Solutions of Ecological models by Homotopy-Perturbation Method (HPM)

Transcription

1 Ques Journals Journal of Research in Alied Mahemaics Volume ~ Issue 0 (0) : 7- ISSN(Online) : 9-07 ISSN (Prin): Research Paer Soluions of Ecological models by Homooy-Perurbaion Mehod (HPM) B. Srinivasa Rao Senior Lecurer in Mahemaics Sir C. R. R. Auonomous College, Eluru Andhra Pradesh, India srinivasrao.banda@gmail.com Received 8 July, 0; Acceed 0 Augus, 0 The auhor(s) 0. Published wih oen access a ABSTRACT: In his aer, He s homooy erurbaion mehod(hpm) is used o find aroximae analyical soluions of a sysem of nonlinear differenial equaions which are he mahemaical models of wo ineracing secies in oulaion dynamics. The HPM mehod is sraigh forward, highly effecive and a romising ool for obaining he aroximae analyical soluion of non-linear ODE's. Three differen models having commensalism as ineracion beween he secies are considered. The funcional resonse differs in each model. He s Homooy-Perurbaion Mehod (HPM) is used in he resen sudy o find aroximae soluions of hese models and he soluion curves are drawn using MATLAB. The soluion curves obained by HPM and R-K h order mehod are deiced in he same grah and comared. Keywords: Homooy-Perurbaion Mehod (HPM), non-linear differenial equaions, commensalism, commensal, hos, Monod model. I. INTRODUCTION The non-linear henomena lay a crucial role in alied mahemaics and science. I is one of he mos simulaing areas of he research. In he research of as few decades [-] grea rogress was made in he develomen of mehods for obaining aroximae analyical soluions for non-linear differenial equaions arising in various fields of Science and Engineering. I is observed ha mos of he mehods require a edious analysis. Comaraively, He s Homooy Perurbaion Mehod (HPM) require less comlicaed analysis and easy o find he aroximae soluions of he non-linear differenial equaions. The Homooy- Perurbaion Mehod (HPM) was iniially roosed by Chinese mahemaician J.H.He [-]. The HPM is useful o obain exac or aroximae soluions of linear and non-linear differenial equaions. The rimary objecive of his mehod is o aroximae he acual soluion from iniial aroximaion as he homooy arameer, say, varies from 0 o. In his mehod a soluion is exressed as a series in which converges o he exac soluion. The key feaure of his HPM is ha linearizaion or discreizaion or round of errors can be evaded. The aroximaions converge raidly o exac soluions []. HPM has been used o find aroximae soluions effecively, easily and accuraely a large class of non-linear roblems in oulaion dynamics, and eidemic models[,,,7,8,9,0]. The objecive of his aer is o exlore he use of HPM for obaining aroximae soluions o biological models of oulaion dynamics having commensalism beween he secies. A comaraive sudy of hese soluions wih he soluions obained using h order Runge- Kua mehod (R-K mehod) is resened. In his aer hree biological models are considered in which he ineracion beween he secies is commensalism. In model- and model- he wo secies have limied food source and each secies follow logisic growh law in he absence of he oher. The model- is more comlex han model- in he sense ha he commensalism is characerised by a funcion of hos secies. Unlike model- and model-, model- he hos secies follows logisic growh law and has limied food sources whereas he commensal secies decline in he absence of hos secies. The commensal secies survive only because of he ineracion of he hos secies. Descriion of hese models is given in he nex secion. II. MODELS. Model-: This model deals wih ineracion of wo secies uilizing he same limied asses for inborn develomen of he secies. I is assumed ha he ineracion of he wo secies benefis one of he secies. The *Corresonding Auhor: B. Srinivasa Rao Senior Lecurer in Mahemaics Sir C. R. R. Auonomous College, Eluru Andhra Pradesh, India 7 Page

