses ae a boad class of ineacions aong oganiss --aensalis involves one ogais affecing anohe negaively wihou any posiive o negaive benefi fo iself. A. V. N. Achayulu and Paabhi Raachayulu [,,3,4] sudied he sabiliy of eney aensal species pai wih liied esouces and B. Shiva Pakash and T. Kaunanihi sudied he ineacion beween schizosacchaoyces pobe and sacchaoyces ceevisiae o pedic sable opeaing condiions in a cheosa [5].. Abou HAM: In 99 Liao eployee he basic idea of hooopy in opology o popose a poweful analyical ehod fo nonlinea pobles naely Hooopy Analysis Mehod [6,7,8,9,]. Lae on M. Ayub, A. Rasheed, T. Haya, Fadi & Awawdeh [,,3] successfully applied his echnique o solve diffeen ypes of non-linea pobles. The HAM iself povides us wih a convenien way o conol adjus he convegence egion and ae of appoxiaion seies. In his pape we popose HAM ehod o invesigae he seies soluions of a wo species aensalis odel wih a cove fo he fis species o poec i fo he aacks of he second species. Exaples: Aavian boulis and ed ides (caused by dino flagellaes) ae ofen exeely oxic fo bids, aine aals, huans; ec. Aensalis ay lead o he pe-epive colonizaion of a habia. Once an oganis esablishes iself wihin a habia i ay peven ohe populaions fo suviving in ha habia. The poducion of lacic acid o siila low-olecula-weigh fay acids is inhibioy o any baceial populaions. Populaions able o poduce and oleae high concenaions of lacic acids, fo exaple, ae able o odify he habia so as o peclude he gowh of ohe baceial populaions. E.coli is unable o gow in he uen, pobably because of he pesence of volaile fay acids poduced hee by anaeobic heeoophic icobial populaions. Fay acids 8

2 B. Sia Rababu e al Adv. Appl. Sci. Res.,, 3(3):8-86 poduced by icooganiss on skin sufaces ae believed o peven he colonizaion of hese habias by ohe icooganiss. Populaions of yeass on skin sufaces ae ainained in low nubes by icobial populaions poducing fay acids. Acids poduced by icobial populaions in he vaginal ac ae pobably esponsible fo pevening infecion by pahogens such as Candida albicans. In he pesen invesigaion a wo species aensalis odel wih liied esouces fo boh he species and a paial cove fo he fis species o poec i fo he aacks of he second species was aken up fo analyic sudy. The odel is epesened by coupled non-linea odinay diffeenial equaions. The seies soluion of he non-linea syse is appoxiaed by Hooopy Analysis Mehod suppoed by nueical exaples. The govening equaions of he syse ae as follows dx = ax( ) αx ( ) α( k) x( ) y( ) dy = a y y ( ) α ( ) (.) Wih he noaion x(),y() ae especively he populaions of species and species, and k is a cove povided fo he species <k<. Basic ideas of HAM: In his pape, we apply he hooopy analysis ehod o he discussed poble. To show he basic idea, le us conside he following diffeenial equaion N(u,) =, (.) Whee N is a nonlinea opeao, denoe independen vaiable, u () is an unknown funcion, especively. Fo sipliciy, we ignoe all bounday o iniial condiions, which can be eaed in he siila way. By eans of genealizing he adiional hooopy ehod, Liao consucs he so-called zeo-ode defoaion equaion ( p) L[ φ( ; p) u ( )] = phh ( ) N[ φ( ; p)], (.) Whee p [,] is he ebedding paaee, h is a nonzeo auxiliay, paaee H is an auxiliay,paaee H is an auxiliay funcion L is an auxiliay linea opeao, u () is an iniial guess of u(),φ(,p) is a unknown funcion, especively. I is ipoan ha one has gea feedo o choose auxiliay hings in HAM. Obviously, When P= and p=, i holds φ (, ) = u (), φ (, ) = u () (.3) Respecively. Thus as p inceases fo o, he soluion φ (; p) vaies fo he iniial guesses u () o he soluion u(). Expanding φ(; p) in Taylo seies wih espec o p,one has φ( ; p) u ( ) u ( ) p Whee + = + (.4) = φ( ; p) u( ) =! p p= If he iniial guess (.3),he auxiliay linea paaee L i,he non-zeo auxiliay paaee h i and he auxiliay funcion H i ae popely choosen,so ha he powe seies (.4) conveges a p=. (.5) 8

