On the System of Three Order Rational Difference Equation

Transcription

1 British Joural of Applie Sciece & Techology 7(3): Article objast27524 ISSN: NLM ID: SCIENCEDOMAIN iteratioal wwwscieceomaiorg O the System of Three Orer Ratioal Differece Equatio Guozhua Hu Faculty of Mathematics a Physics Huaiyi Istitute of Techology Huaia Jiagsu 2233 Chia Author s cotributio The sole author esige aalyze a iterprete a prepare the mauscript Article Iformatio DOI: 9734/BJAST/26/27524 Eitor(s): () Qig-We Wag Departmet of Mathematics Shaghai Uiversity PR Chia Reviewers: () Mohame Mosleh Elysaa Abu Samak Uiversiti Kebagsaa Malaysia Malaysia (2) Qiahog Zhag Guizhou Uiversity of Fiace a Ecoomics PR Chia (3) Qamar Di The Uiversity of Pooch Rawalakot Pakista (4) Jayakumar J Poicherry Egieerig College Poicherry Iia (5) Iyakio P Akpa College of Agriculture Lafia Nigeria Complete Peer review History: Receive: 6 th Jue 26 Accepte: 3 th July 26 Origial Research Article Publishe: 3 st August 26 Abstract This paper is cocere with the local a global asymptotic behavior of positive solutio for a system of three orer ratioal ifferece equatios + + α + β + where α β ( ) a the iitial values x x ( ) y y ( ) Fially some umerical examples are provie to illustrate theoretical results obtaie Keywors: Differece equatios; equilibrium poit; rate of covergece; global asymptotic behavior 2 Mathematics Subject Classificatio: 39A *Correspoig author: hugzmath@63com;

2 Hu; BJAST 7(3) -7 26; Article objast27524 INTRODUCTION Differece equatios or iscrete yamical systems are iverse fiels which impact almost every brach of pure a applie mathematics Every yamical system + f( ) etermies a ifferece equatio a vise versa Recetly there has bee great iterest i stuyig ifferece equatios systems Oe of the reasos for this is a ecessity for some techiques which ca be use i ivestigatig equatios arisig i mathematical moels escribig real life situatios such as populatio biology [ 2] ecoomic probability theory geetics psychology etc I particular ratioal ifferece equatios have appeale more a more scholars ue to their wie applicatio For etail see [ ] Kurbali [3] stuie a three-imesioal system of ratioal ifferece equatios + y + z + z z where the iitial coitios are arbitrary real umbers Ciar et al [4] has obtaie the positive solutio of the ifferece equatio system + m + p Ciar [5] has obtaie the positive solutio of the ifferece equatio system + + Also Ciar [6] has obtaie the positive solutio of the ifferece equatio system + + z + z Ozba [7] has ivestigate the positive solutios of the system of ratioal ifferece equatios + + k m m+k Papaschiopoulos et al [8] ivestigate the global behavior for a system of the followig two oliear ifferece equatios + A + + A + y p x q where A is a positive real umber p q are positive itegers a x p x y q y are positive real umbers I 22 Zhag Yag a Liu [9] ivestigate the global behavior for a system of the followig thir orer oliear ifferece equatios B + 2 A + 2 where A B ( ) a the iitial values x i y i ( ) i 2 Although ifferece equatios are sometimes very simple i their forms they are extremely ifficult to uersta thoroughly the behavior of their solutios I book [25] Kocic a Laas have stuie global behavior of oliear ifferece equatios of higher orer Similar oliear systems of ratioal ifferece equatios were ivestigate (see [26]) Other relate results reaer ca refer [ ] Motivate by above iscussio our goal i this paper is to ivestigate the solutios of the two-imesioal system of three orer ratioal oliear ifferece equatios i the form + + α + β + () where α β ( ) a the iitial values x x y a y ( ) Moreover we have stuie the local stability global stability boueess of solutios We will cosier some special cases of () as applicatios Fially we give some umerical examples 2