2 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) secies which ges benefi is called commensal secies and he secies which hels for he growh of he oher is called hos secies. Suose N N, N N reresens he size of oulaion of he commensal and hos secies resecively a any ime. The commensal secies (N ), in sie of he limiaion of is naural resources, flourishes by drawing srengh from he hos secies (N ). Suose a, a resecively reresen he inrinsic growh raes; a, a reresen he raes of reducion due o he limiaion of naural sources of he commensal and hos secies. a reresen beneficial facor of he commensal secies by ineracion wih hos secies. All hese arameers assume osiive values.the mahemaical model governing he commensalism beween wo secies is given by couled non-linear differenial equaions, dn a N a N a N N d dn a N a N () d Phani Kumar.eal. [] esablished ha he sysem has four equilibrium saes and he co-exisen sae aa aa a, is he only sable sae under he assumion a a a (A) aa aa aa. The global sabiliy is analysed by a suiably consruced Liaunov s funcion.. Model-: This model is concerned wih he ineracion of he wo secies having commensalism beween he secies. Commensal secies N is weak o susain, desie of he suor of he oher hos seciesn. The hos secies has heir limied food source and he secies N benefis by he ineracion wih he hos secies N. Mahemaically his model is reresened by: dn d N a N a N N d dn a N a N () d Here d reresens he naural deah rae of he commensal secies in he absence ofn and all oher arameers have same meaning as menioned in model-. Seshagiri Rao e.al [] sudied his model exensively and concluded ha (i) N will susain forever in he absence of Nand end o he equilibrium oin 0, a a condiion (B) (ii) In he coexisen sae he equilibrium oin d aa. aa The global sabiliy is analysed by a suiably consruced Liaunov funcion. a a d a a a, a a a is sable wih he. Model-: In his model he commensalism is reresened by FN,a funcion of he hos secies of he form FN N N, insead of a linear funcion a N in he model discussed in secion.. I is clear ha FN is bounded and FN a consan 0 as N. Furher 0 is a arameer which signifies he srengh of he commensalism. The commensalism is srong, weak or neural according as 0 ai 0 or 0.The raio Ki, i, is he carrying caaciy of commensal and hos secies. aii This mahemaical reresenaion of his model is dn a N a N N F( N) d, *Corresonding Auhor: B. Srinivasa Rao 8 Page

3 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) dn a N a N () d Phani Kumar and Paabhiramacharyulu [] esablished ha he sysem has four equilibrium saes and he coexisen sae (C) K a N, N = a K K K a K, K K K a. The global sabiliy is esablished by adoing Liaunov second mehod. is always sable under he assumion III. HOMOTOPY-PERTURBATION METHOD (HPM) The Homooy-Perurbaion Mehod is a combinaion of he classical erurbaion echnique and homooy echnique. To exlain he basic idea of homooy-erurbaion mehod, consider a non-linear differenial equaions. Au f r 0, r () u subjec o he boundary condiion B u, 0, r n boundary oeraor, where A is a general differenial oeraor, B is a f r is a known analyic funcion, is he boundary of he domain and n denoes differeniaion along he normal drawn ouwards from. In general he oeraor A be divided ino a linear ar L and a non-linear ar N. The equaion () can be wrien as Lu N u f r 0 (5) By he Homooy echnique [], consruc a funcion vr, : 0, which saisfies H v, Lv Lu0 Av f r 0, 0,, r H v, Lv Lu Lu N v f r 0 () 0 0 (7). 0, is an imbedding arameer. u0 is iniial aroximae soluion o equaion (),which saisfies he boundary condiions. From he equaions (), (7) H v Lv Lu0 and H v Av f r Thus i can be observed ha as he arameer P varies from zero o uniy vr, varies from,0 0 (8), 0 (9) Assume vr, as a ower series in u0 r o v v0 v v v v (0) The limi as v aroaches he soluion of equaion () u L v v v v v v () 0 ur.. Soluions Of The Models By HPM In his secion we aly he HPM differenial equaions (), () and (). exlained in secion o he sysem of non-linear ordinary... Soluion Of Model- Consider he equaions () wih iniial aroximaions v N v 0 c v N v c 0 0 As exlained in secion., he equaions () can be wrien as v N N a v a v a v v v N N a v a v *Corresonding Auhor: B. Srinivasa Rao () () 9 Page

