Uniform asymptotical stability of almost periodic solution of a discrete multispecies Lotka-Volterra competition system

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1 Iteratoal Joural of Egeerg ad Advaced Research Techology (IJEART) ISSN: , Volume-2, Issue-1, Jauary 2016 Uform asymptotcal stablty of almost perodc soluto of a dscrete multspeces Lotka-Volterra competto system Hu Zhag Abstract I ths paper, we study a dscrete multspeces Lotka-Volterra competto system. Assume that the coeffcets the system are almost perodc sequeces, we obta the suffcet codtos for the exstece of a uque almost perodc soluto whch s uformly asymptotcally stable by costructg a sutable Lapuov fucto. Oe example together wth umercal smulato dcates the feasblty of the ma results. Idex Terms Almost perodc soluto, Dscrete, Lotka-Volterra competto system, Permaece, Uformly asymptotcally stable Wth the help of the methods of the Lyapuov fucto, some aalyss techques, ad prelmary lemmas, they establsh a crtero for the exstece, uqueess, ad uformly asymptotc stablty of postve almost perodc soluto of the system. However, few work has bee doe prevously o a almost perodc verso whch s correspodg to system (1.2). The, we wll further vestgate the global stablty of almost perodc soluto of system (1.2). Deote as Z ad Z + the set of tegers ad the set of oegatve tegers, respectvely. For ay bouded sequece {g()} defed o Z, defe I. INTRODUCTION I paper [1], Che ad Wu had vestgated the dyamc behavor of the followg dscrete -speces Glp-Ayala competto model = 1, 2,, ; x (k) s the desty of competto speces at k-th geerato. a j (k) measures the testy of traspecfc competto or terspecfc acto of competto speces, respectvely. b (k) represets the trsc growth rate of the competto speces x. θ j are postve costats. b (k), a j (k),, j = 1, 2,, are all postve sequeces bouded above ad below by postve costats. Obvously, whe θ j 1, system (1.1) reduces to the tradtoal dscrete multspeces Lotka-Volterra competto model For geeral o-autoomous case, suffcet codtos whch esure the permaece ad the global stablty of system (1.1) ad (1.2) are obtaed; For perodc case, suffcet codtos whch esure the exstece of a uque globally stable postve perodc soluto of system (1.1) ad (1.2) are obtaed. Notce that the vestgato of almost perodc solutos for dfferece equatos s oe of most mportat topcs the qualtatve theory of dfferece equatos due to ts applcatos bology, ecology, eural etwork, ad so forth(see [2 13] ad the refereces cted there). Wag ad Lu [3] studed a dscrete Lotka-Volterra compettve system Throughout ths paper, we assume that: (H1) a j (k) ad b (k) are bouded postve almost perodc sequeces such that From the pot of vew of bology, the sequel, we assume that x(0) = (x 1 (0), x 2 (0),, x (0)) > 0. The t s easy to see that, for gve x(0) > 0, the system (1.1) has a postve sequece soluto x(k) = (x 1 (k), x 2 (k),, x (k))(k Z + ) passg through x(0). The remag part of ths paper s orgazed as follows: I Secto 2, we wll troduce some deftos ad several useful lemmas. I Secto 3, by applyg the theory of dfferece equalty, we preset the permaece results for system (1.2). I Secto 4, we establsh the suffcet codtos for the exstece of a uque uformly asymptotcally stable almost perodc soluto of system (1.2). The ma results are llustrated by a example wth a umercal smulato the last secto. II. PRELIMINARIES Frst, we gve the deftos of the termologes volved. Defto 2.1([14]) A sequece x: Z R s called a almost perodc sequece f the ε-traslato set of x s a relatvely dese set Z for all ε > 0; that s, for ay gve ε > 0, there exsts a teger l(ε) > 0 such that each terval of legth l(ε) cotas a teger τ E{ε, x} wth τ s called a ε-traslato umber of x(). Lemma 2.1([15]) If {x()} s a almost perodc sequece, the {x()} s bouded. Lemma 2.2([16]) {x()} s a almost perodc sequece f ad oly f, for ay sequece m Z, there exsts a subsequece {m k } {m } such that the sequece {x( +m k )} 29