3 B. Sia Rababu e al Adv. Appl. Sci. Res.,, 3(3):8-86 Then we have unde hese assupions he soluion seies + u( ) u ( ) u ( ) = + (.6) = Which us be one of soluion s of oiginal non linea equaion, as poved by Liao [5].As h =- and H()=,Eq (.) becoes ( p) L[ φ( ; p) u ( )] + pn[ φ( ; p)] =, (.7) which is used osly in he hooopy peubaion ehod, wheeas he soluion obained diecly, wihou using Taylo seies which is explained by H. Jafai, M. Zabihi and M. Saidy [4] and J. H. He [5,6] and S. J. Liao [7] copae he HAM and HPM. Accoding o he definiion, he govening equaion can be deduced fo he zeoode defoaion equaion (.7). Define he veco u = { u u,... u } k, k Diffeeniaing Eq. (.7), ies wih espec o ebedding paaee p and hen seing p= and finally dividing he by!, we have he so-called h-ode defoaion equaion L[ u ( ) u ( )] hh ( ) R ( u ), χ = Whee N[ φ( ; p)] R ( u ) =, ( )! p and p=,, χ =, >. (.9) 3. Applicaion: Conside he nonlinea diffeenial equaion(.) wih iniial condiions.we assue he soluion of he syse(.),x(),y() can be expessed by following se of base funcions in he fo + + (3.) = = x( ) = a, y( ) = b Whee a,b ae coefficiens o be deeined. This povides us he so called ule of soluion expession i.e., he soluion of (.) us be expessed in he sae fo as (3.) and he ohe expessions us be avoided. Accoding o (.) and (3.) we chose he linea opeao. To solve he syse of Eqs.(.),Hooopy analysis ehod is eployed. We conside he following iniial appoxiaions x ( ) = x( = ) = x y ( ) = y( = ) = y (3.) The linea and non-linea opeaos ae denoed as follows. dx( ; p) dy( ; p) L[ x( ; p)], L[ y( ; p)], (3.3) dx( ; p) N[ x(, p)] = a x( ; p) + αx ( ; p) + α( k) x( ; p) y( ; p) (3.4) (.8) 83

4 B. Sia Rababu e al Adv. Appl. Sci. Res.,, 3(3):8-86 dy( ; p) N [ y(, p)] = a y( ; p) + α y ( ; p) (3.5) Using above definiion he zeo ode defoaion equaion can be consuced as ( p) L [ x( ; p) x ( )] = ph N [ x, y],( p) L [ y( ; p) y ( )] = ph N [ x, y] (3.6) When p= and p=,fo he zeo-defoaion equaions one has, x( ;) = x ( ) x( ;) = x( ), y( ;) = y ( ) y( ;) = y( ) And expanding x(;p) and y(;p) in Taylos seies, wih espec o ebedding paaee p, one obains + + (3.8) = = x( ; p) = x ( ) + x ( ) p, y( ; p) = y ( ) + y ( ) p Whee d x( ; p) d y( ; p) x ( ) =, y ( ) = (3.9)! dp! dp p= p= + + p = { x ( ) = x ( ) + x ( ), y ( ) = y ( ) + y ( ) = = (3.7) (3.) Define he veco x = [ x ( ), x ( ),... x ( )], y = [ y ( ), y ( ),... y ( )] (3.) And apply he pocedue saed befoe. The following h -ode defoaion Eq will be achieved. L[ x( ) χx ( )] = h H( ) R ( x, y ), (3.) L [ y ( ) χ y ( )] = h H ( ) R ( x, y ), Le us conside H ( ) = H ( ) = and he iniial condiions x ( ) = x( = ) = x y ( ) = y( = ) = y in above equaions d d R ( x, y ) = N[ x(, p)] = x ( ) a x + α x ( ) x ( ) + α ( k) x ( ) y ( ) n n n n ( )! dp n= n= d d R ( x, y ) = N[ y(, p)] = y ( ) a y + α y ( ) y ( ) (3.3) n n ( )! dp n= The following will be obained successively 84