4 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) Assume he aroximae soluions of N, N as v v,0 v, v, v, v, () v v,0 v, v, v, v, vij, i,; j,,,,... are o be deermined by subsiuing equaions () and () ino equaion () and arranging he erms in he increasing owers of v, N 0 av,0 av,0 av,0 v,0 v, av, av,0 v, av,0 v, av, v,0 v, av, av,0 v, av, av,0 v, av, v, av, v,0 v, av, av,0 v, av, v, av,0 v, av, v, av, v, av, v,0 v, N 0 av,0 av,0 v, av, av,0 v, 0 v, av, av,0 v, av, v, av, av,0 v, av, v, 0 (5) To obain he unknowns vij,, i,; j,,, solve he following sysem of linear differenial equaions wih he iniial condiions given in () From (5) v N a v a v a v v 0, v 0 0 (), 0,0,0,0,0,, 0,0,0, v N a v a v 0, v 0 0 (7) v a v a v v a v v a v v 0, v 0 0 (8),,,0,,0,,,0, v a v a v v 0, v 0 0 (9),,,0,, v a v a v v a v a v v a v v a v v 0, v 0 0 (0),,,0,,,0,,,,,0,,,,0,,, v a v a v v a v 0, v 0 0 () v a v a v v a v v a v v a v v a v v a v v 0, v 0 0 (),,,0,,,,0,,,,,,,0, v a v a v v a v v 0, v 0 0 (),,,0,,,, Soluions of he differenial equaions ()-() are given by () v a v d a v d a v v d a a c a c c,,0,0,0,0,,0,0 (5) v a v d a v d a a c c 0 0 v, a v, d a v,0 v, d a v,0 v,d a v, v,0d 0 = a ac ac ac ac ac c a a ac cc,,,0, 0 0 (7) v a v d a v v d a a c c a a c () *Corresonding Auhor: B. Srinivasa Rao 0 Page

5 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) v, a v, d a v,0 v, d a v, d a v,0 v,d a v, v,d a v, v,0d = a ac ac a ac ac ac ac ac c a a ac cc + a ac ac c a a ac c ac a ac ac ac c a ac a ac v a v d a v v d a v d a a c c a a c a c a a c,,,0,, v, a v, d a v,0 v, d a v, v, d a v,0 v,d a v, v,d a v, v,d a v, v,0d 0 a ac ac a ac ac ac ac ac c a a ac cc = a ac ac c a a ac c ac a ac ac a cc a ac a ac + a ac ac c ac a ac a ac a a ac ac ac ac ac c a a ac cc 8 + ac c a ac a ac ac a ac + ac a ac a ac ac ac ac ac c a a ac cc 8 a a c a c v, a v,d a v,0v,d a v,v, d = a ac a ac c a ac 8ac a ac The aroximae soluion of,, k k 0 N N is N L v v () N L v v (), k k0 which yield N c a ac ac c a ac ac ac ac acc a a ac cc a ac ac a ac ac ac ac acc a a ac cc + +a ac ac c a a ac c ac a ac ac acc a ac a ac + a ac ac a ac ac ac ac acc a a ac cc a ac ac c a a ac c ac a ac ac acc a ac a ac +a ac ac c ac a ac a ac a a ac ac ac ac acc a a ac cc + acc a ac a ac ac a ac +ac a ac a ac ac ac ac acc a a ac cc a a c a c (8) (9) (0) () ) *Corresonding Auhor: B. Srinivasa Rao Page

6 N N Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) N c a ac c a ac c a ac a ac c a ac ac a ac + a ac a ac c a ac 8ac a ac (5) The values of he arameers in equaion() which saisfy condiion (A) (see secion.) are aken as a.5;a 0.05;a 0.;a.78;a 0.0; N0 ; N0 The aroximae soluions of equaion () are obained by using equaions () and (5) and heir grahical reresenaions are given in figure and Figure. In hese figures soluion obained by h order Runge- Kua mehod are also deiced. I is observed from he figures and ha he soluions obained by he wo mehods agree u o he calculaed degree of accuracy..5 HPM Mehod R K Mehod ime() Fig.. For he values a.5;a 0.05;a 0.;a.78;a 0.0; N0 ; N0,Comarison beween he soluions of N (Eq()) obained by HPM ( calculaed u o erms) and R-K mehod. 5.5 HPM Mehod R K Mehod ime() Fig.. For he values a.5;a 0.05;a 0.;a.78;a 0.0; N0 ; N0,Comarison beween he soluions of N (Eq()) obained by HPM ( calculaed u o erms) and R-K mehod.... Soluion of model- Consider he sysem of equaions () wih iniial aroximaions v N v 0 c v N v c 0 0 As exlained in secion., he equaions () can be wrien as () *Corresonding Auhor: B. Srinivasa Rao Page