2 Uform asymptotcal stablty of almost perodc soluto of a dscrete multspeces Lotka-Volterra competto system coverges uformly for all Z as k. Furthermore, the lmt sequece s also a almost perodc sequece. Lemma 2.3([15]) Suppose that {p 1 ()} ad {p 2 ()} are almost perodc real sequeces. The {p 1 ()+p 2 ()} ad {p 1 ()p 2 ()} are almost perodc;1/p 1 () s also almost perodc provded that p 1 () 0 for all Z. Moreover, f ε > 0 s a arbtrary real umber, the there exsts a relatvely dese set that s ε almost perodc commo to {p 1 ()} ad {p 2 ()}. Lemma 2.4( [17]) Assume that sequece {x()} satsfes x() > 0 ad for N, a() ad b() are o-egatve sequeces bouded above ad below by postve costats. The soluto (x 1 (k), x 2 (k),, x (k)) of system (1.2) satsfes Proposto 3.2 Assume that (H1) ad hold for all = 1, 2,,, M, = 1, 2,, are defed by (3.1). The for every soluto (x 1 (k), x 2 (k),, x (k)) of system (1.1) satsfes Lemma 2.5( [17]) Assume that sequece {x()} satsfes ad x(n 0 )>0, a() ad b() are o-egatve sequeces bouded above ad below by postve costats ad N 0 N. The Cosder the followg almost perodc dfferece system: f : Z + S B R K, S B = {x R k : x < B}, ad f(, x) s almost perodc uformly for x S B ad s cotuous x. The product system of (2.1) s the followg system: ad Zhag [18] obtaed the followg Theorem. Theorem 2.6( [18]) Suppose that there exsts a Lyapuov fucto V (, x, y) defed for Z +, x < B, y < B satsfyg the followg codtos: ad a s creasg}; s a costat, ad Moreover, f there exsts a soluto φ() of (2.1) such that φ() B < B for Z +, the there exsts a uque uformly asymptotcally stable almost perodc soluto p() of system (2.1) whch s bouded by B. I partcular, f f(, x) s perodc of perod ω, the there exsts a uque uformly asymptotcally stable perodc soluto of system (2.1) of perod ω. III. PERMANENCE I ths secto, we establsh a permaece result for system (1.2), whch ca be foud by Lemma 2.4 ad 2.5. Proposto 3.1 Assume that (H1) holds. The ay postve Theorem 3.3 Assume that (H1) ad (H2) hold, the system (1.1) s permaet. The ext result tells us that there exst solutos of system (1.2) totally the terval of Theorem 3.3. We deote by Ω the set of all solutos (x 1 (k), x 2 (k),, x (k)) of system (1.2) satsfyg m x (k) M ( =1, 2,, ) for all k Z +. Proposto 3.4 Assume that (H1) ad (H2) hold. The Ω Φ. Proof. By the almost perodcty of {a j (k)} ad {b (k)}, there exsts a teger valued sequece {δ p } wthδ p + as p + such that Let ε be a arbtrary small postve umber. It follows from Theorem 3.3 that there exsts a postve teger N 0 such that Wrte x p (k) = x (k + δ p ) for k N 0 δ p ad p = 1, 2,. For ay postve teger q, t s easy to see that there exsts a sequece {x p (k) : p q} such that the sequece x p (k) has a subsequece, deoted by {x p (k)} aga, covergg o ay fte terval of Z as p. Thus we have a sequece {y (k)} such that Ths, combed wth gves us We ca easly see that (y 1 (k), y 2 (k),, y (k)) s a soluto of system (1.2) ad m ε y (k) M + ε for k Z +. Sce ε s a arbtrarly small postve umber, t follows that m y (k) M ad hece we complete the proof. 30

3 Iteratoal Joural of Egeerg ad Advaced Research Techology (IJEART) ISSN: , Volume-2, Issue-1, Jauary 2016 IV. ALMOST PERIODIC SOLUTION The ma results of ths paper cocer the exstece of a uque uformly asymptomatcally stable almost perodc soluto of system (1.2) by costructg a o-egatve Lyapuov fucto. Theorem 4.1 Assume that (H1), (H2) ad we have Moreover, for ay hold, = 1, 2,,. The there exsts a uque uformly asymptotcally stable almost perodc soluto (x 1 (k), x 2 (k),, x (k)) of system (1.2) whch s bouded by Ω for all k Z +. Proof. Let p (k) = l x (k), = 1, 2,,. From system (1.2), we have From Proposto 3.4, we kow that system (4.1) have bouded soluto (p 1 (k), p 2 (k),, p (k)) satsfyg ad Thus, codto () of Theorem 2.6 s satsfed. Fally, calculatg the ΔV(k) of V(k) alog the solutos of system (4.2), we have Hece, p (k) A, A = max{ l m, l M }, = 1, 2,,. For X R, we defe the orm X x. Cosder the product system of system (4.1) We assume that Q = (p 1 (k), p 2 (k),, p (k)), W = (q 1 (k), q 2 (k),, q (k)) are ay two solutos of system (4.1) defed o Z + S ; the, Q B, W B, B = {A B }, S = {(p 1 (k), p 2 (k),, p (k)) l m p () l M, = 1, 2,,, k Z + }. Let us costruct a Lyapuov fucto defed o Z + S S as follows: By the mea value theorem, t derves that ξ (k) les betwee e p(k) ad e q(k). The, we have It s obvous that the orm Q W = p (k) q (k) s equvalet to Q W = (p (k) q (k)) 2 ] 1/2 ; that s, there are two costats c 1 > 0, c 2 > 0, such that the Let the, codto () of Theorem 2.6 s satsfed. 31