5 B. Sia Rababu e al Adv. Appl. Sci. Res.,, 3(3):8-86 L ( x ( ) χx ( ) ) = h x ( ) ax ( ) + αx ( ) + α( k) x ( ) y( ) χ ( ) =, and, > ( ) L x ( ) = h ax ( ) + αx ( ) + α( k) x ( ) y( ) x ( ) = h ax + αx + α( k) x y L ( y( ) χy( ) ) = h y( ) a y( ) + α y ( ) ( ) L y( ) = h a y( ) + α y ( ) y( ) = h a y + α y L ( x ( ) χ x ( ) ) = h ( ) ( ) ( ) ( ) ( ) ( ) ( ) x a x + α x x + α k x y n n n n n = n = x ( ) = ( h + h ) M + a h M + α x h M + α ( k) y M + haα ( k) x whee M = a x + α x + α ( k) x y L y ( ( ) χ y ( ) ) = h ( ) ( ) ( ) ( ) y a y + α y y n n n = h y y ( ) = ( h + h ) a y + α y + ( a 3a α y + α y ) L ( x3 ( ) χ3x( ) ) = h x( ) a x ( ) + α xn ( ) x n( ) + α ( k) xn ( ) y n( ) n= n= x3 ( ) = ( + h )( h + h ) M + ( + h ) M + h α y( h + h ) M + α h x ( h + h )( a y + α y ) a h M + h αm x + αh M + hα h x y ( a 3a α y + α y ) + h hα M( a y + α y ) + h αm y 3 M = ah ( k) M + h αx M + α ( k) ym + hα α x ( k) y hα a x ( k) L ( y ( ) χ y ( ) ) = h y ( ) a y ( ) + α 3 3 n n n= y ( ) y ( ) y h k h k a h k h k y a h k h k y α h k ( ) = ( + ) 3 + [( + ) α ] α 4 + ( h + h ) 3 Whee 85

6 B. Sia Rababu e al Adv. Appl. Sci. Res.,, 3(3):8-86 k = a x + αx + αx y h x k = h a x y a x y h a y y k3 = ( h + h ) a y + α y ( α α )( α α ) α ( α ) h y k a a y y 4 = + ( 3 α α ) The hee es appoxiaion o he soluion will be consideed as x( ) x + x ( ) + x ( ) + x ( ) 3 y( ) y + y ( ) + y ( ) + y ( ) 3 REFERENCES [].Achayulu K.V.L.N. & Paabhi Raachayulu. N.Ch. In.J.Open pobles Cop.Mah (IJOPCM), Vol.3.No..pp.73-9., Mach. []. Achayulu K.V.L.N. & Paabhi Raachayulu. Inenaional jounal of copuaional Inelligence Reseach (IJCIR), Vol.6, No.3; pp , June. [3]. Achayulu K.V.L.N. & Paabhi Raachayulu. N.Ch; Inenaional Jounal of Applied Maheaical Analysis and Applicaions.vol 4,No,pp.49-6,July 9. [4]. Achayulu K.V.L.N. & Paabhi Raachayulu. N.Ch. ; On An Aensal-Eney Ecological Model Wih Vaiable Aensal Coefficien is acceped fo publicaion in Inenaional Jounal of Copuaional cogniion(ijcc),yang s Scienific Reseach Insiue, USA(in pess). [5] Shiva Pakash & T.Kaunanihi; Inenaional Jounal of Che. Tech Reseach Coden (USA). [6] S.J.Liao,The poposed hooopy analysis echnique fo he soluion of nonlinea pobles,phd hesis,shanghai Jiao Tong Univesiy 99. [7] S.J.Liao Beyond peubaion: inoducion o he hooopy analysis ehod.crc Pess, Boca Raon: Chapan & Hall (3) [8] S.J.Liao. Appl.Mah. Copa., 47(4), [9] S.J.Liao. Appl Mah Copu.,69(5), [] S.J.Liao In J Hea Mass Tansfe.,.48(5), [] M.Ayub, A.Rasheed, T.Haya. In J Eng Sci..4(3), 9-3. [] T.Haya, M.Khan. Nonlinea Dyn.,4:(5), [3] Fadi Awawdeh, H.M. Jaada, O. Alsayyed, Solving syse of DAEs by hooopy analysis ehod, Chaos, Soluions and Facals 4 (9) 4 47,Elsevie. [4] H.Jafai, M.Zabihi and M.Saidy. Appl.Mah.sci.,, 8), [5] J.H.He. Phys Le A.35 ()(6), [6] J.H.He, Appl.Mah. Mehod, 4(4) [7] SJ.Liao, Appl.Mah.Copu. 69 (5) [9]Shiva Reddy. K and Paabhiaachayulu, N.Ch. Adv. Appl. Sci. Res.,,, 3, 8-8. [] R. Silaha, R. Ravinda Reddy & N.Ch Paabhiaachayulu, Adv. Appl. Sci. Res.,,, 3, 5-65 [] R. Silaha, & N.Ch Paabhiaachayulu, Adv. Appl. Sci. Res.,,, 3, [] K. Madhusudhan Reddy and Lakshi Naayan.K Adv. Appl. Sci. Res.,,, 4, [3] B. Hai Pasad and N.Ch Paabhiaachayulu, Adv. Appl. Sci. Res.,,, 5,

Application of Homotopy Analysis Method for Solving various types of Problems of Partial Differential Equations

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