7 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) v N N d v a v a v v v N N a v a v Assume he aroximae soluions of N, N as v v,0 v, v, v, v, (8) v v,0 v, v, v, v, vij, i,; j,,,,... are o be deermined by subsiuing equaions () and (8) ino equaion (7) and arranging he erms in he increasing owers of. v, N 0 dv,0 av,0 av,0 v,0 v, dv, av,0 v, av,0 v, av, v,0 v, dv, av,0 v, av, av,0 v, av, v, av, v,0 v, dv, av,0 v, av, v, av,0 v, av, v, av, v, av, v,0 0 v, N 0 av,0 av,0 v, av, av,0 v, v, av, av,0 v, av, v, av, av,0 v, av, v, 0 (9) To obain he unknowns vij,, i,; j,,, solve he following sysem of linear differenial equaions wih he iniial condiions given in () From (9), 0,0,0,0,0, v N d v a v a v v 0, v 0 0 (0), 0,0,0, v N a v a v 0, v 0 0 () v d v a v v a v v a v v 0, v 0 0 (),,,0,,0,,,0, v a v a v v 0, v 0 0 (),,,0,, v d v a v v a v a v v a v v a v v 0, v 0 0 (),,,0,,,0,,,,,0,,,,0,,, v a v a v v a v 0, v 0 0 (5) v d v a v v a v v a v v a v v a v v a v v 0, v 0 0 (),,,0,,,,0,,,,,,,0, v a v a v v a v v 0, v 0 0 (7),,,0,,,, Soluions of he differenial equaions (0)-(7) are given by (8) v d v d a v d a v v d d a c a c c,,0,0,0,0,,0,0 (9) v a v d a v d a a c c 0 0 v, d v, d a v,0 v, d a v,0 v,d a v, v,0d 0 = d ac ac dc ac ac c a a ac cc,,,0, 0 0 (5) v a v d a v v d a a c c a a c (7) (50) *Corresonding Auhor: B. Srinivasa Rao Page

8 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM),,,0,,,0,,,,,0 v d v d a v v d a v d a v v d a v v d a v v d = d a c a c d a c a c d c a c a c c a a a c c c + d ac ac c a a ac c ac d ac ac ac c a ac a ac (5) v, a v,d a v,0v,d a v, d a ac c a ac ac a ac (5) v, d v, d a v,0 v, d a v, v, d a v,0 v,d a v, v,d a v, v,d a v, v,0d 0 d ac ac d ac ac dc ac ac c a a ac cc = d ac ac d ac ac c a a ac c ac d ac ac ac c a ac a ac + d ac ac c ac a ac a ac a d ac ac dc ac ac c a a ac cc 8 + ac c a ac a ac ac a ac + ac a ac d ac ac dc ac ac c a a ac cc 8 v, a v,d a v,0v,d a v,v, d = a ac a ac c a ac 8ac a ac The aroximae soluion of N, N is given by, k k 0 N L v v (5) N L v v (57), k k0 which yield N c d ac ac c d ac ac dc ac acc a a ac cc d ac ac d ac ac dc ac acc a a ac cc + +d ac ac c a a ac c ac d ac ac acc a ac a ac + d ac ac d ac ac dc ac acc a a ac cc d ac ac c a a ac c ac d ac ac acc a ac a ac +d ac ac c ac a ac a ac a d ac ac dc ac acc a a ac cc + acc a ac a ac ac a ac +ac a ac d ac ac dc ac acc a a ac cc d a c a c (58) (5) (55) *Corresonding Auhor: B. Srinivasa Rao Page

