A Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique

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1 Inrnionl hmil orum no Sud of h Soluions of h o Volrr r rdor Ssm Using rurion Thniqu D.Vnu ol Ro * D. of lid hmis IT Collg of Sin IT Univrsi Vishnm.. Indi Y... Thorni D. of lid hmis IT Collg of Sin IT Univrsi Vishnm.. Indi sr This r dvlos roxim nlil soluions of h gnrl form of o Volrr r rdor ssm using h rurion hniqu. Hr i is shown h rsuls of h rvious wor [] om riulr s of h rsn wor. hmis suj lssifiions: D 799 Kwords: rurion hniqu o Volrr r rdor * orrsonding uhor -mil ddrss: Inroduion Th mhmil sud of r rdor ssms in oulion dnmis hs n h suj of svrl rn rs sring wih h wor of o Volrr. In h sud of non linr ssm of diffrnil quions suh s in h o Volrr quion nlil soluions r usull unnown. In his s

2 668 D.Vnu ol Ro nd Y... Thorni in ordr o nlz h hviour of h ssm w usull rsor o numrill ingrd hniqus suh s rurion hniqu. Th rurion hniqu dnds on h xisn of smll or lrg rmrs in h non linr rolms. Th gnrl form of quions whih w onsidrd hr is x& x x x. x& x x x hr x is r oulion nd x is rdor oulion whr r ll osiiv onsns wih ngiv fd i.. >. Vrm [] oind x soluions of h quions. x& x x x & x x. undr h ssumion. Wilson [6] gv h form of x soluions of. wih h ssumion nml whr r funions of im whr is onsn. urnsid [] gv h form of x soluions wih n ssumion & & K..urh nd D.V..Ro [] oind roxim nlil soluions of h quions. x& x x x. x & x x Ths r dvlos roxim nlil soluions of h gnrl o Volrr quions. using rurion hniqu. Hr i is shown h h soluions of. om riulr s of... rurion hod Th rurion mhod [] hs n usd widl in non linr mhnis. This mhod n lid o ir of firs ordr diffrnil quions of h. x & f x μ g x. & f x μ g x wih h iniil ondiions x nd. Th linr rms r wrin in funions f nd f. Th non linr rms r wrin in funions g nd g. Th rmr μ idll smll rmr is ssoid wih g nd g. Th soluion is found s owr sris in μ. This sris will onvrg ridl if μ is smll. In ri μ is inrodud rifiill. Th sris soluion sough is of h form

3 Soluions of h o Volrr r rdor ssm 669 x x μ x μ x.... μ μ... Th rms x nd r lld gnring rms nd r h x soluions of h linr quions x & f x & f x. Th rms x x r lld orrion rms. n imorn fur of h rurion mhod is h oh gnring nd orrion rms in h soluions r oind solving linr diffrnil quions onl.. liion o gnrl r rdor ssm of quions Th o Volrr quions. hv h quilirium oin. sids h quilirium oins Dfining h vrils x x. w n oin h diffrnil quions in nd s givn low. Th vrils nd rrsn r nd rdor oulion dviions from h quilirium vlu givn in quion. & &. Inroduing h rmr µ ino h nonlinr rms of h ov quions ling h rurion hniqu. & μ μ μ μ &. Soluions r sough in h form μ μ.... μ μ... Th rmr μ will s qul o uni fr h soluions r drmind. susiuing. ino. nd quing qul owrs of μ on oh sids w oin &.6 & & &.7

4 67 D.Vnu ol Ro nd Y... Thorni Th iniil ondiions r lid o h gnring soluions nd. Th diffrnil quions for orrsonding rms hn hv zro iniil ondiions. Th soluions of h linr quions.6 r oind s [ ].8 [ ].9 whr whr wih whr nd r iniil ondiions of nd rsivl. Hving drmind nd quions.7 w nx solvd for nd o ild.. whr Q R R

5 Soluions of h o Volrr r rdor ssm 67 Q Q R Q R R Q Q R nd Q R

6 67 D.Vnu ol Ro nd Y... Thorni Using hs firs orrion rms n rori soluion o. is. In rms of h originl r nd rdor oulions x nd x. h soluions r x. x. Susiuing in.. h quilirium oin of. is wih Thn w g roxim nlil soluions of. of h form x x whr i i onsns oind ing in. nd..

7 Soluions of h o Volrr r rdor ssm 67 Hr i is shown h h soluions of h o volrr quions. m riulr s of h gnrl o Volrr quions.. us of h ddiionl orrion rms h ov soluions n xd o mor ur nd vlid for lrgr dviions from h quilirium oin. Th ur of h soluions rurion mhod n imrovd furhr drmin sond nd highr ordr orrion rms.. Conlusions Th roxim nlil soluions of h gnrl o Volrr quions hv n drmind using rurion mhod. Th rurion mhod n lid o ohr non linr mhmil modls of oulion dnmis. Rfrns [] R.R.urnsid no on x soluions of h r-rdor quions ull. h iol [] W.J. CunningHm Inroduion o nonlinr nlsis rw Hill w Yor 98. [] K..ur nd D.V..Ro roxim nlil soluions of nrl o Volrr quions Journl of hmil nlsis nd liions w Yor Vol o [].. ilov n Inroduion of hmil olog Wil Inr sin w Yor 969. [] V.S.Vrm x Soluions of Sil r-rdor or Coming Sis Ssm ull h. iol [6].J.Wilson Vrm s r-rdor rolm ull h iol Rivd: ril

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