3 Hu; BJAST 7(3) -7 26; Article objast MAIN RESULTS Let I x I y be some itervals of real umber a f : Ix 2 Iy 2 I x g : Ix 2 Iy 2 I y be cotiuously ifferetiable fuctios The for every iitial coitios (x i y i) I x I y(i ) the system of ifferece equatios + f( ) + g( ) 2 (2) has a uique solutio {( )} A poit ( x ȳ) I x I y is calle a equilibrium poit of (2) if x f( x x ȳ ȳ) ȳ g( x x ȳ ȳ) i e ( ) ( x ȳ) for all Defiitio 2 Assume that ( x ȳ) be a fixe poit of (2) The (i) ( x ȳ) is sai to be stable relative to I x I y if for every ε > there exists δ > such that for ay iitial coitios (x i y i) I x I y(i ) with i xi x < δ i yi ȳ < δ implies x < ε ȳ < ε (ii) ( x ȳ) is calle a attractor relative to I x I y if for all (x i y i) I x I y(i ) lim x lim ȳ (iii) ( x ȳ) is calle asymptotically stable relative to I x I y if it is stable a a attractor (iv) Ustable if it is ot stable Theorem 2 [25] Assume that X( + ) F (X()) is a system of ifferece equatios a X is the equilibrium poit of this system ie F (X) X If all eigevalues of the Jacobia matrix J F evaluate at X lie isie the ope uit isk λ < the X is locally asymptotically stable If oe of them has moulus greater tha oe the X is ustable Theorem 22 Assume that α < β < The the followig statemets are true (i) The equilibrium ( ) is locally ustable (ii) If α β the the system has ifiite positive equilibrium poits ( x ȳ) such that xȳ α which are locally ustable Proof (i) We ca easily obtai that the liearize system of () about the equilibrium ( ) is Φ + DΦ (22) where Φ ( ) T α D ( ij ) 4 4 β (23) The characteristic equatio of (22) is ( f(λ) λ 2 λ ) ( λ ) (24) α β This shows that the roots of characteristic equatio λ a λ lie outsie uit α β isk So the uique equilibrium ( ) is locally ustable (ii) If α β We ca easily obtai that system () has ifiite positive equilibrium poits ( x ȳ) such that xȳ α The liearize system about equilibrium poit ( x ȳ) of system () is Φ + GΦ (25) where Φ ( ) T α x 2 G ȳ 2 β Let λ λ 2 λ 3 λ 4 eote the 4 eigevalues of Matrix G Let D iag( 2 3 4) i (i 2 3 4) be a iagoal matrix Clearly D is ivertible Computig DGD we obtaie 2 (α ) 4 x 2 DGD ȳ (β ) 4 3 It is well kow that G has the same eigevalues as DGD we obtai that max λ k DGD k 4 { max ( α) + 4 x ( β) + } 3 ȳ > It follows from Theorem 2 [25] that the positive equilibrium poits ( x ȳ) is locally ustable 3

4 Hu; BJAST 7(3) -7 26; Article objast27524 Theorem 23 Assume that α > β > The the equilibrium ( ) is globally asymptotically stable Proof For α > β > from (i) of Theorem 22 the equilibrium ( ) is locally asymptotically stable From () it is easy to see that every positive solutio ( ) is boue i e x y Now it is sufficiet to prove that( ) is ecreasig From () we have + α + α < + β + β < This implies that the sequeces { } a { } are ecreasig Hece lim lim Therefore the equilibrium ( ) is globally asymptotically stable Theorem 24 Assume that α β The the followig statemets are true (i) the system () exist ifiite equilibrium poits ( y) a (x ) (ii) every positive solutio ( ) of () coverges ( ) Proof () For α β followig system x we cosier the x + xy y y + xy (26) It is clear to see that the system (26) has ifiite equilibrium poits ( y) a (x ) (ii) Sice the iitial values x x y y are positive real umber It is similar to the proof of Theorem 23 we ca easily get the positive solutio ( ) coverges the equilibrium ( ) 3 RATE OF CONVERGENCE I orer to stuy the rate of covergece of positive solutios of () which coverge to equilibrium poit ( ) of this system first we cosier the followig results that gives the rate of covergece of solutio of a system of ifferece equatios X + [A + B()]X (3) where X is m imesioal vector A C m m is a costat matrix B : Z + C m m is a matrix fuctio satisfyig B() (32) as where is ay matriorm which is associate with the vector orm (x y) x 2 + y 2 Propositio 3 (Perros Theorem)[27] Suppose that coitio (32) hols If X is ay solutio of (3) the X for all large or X + ρ lim X (33) exists a is equal to the moulus of oe of the eigevalues of matrix A Propositio 32 [27] Suppose that coitio (32) hols If X is ay solutio of (3) the X for all large or ρ lim X+ (34) exists a is equal to the moulus of oe of the eigevalues of matrix A Let( ) be a arbitrary positive solutio of system () such that lim lim It follows from () that + a + α + α + β + β + Let E E 2 the we have E + A E +B E 2 E 2 + C E +D E 2 where A Moreover α + B C D lim A α β + lim D β 4