9 N N Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) a a c a a c c a a c a c a a c N c a ac c a ac c a ac a ac c a ac ac a ac + 8 (59) The values of he arameers in equaion() which saisfy condiion (B) (see secion.) are aken as d 0.;a 0.;a 0.007;a.;a 0.5; N0 ; N0 The aroximae soluions of equaion () are obained by using equaions (58) and (59) and heir grahical reresenaions are given in figure and Figure. In hese figures soluion obained by h order Runge- Kua mehod are also deiced. I is observed from he figures and ha he soluions obained by he wo mehods agree u o he calculaed degree of accuracy HPM Mehod R K Mehod ime() Fig..For he values d 0.;a 0.;a 0.007;a.;a 0.5; N0 ; N0,Comarison beween he soluions of N (Eq()) obained by HPM ( calculaed u o erms) and R-K mehod.8 HPM Mehod R K Mehod ime() Fig..For he values d 0.;a 0.;a 0.007;a.;a 0.5; N0 ; N0,Comarison beween he soluions of N (Eq()) obained by HPM ( calculaed u o erms) and R-K mehod.. Soluion of Model- Consider he sysem of equaions () wih iniial aroximaions v N v 0 c v N v c 0 0 As exlained in secion., he equaions () can be wrien as (0) vv v N 0 N 0 av av 0 v *Corresonding Auhor: B. Srinivasa Rao 5 Page

10 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) v N N a v a v () Assume he aroximae soluions of soluions of N, N as v v,0 v, v, v, v, () v v,0 v, v, v, v, vij, i,; j,,,,... are o be deermined by subsiuing equaions (0) and () ino equaion () and arranging he erms in he increasing owers of. v, N 0 av,0 av,0 v,0 v,0 v,0 v,0 v,0 v,0 v,0 v,0 v,0 v, v, v,0 v,0 v,0 v, v, v,0 + v, av, av,0 v, v,0 v, v,0 v, v,0 v,0 v,0 v, v, v,0 v,0 v,0 v, v,0 v, v,0 v, v, v, v, v,0,,0,,,0 v v v v v + v, av, av,0 v, av, v,0 v, v,0 v,0 v,0 v, v,0 v,0 v, v,0 v, v,0 v, v, v,0 v, v,0 v, v,0 v, v, v,0 v,0 v,0 v, v,0 v, v, v, v,0 v, v v v v v v v v,0 v, v, v, v, v, v, v,0,,,,0,,,0 + v, av, av,0 v, av, v, v,0 v, v,0 v,0 v,0 v, v, v,0 v, v, v, v,0 v, v,0 v, v, v, v,0 v, v,0 v,0 v,0 v, v,0 v, v, v,0 v,0 v, v,0 v, v,0 v, v, v, v,0 v, v,0 v, v, v,0 0 v, N 0 av,0 av,0 v, av, av,0 v, v, av, av,0 v, av, v, av, av,0 v, av, v, 0 () To obain he unknowns vij,, i,; j,,, solve he following sysem of linear differenial equaions wih he iniial condiions given in () From () v, N 0 av,0 av,0 v,0 v,0 v,0 v,0 v,0 v,0 v,0 v,0 0, v, 0 =0 (), 0,0,0, v N a v a v 0, v 0 0 (5) v,0 v, v, v,0 v,0 v,0 v, v, v,0 v, av, av,0 v, 0, v, 0 0 v,0 v, v,0 v, v,0 v,0 v,0 v, v, v,0 () v a v a v v 0, v 0 0 (7),,,0,, *Corresonding Auhor: B. Srinivasa Rao Page

11 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) v,0 v,0 v, v,0 v, v,0 v, v, v, v, v,0 v, v,0 v, v, v,0 v, av, av,0 v, av,, 0, v 0 0 v,0 v, v,0 v,0 v,0 v, v,0 v,0 v, v,0 v, v,0 v, v, v,0 v, v,0 v, v,0 v, v, v,0 (8),,,0,,, v a v a v v a v 0, v 0 0 (9) v,0 v,0 v, v,0 v, v, v, v,0 v, v v v v v v v v,0 v, v, v, v, v, v, v,0,,,,0,,,0 v, av, av,0 v, av, v, v,0 v, v,0 v,0 v,0 v, v, v,0 v, 0, v, 0 0 v, v, v,0 v, v,0 v, v, v, v,0 v, v,0 v,0 v,0 v, v,0 v, v, v,0 v,0 v, v,0 v, v,0 v, v, v, v,0 v, v,0 v, v, v,0 v a v a v v a v v 0, v 0 0 (7),,,0,,,, Soluions of he differenial equaions ()-(7) are given by v, a v,0 d a v,0 d v,0 v,0d v,0 v,0 d v,0v,0 d v,0v,0 (7) = c a ac cv,,0,0 0 0 (7) v a v d a v d a a c c v,0 v,d v, v,0d v,0 v,0v,d v, v,0 d v, a v, d a v,0 v, d 0 0 v,0v,v,0 d v, v,0 d v,0v,0 v,d v, v,0 d = c a ac cv a ac cv cc V a ac,,,0, 0 0 (75) v a v d a v v d a a c c a a c v,0 v,0v,d v,0 v, d 0 0 v,0 v,d v, v,d v, v,0d v, v,0v,d v, v,0 d 0 0 v, a v, d av,0 v, d av, d v,0 v,v,0 d v,0 v,0v, d v,0 v,0 v,d v,0 v, v,0 d v, v,v,0 d v, v,0 d v, v,0 v,d v, v,0 d = a ac cv c a ac cv a ac cv cc V a ac a c a ac cv cc V a ac a + a c c c V a a c a ac cv cc V a ac (70) (7) (7) *Corresonding Auhor: B. Srinivasa Rao 7 Page

