An Extensive Study of Approximating the Periodic. Solutions of the Prey Predator System

Transcription

1 pplid athmatical Scincs Vol. 00 no n xtnsiv Study of pproximating th Priodic Solutions of th Pry Prdator Systm D. Vnu Gopala Rao * ailing addrss: Plot No.59 Sctor-.V.P.Colony Visahapatnam Dsignation: Dpt. of pplid athmatics GI Collg of Scinc GI Univrsity Visahapatnam.P. India * corrsponding author. -mail: dvgrgitam@gmail.com P. Sri Hari Krishna Dpt of pplid athmatics GI Collg of Scinc GI Univrsity Visahapatnam.P. India bstract In this papr an approximation to th priodic solutions of th gnral Lota Voltrra pry prdator systm is obtaind using th Poincar Lindst mthod. In [] th proprtis of gnral solutions wr obtaind without considration of ral world complxitis. Hnc w analyz th systm aftr modifying it to bttr approximat th ral lif scnario and prov that th wor don by [] is a particular cas of our modl. athmatics subjct classifications: B5 D0 799 Kywords: Poincar - Lindst prturbation Lota Voltrra Pry Prdator. Introduction h mathmatical study of pry prdator systm in population dynamics has bn th subjct of svral rcnt paprs starting with th wor of Lota Voltrra. In th study of non linar systm of diffrntial quations such as th Lota Voltrra

2 5 D. Vnu Gopala Rao and P. Sri Hari Krishna quations analytical quations ar usually unnown. In this cas in ordr to analyz th bhaviour of th systm w usually rsort to intgral tchniqus such as prturbation tchniqu. Prturbation tchniqu dpnds on th xistnc of small or larg paramtrs in th non linar problms. Vrma Wilson and Burnsid [0] [] [] obtaind xact solutions of dx X a by d dy Y C X - d. d with various assumptions. K.N.urty and D.V.G.Rao [7] obtaind approximat analytical solutions of th quations dx a - b X - cy d X dy - a c X Y. d whr a b c a b ar positiv constants with out maing any of th abov assumptions on th constants.. Grozdanovsi and J.J. Shphrd [] considrd a simpl modl of pry prdator systm. and obtaind approximat priodic solutions. In th ral world to analyz th bhaviour of th solutions such simpl modl is of not much us. In som situations ach spcis may hav a ngativ fd bac rsulting in stabl coxistnc of both spcis. For our analysis w introduc ngativ fd bac for th pry population and dvlop approximat priodic solutions of th gnral Lota Voltrra pry prdator systm. using th Poincar Lindst mthod and analyz th bhaviour of th solutions nar th non trivial critical point. Hr it is shown that th rsults of [] ar particular cas of th prsnt wor. h phas plan rprsntation of th rsults is also givn. h gnral form of Lota Voltrra systm is considrd introducing ngativ fd bac for th pry population givn in. Introducing a chang in th systm by dfining th variabls c cc x t X yt Y a a c a b a ba t a. a c a For a stabl and positiv quilibrium point of th systm w rquir th inquality. a c < which implis 0 < <. a b hn th systm bcoms d xt xt x t x t yt

3 pproximating th priodic solutions 5 d y t yt xt yt. h non zro critical point of th systm is obtaind at xt yt. Nar th nontrivial quilibrium w introduc nw variabls that masur our distanc from. x t ut yt νt whr is small.5 s a rsult du t u t u t ν t u t ν t dv t u t u t ν t.. Priodic natur of th scond ordr quations. h systm of quations by taing 0 or ignoring th non linar trms in. du t u t ν t du t u t. combining th quations and writing thm as a scond ordr quation d ut dut ut 0. Lt ut b any solution of. considr a transformation ut mt yt whr m t yt ar twic continuously diffrntiabl functions. h computations ofu u and thir substitutions in. lads to m y m m y m m mt yt 0 hn quating th cofficint of y to zro it is sn that mt xp t hrfor yt satisfis a diffrntial quation y yt 0. whr is a constant and 0 < <. So it is concludd that if ut is a solution of. thn yt ut xp t is a solution of.. Similarly if yt is a solution of. thn ut yt xp t is a solution of. i... is oscillatory if and only if. is oscillatory. Hnc th priodic natur of. is provd.

4 5 D. Vnu Gopala Rao and P. Sri Hari Krishna. h Poincar Lindst mthod Introducing a tim scal wt whr w dpnds on w0. h non linar wdu systm. bcoms u u ν - u ν. d wd ν u u ν.. d and solutions can b xpctd as u u 0 u u.. ν ν 0 ν ν... w K w w.. Whr u i u i and ν i ν c ar priodic in. Now substituting ths xprssions into. and collcting li powrs of w obtain a squnc of diffrntial quations. Collcting th trms non containing w gt u o u o ν o ν o u o. quating th cofficint of on bothsids u wu o u uo ν u0ν 0 ν wν o u u. 0ν 0 Collcting th cofficint of on both sids w u o w u u u u0u ν u0ν uν 0 ν wν wν 0 u u0ν uν 0.5 h systm. has solutions u cos. 0 ν 0 sin cos Whr ar arbitary constants. By giving valus to initial conditions ar obtaind. On substituting th solutions. into. th following diffrntial quations ar obtaind. u u ν cos sin

5 pproximating th priodic solutions 55 w sin ν cos u [ sin cos ] w cos.7 Hr th following simpl rsult is usful for th systm of quations to hav a priodic solutions. dx For th systm y sin t B cos t highr harmonics dy x C sin t D cos t highr harmonics. to hav priodic solutions it is ncssary and sufficint that D 0 and B C 0. pplying th abov rsult. th systm.7 will hav priodic solutions only if w choos w 0 giving whr u [ Pcos Q sin R] [ cos U sin S] v.9 [ ] P 5 Q [ ] R S [ QK P ] U [ Q P ] Hr.9 is a particular solution of.7 and no complmntary solution is includd. Sinc th gnral solution of. will involv two arbitrary constants. Substituting.9 in to.5 and noting that w 0 w obtain