5 Hu; BJAST 7(3) -7 26; Article objast27524 Now the limitig system of error terms ca be writte as ( ) ) ( E + E E 2 + ( /α /β E 2 ) which is similar to liearize system of () about the equilibrium poit ( ) & Usig Propositio 3 a Propositio 32 we have followig result Theorem 3 Assume that ( ) be a positive solutio of () such that lim lim the the error vector E (E E 2 ) T of every solutio of () satisfies the followig asymptotic relatios Fig The fixe poit () is globally asymptotically stable lim E λ 2 F J ( ) lim E + E λ 2F J ( ) where λ 2F J( ) or α characteristic of Jacobia matrix F J( ) 4 NUMERICAL EXAMPLES β are the & I orer to illustrate the results of the previous sectios a to support our theoretical iscussios some iterestig umerical examples are cosiere i this sectio These examples represet ifferet types of qualitative behavior of solutios to the system of oliear ifferece equatios Example 4 If the iitial coitios x 7 x 8 y 9 y 5 a α 4 β 2 we have the followig system It is clear that α > β > The the equilibrium ( ) is globally asymptotically stable(usig MATLAB software See Theorem 23 Fig ) Example 42 If the iitial coitios x 98 x 72 y 96 y 62 a α 8 β 7 we have the followig system It is clear that α < β < The equilibrium ( ) a ( x ȳ) are ustable(usig MATLAB software see Theorem 22 Fig 2) & Fig 2 The fixe poit () a ( x ȳ) is ustable Fig 3 the positive solutio ( ) of system () coverges the equilibrium ( ) Example 43 If the iitial coitios x 7 x 8 y 6 y 3 a α β we have the followig system

6 Hu; BJAST 7(3) -7 26; Article objast27524 It is clear that α β The the positive solutio ( ) of system () coverges the equilibrium ( ) (Usig MATLAB software see Theorem 24 Fig 3) 5 CONCLUSION This paper is cocere with the behavior of positive solutio to system () uer some coitios The results obtaie are as follows: (i) If α > a β > the system () has a uique equilibrium ( ) which is globally asymptotically stable (ii) If α < a β < the system () has equilibrium ( ) which is ustable Furthermore if α β < the system has ifiite positive equilibrium ( x ȳ) such that xȳ α which are locally ustable (iii) If α β system () has ifiite equilibrium poit ( y) a (x ) a every positive solutio ( ) coverges equilibrium poit ( ) ACKNOWLEDGEMENT The author woul like to thak the Eitor a the aoymous referees for their careful reaig a costructive suggestios COMPETING INTERESTS Author has eclare that o competig iterests exist Refereces [] Hassell MP Comis HN Discrete time moels for two-species competitio Theoretical Populatio Biology 976;9(2):22-22 [2] Ibrahim TF Two-imesioal fractioal system of oliear ifferece equatios i the moelig competitive populatios Iteratioal Joural of Basic & Applie Scieces 22;2(5):3-2 [3] Kurbali AS O the behavior of solutios of the system of ratioal ifferece equatios: + /( ) Y + /( ) a z + z /( z ) Discrete Dyamics i Nature a Society; 2 Article ID pages 2 [4] Ciar C Yalcikaya I Karatas R O the positive solutios of the ifferece equatio system + m py + J Ist Math Comp Sci 25;8:35-36 [5] Ciar C O the positive solutios of the ifferece equatio system + y + Applie Mathematics a Computatio 24;58:33-35 [6] Ciar C Yalcikaya I O the positive solutios of the ifferece equatio system x + z + z + Iteratioal Mathematical Joural 24;5: [7] Ozba AY O the positive solutios of the system of ratioal ifferece equatios y + k + m m k J Math Aal Appl 26;323:26-32 [8] Papaschiopoulos G Schias CJ O a system of two oliear ifferece equatios J Math Aal Appl 998;29: [9] Zhag Q Yag L Liu J Dyamics of a system of ratioal thir-orer ifferece equatio Avaces i Differece Equatios 22;36:-8 [] Kurbali AS Ciar C Yalcikaya I O the behavior of positive solutios of the system of ratioal ifferece equatios: + /( ) Y + /( ) Mathematical a Computer Moellig 2;53: [] Kurbali AS Ciar C Simssek D O the perioicty of solutios of the system of ratioal ifferece equatios + /( ) Y + /( ) Applie Mathematics 2;2:4-44 [2] Liu K Zhao Z Li X Li P More o threeimesioal systems of ratioal ifferece equatios Discrete Dyamics i Nature a Society; 2 Article ID pages [3] Zhag Q Zhag W O a system of two high-orer oliear ifferece equatios Avaces i Mathematical Physics; 24 Article ID pages 6

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