12 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) (77) v a v d a v v d a v d a a c c a a c a c a a c,,,0,, v,0 v,d v, v,d v, v,d v, v,0d 0 v,0 v,0v,d v,0 v,v, d v, v,0v,d v, v, d v, v,0v,d v,v,0 d v, av,d av,0 v, d av, v, d v,0v,v,0 d v,0 v,0v,v, d v,0 v, d v, v,v,0 d v, v,0v, d v, v,v,0 d v, v,0 d 0 v, 0v,0 v,d v,0 v,v, v,0 d v,0 v, v,0d v, v,0 v,d v, v, v,0 d v, v,0 v,d v, v,0 d 0 a ac cv c a ac cv a ac cv cc V a ac = a ac cv ac a ac cv cc V a ac a a c cv a ac cv a ac c cc V a ac ac a ac cv c a ac cv a ac cv cc V a ac 8 cc V a ac a ac ac a ac c V a ac cv c a ac a ac 8 + V c a ac cv a ac cv cc V a ac c a ac 8 cv a ac c a a c c V a ac cv a ac c 8 + c c c a ac v, a v,d a v,0v,d a v,v, d = a ac a ac c a ac 8ac a ac V c c c ; V c c c ; V c c where The aroximae soluion of N, N is given by (by considering he firs erms), k k 0 N L v v (80) N L v v (8), k k0 (78) (79) which yield *Corresonding Auhor: B. Srinivasa Rao 8 Page

13 N Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) N c c a ac cv c a ac cv a ac cv cc V a ac a ac cv c a ac cv a ac cv cc V a ac ac a ac cv + cc V a ac a ac cc V a ac a ac cv c c V a ac a ac cv c a ac cv a ac cv cc V a ac ac a ac cv cc V a ac a ac cv a ac cv a ac c c c V a ac ac a ac cv c a ac cv a ac cv cc V a ac + a ac cv c cv a ac a ac ac a ac c V a a c cv c a ac a ac + V c a ac cv a ac cv cc V a ac c a ac cv a ac c a ac c V a a c cv a ac c + c c c a ac a a c a a c c a a c a c a a c N c a ac c a ac c a ac a ac c a ac ac a ac + 8 (8) The values of he arameers in equaion() which saisfy condiion (C) (see secion.) are aken as a ;a.; ; q.;a 0.0;a 0.5; N ; N 0 0 The aroximae soluions of equaion () are obained by using equaions (8) and (8) and heir grahical reresenaions are given in figure 5 and Figure. In hese figures soluion obained by h order Runge- Kua mehod are also deiced. I is observed from he figures5 and ha he soluions obained by he wo mehods agree u o he calculaed degree of accuracy. (8) 0.98 HPM Mehod R K Mehod ime() Fig.5. For he values a ;a.; ; q.;a 0.0;a 0.5; N0 ; N0,Comarison beween he soluions of N (Eq()) obained by HPM ( calculaed u o erms) and R-K mehod *Corresonding Auhor: B. Srinivasa Rao 9 Page