6 5 D. Vnu Gopala Rao and P. Sri Hari Krishna K u u θ ν [ cos θ sin cos θ sin θ ] W sin θ cos ν u [ cos θ sin θ w cos θ sin θ ] cos θ.0 whr [ S P R KQ] [ U Q P R ] [ P KQ] [ U Q P] [ QK { R P }] [ R U P U Q ] [ QK P ] [ P U U Q ] pplying th rsult. to.0 it obsrvd that to gt π priodic solutions w must choos w G. K whr G [ P K QK R KU ] and so w obtain th following particular solutions u sin KG K

7 pproximating th priodic solutions 57 cos } K KG K K { } [ sin N N K { } N N cos ] cos 9 ν sin. Whr ] 9 [ KQ U P P U KQ L } 9 { } { U P Q Q P U L 0 } { L L 0 } { L L [ ] N [ N Substituting u i v i & w i i in x u y ν w obtain cos { cos P t x [ G R Q { } sin G { }sin }] } Cos

8 5 D. Vnu Gopala Rao and P. Sri Hari Krishna [ } sin { N N } { Cos N N. a } sin { Cos t y } sin { U Cos S }cos [{ 9 }sin {.b G W. Whr is vry small amplitud of th dviations of x y from h rsults of. giv an approximation to th priod of ths orbits ] [ ω G Π Π.5 givs a highr ordr approximations to th priod for small amplitud. Put 0 U O S O R Q p thn L L G N N

9 pproximating th priodic solutions 59 W in.a.b and. thn w gt approximat priodic solutions of simpl modl q.. ar of th form. a x a cos sin Cos a a a Sm cos sin a Cos o a a y a sin sin cos a sin cos O a a W whr a 0 a Hr it is shown that th solutions of th modl. bcom a particular cas of th gnral Lota Voltrra quations with ngativ fd bac for th pry population... RSULS Using th approximations.a and.b w plot th approximat solutions of x and y for various initial conditions. Figur shows th rsults for Є 0.05 and 0. for. Figur shows th rsults for Є 0.5 and 0. for. Figur shows th rsults for Є 0.05 and 0.5 for. Figurs - show that th orbits associatd with our modl ar curvs that spiral in toward th critical point. If th valu of th amplitud Є is incrasd w obsrv distortion in th path of th spiral as shown in Fig whn compard to that in Fig. Similarly with incras in valu of at constant amplitud w obsrv a mard diffrnc in th path travrsd by th spiral.

10 0 D. Vnu Gopala Rao and P. Sri Hari Krishna Y X Fig : Plot of th approximat priodic solution for amplitud Є Y X Fig : Plot of th approximat priodic solution for amplitud Є

11 pproximating th priodic solutions Y X Fig : Plot of th approximat priodic solution for amplitud Є X t Fig : Plot of x and t for th approximat solution shown in Figur

12 D. Vnu Gopala Rao and P. Sri Hari Krishna Y t Fig 5: Plot of y and t for th approximat solution shown in Figur XY t Fig : Plot of th product xy and t for th approximat solution shown in Figur

13 pproximating th priodic solutions XY t Fig 7: Plot of xsolid lin and ydottd lin for th approximat priodic solution obtaind in Figur Figurs -7 compar th approximations for x y and xy obtaind from.a and.b. In Figur 7 th x approximation is qual in amplitud with th y approximation and hnc rsultd in th propr spiral shown in Figur. 5. Conclusions h approximat priodic solutions of th gnral Lota Voltrra pry prdator systm wr obtaind using th Poincar Lindst mthod aftr introducing ngativ fdbac in th pry population as th additional complxity thrby giving a bttr approximation to th ral lif scnario. W also provd that th wor don by [] is a particular cas of our modl. Rfrncs [] R.R. Burnsid not on xact solutions of th pry prdator quations. Bull. ath Biol

14 D. Vnu Gopala Rao and P. Sri Hari Krishna [] W.J. Cunningham Introduction to nonlinar nalysis. c Graw Hill Nw Yor 95. [] R. Grimshaw Nonlinar ordinary diffrntial quations. CRC prss US 99. []. Grozdanovsi and J.J. Shphrd pproximating th priodic solutions of Lota - Voltrra quations. ustralian mathmatical soc. 00. nziam J. 9 C- 007 PP C-C [5].J. Lota lmnts of physical biology. Williams & Wilins Batismor aryland US 95. [] J.D. urray athmatical Biology n introduction. Springr Vrlog Brlin Hidlbrg 00. [7] K.N. urty and D.V.G.Rao pproximat analytical solutions of th gnral Lota Voltrra quations. Journal of mathmatical analysis and applications [].G. Pilov n introduction to mathmatical cology. Wily Intrscinc Nw Yor 99. [9] F. Roth h priods of th Voltrra Lota systms. J.Rsn. ngrw ath [0] V.S. Varma xact solutions for a spcial pry-prdator compting spcis systm. Bull. ath. Biol [] V. Voltrra Variations and fluctuations of th numbr of individuals in animal spcis living togthr. ranslatd from 9 dition by R.N.Chapman. nimal cology rno Nw Yor US 9. [] J. Waldvogl h priod in th Voltrra - Lota prdator pry modl. SI J. Numrical nalysis [] J. Waldvogl h priod in th Voltrra - Lota systm is monotonic. J. ath nalysis and pplications C C55. [].J. Wilson In Varma s pry-prdator problms. Bull. ath. Biol Rcivd: pril 00