14 N Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) 0.99 HPM Mehod R K Mehod ime() Fig.5. For he values a ;a.; ; q.;a 0.0;a 0.5; N0 ; N0,Comarison beween he soluions of N (Eq()) obained by HPM ( calculaed u o erms) and R-K mehod IV. CONCLUSIONS AND RESULTS. In his aer using He s homooy erurbaion mehod, aroximae analyical soluions of a sysem of nonlinear differenial equaions which reresen he ineracion beween wo secies are evaluaed. Three differen models of oulaion dynamics wih commensalism beween he secies are considered and heir aroximae soluions (firs four erms of he series soluions, see secions..-..) are obained. The soluions of he oulaion models are also evaluaed by R-K mehod of h order and comared wih ha of he soluions obained by uilising HPM. In Figs.and he soluion curves of he mahemaical model of oulaion dynamics (equaion ()) obained by he HPM and R-K mehods are deiced. I can be observed in Fig & ha here is a very close aroximaion beween he soluions for N (Commensal oulaion) and N (Hos oulaion) in he ime inerval [0, 0.5] by using only erms of he series given by Eq. (), which indicaes ha he seed of convergence of HPM is very fas. A beer aroximae analyical soluion for 0.5 can be achieved by adding more erms o he series in Eq. (). In he similar way for he oher wo models (Eqns()&()) are also deiced in figs,,5and. The grahs indicae ha he aroximae soluion curves obained HPM (by considering he firs four erms in he series) agree wih he soluion curves obained by R-K mehod of h order Figs. and indicae ha he soluion curves of Eq.() by HPM for N and N almos idenical wih he soluion curves by R-K mehod in he ime inerval [0, 0.5]. Figs.5 and demonsrae ha he soluion curves by four-erm HPM and R-K mehod of he sysem in Eq. () are indisinguishable in he ime inerval [0, 0.8] These numerical soluions are obained by using ode5, an ordinary differenial equaion solver of MATLAB. REFERENCES []. Serdal Pamuk, Nevin Pamuk, He s homooy erurbaion mehod for coninuous oulaion models for single and ineracing secies, Comuers and Mahemaics wih alicaions 59 (00) -. []. I.Hasim, M.S.H.Chowdhury, Adaaion of homooy-erurbaion mehod for numeric-analyic soluion of he sysem of ODE s, Physics Leers A 7 (008) []. J.H.He, Homooy erurbaion mehod for solving boundary value roblems, Physics Leers A 50 (00) []. J.H.He, Homooy erurbaion mehod: A new non-linear analyical echnique, Alied Mahemaics and Comuaion 5 (00) [5]. I.Hasim, M.S.M.Noorani, R.Ahmad, S.A.Bakar, E.S.Ismail, A. M.Zakaria, Chaos soluion fracals 8(00) 9 []. M.Rafei, D.D.Ganji, H.Daniali, Soluion of he eidemic model by homooy erurbaion mehod, Alied Mahemaics and Comuaion 87 (007) 05-0 [7]. DenizAgirseven, Turgu Ozis, An analyical sudy for Fisher ye equaions by using homooy erurbaion mehod, Comuers and Mahemaics wih Alicaions 0 (00) 0-09 [8]. Ji-Hunan He, An elemenary inroducion o he homooy erurbaion mehod, Comuers and Mahemaics wih Alicaions 57 (009) 0-. [9]. Franciso M.Fernandez, on some aroximae mehods for non-linear models, Alied Mahemaics and comuaion, 5 (009) 8-7. [0]. HosseinAminikhah, Jafar Biazar, A New Analyical Mehod for Sysem of ODEs, wiley InerScience-9. []. J.D.Murray, Mahemaical Biology, Sringer, Berlin, 99. []. N.Phani kumar, N.Ch.Paabhiramacharyulu, on a commensal-hos ecological model wih a variable commensal coefficien, Aca Cincea Indica, Meeru, (00) *Corresonding Auhor: B. Srinivasa Rao 0 Page

15 Soluions Of Ecological Models By Homooy-Perurbaion Mehod (HPM) []. N.Phani kumar,n.seshagirirao, N.Ch.Paabhiramacharyulu, On he sabiliy of a hos-a flourishing commensal secies air wih limied resources Inernaional Journal of Logic Based Inelligen Sysems, Vol., No., January June 009, []. N.SeshagiriRao,N.Ch.Paabhiramacharyulu,N.Phani kumar, On he sabiliy of a hos-a declining ommensal secies air wih limied resources Inernaional Journal of Logic Based Inelligen Sysems, Vol., No., January June 009, *Corresonding Auhor: B. Srinivasa Rao Page