4 Uform asymptotcal stablty of almost perodc soluto of a dscrete multspeces Lotka-Volterra competto system The, we have V. NUMERICAL SIMULATION I ths secto, we gve the followg examples to check the feasblty of our results. Example 5.1 Cosder the dscrete multspeces Lotka-Volterra competto system: A computato shows that ad moreover, we have Hece, we have that 0 < m{β 1, β 2, β 3 } < 1. It s easy to see that the codto (H2) ad (H3) are satsfed. Hece, there exsts a uque uformly asymptotcally stable almost perodc soluto of system (5.1). Our umercal smulatos support our results(see Fgs.1,2 ad 3). β = m {β }. That s, there exsts a postve costat 0 1 < β < 1 such that From 0<β <1, the codto () of Theorem 2.6 s satsfed. So, accordg to Theorem 2.6, there exsts a uque uformly asymptotcally stable almost perodc soluto (p 1 (k), p 2 (k),, p (k)) of system (4.1) whch s bouded by S for all k Z +. It meas that there exsts a uque uformly asymptotcally stable almost perodc soluto (x 1 (k), x 2 (k),, x (k)) of system (1.2) whch s bouded by Ω for all k Z +. Ths completed the proof. 2 Remark 4.2 If = 2, the codtos of Theorem 4.1 ca be smplfed. Therefore, we have the followg results. Corollary 4.3 Let = 2, assume that (H1), (H2) ad FIGURE1: Dyamc behavor of the frst compoet x 1 (k) of the soluto (x 1 (k), x 2 (k), x 3 (k)) to system (5.1) wth the tal codtos (0.87,1.02,1.03), (0.93,1.13,0.86) ad hold,, j = 1, 2, j. The there exsts a uque uformly asymptotcally stable almost perodc soluto (x 1 (k), x 2 (k)) of system (1.2) whch s bouded by Ω for all k Z +. FIGURE2: Dyamc behavor of the secod compoet x 2 (k) 32

5 Iteratoal Joural of Egeerg ad Advaced Research Techology (IJEART) ISSN: , Volume-2, Issue-1, Jauary 2016 of the soluto (x 1 (k), x 2 (k), x 3 (k)) to system (5.1) wth the tal codtos (0.87,1.02,1.03), (0.93,1.13,0.86) ad [13] Hu Zhag, Feg Feg, B Jg, Ygq L, Almost perodc soluto of a multspeces dscrete mutualsm system wth feedback cotrols, Dscrete Dyamcs Nature ad Socety, Volume 2015, Artcle ID , 14 pages. [14] A.M. Fk, G. Sefert, Lapuov fuctos ad almost perodc solutos for almost perodc systems, Joural of Dfferetal Equatos, 5(1969) [15] A.M. Samoleko, N.A. Perestyuk, Impulsve Dfferetal Equatos : World Scetfc Seres o Nolear Scece, World Scetfc, Sgapore, [16] Shua Zhag, G. Zheg, Almost perodc solutos of delay dfferece systems, Appled Mathematcs ad Computato, 131(2002) [17] Fegde Che, Permaece for the dscrete mutualsm model wth tme delay, Mathematcal ad Computer Modellg, 47(2008) [18] Shua Zhag, Exstece of almost perodc soluto for dfferece systems, Aals of Dfferetal Equatos, 16(2000) Hu Zhag s a lecturer of X a Research Isttute of Hgh-tech Hogqg Tow. Hs major s almost perodcty of cotuous ad dscrete dyamc system. FIGURE3: Dyamc behavor of the thrd compoet x 3 (k) of the soluto (x 1 (k), x 2 (k), x 3 (k)) to system (5.1) wth the tal codtos (0.87,1.02,1.03), (0.93,1.13,0.86) ad ACKNOWLEDGEMENT The author declares that there s o coflct of terests regardg the publcato of ths paper, ad there are o facal terest coflcts betwee the author ad the commercal detty. REFERENCES [1] Fegde Che, Lpg Wu, Zhog L, Permaece ad global attractvty of the dscrete Glp-Ayala type populato model, Computers ad Mathematcs wth Applcatos, 53(2007) [2] Zhog L, Fegde Che, Almost perodc soluto of a dscrete almost perodc logstc equato, Mathematcal ad Computer Modellg, 50(2009) [3] Qglog Wag, Zhju Lu, Uformly Asymptotc Stablty of Postve Almost Perodc Solutos for a Dscrete Compettve System, Joural of Appled Mathematcs, vol.2013, Artcle ID , 9 pages. [4] Hu Zhag, Ygq L, B Jg. Global attractvty ad almost perodc soluto of a dscrete mutualsm model wth delays, Mathematcal Methods the Appled Sceces, 2013, DOI: /mma [5] Chegyg Nu, Xaoxg Che, Almost perodc sequece solutos of a dscrete Lotka-Volterra compettve system wth feedback cotrol, Nolear Aalyss:Real World Applcatos, 10(2009) [6] Zhog L, Fegde Che, Megx He, Almost perodc solutos of a dscrete Lotka-Volterra competto system wth delays, Nolear Aalyss:Real World Applcatos, 12(2009) [7] Yogku L, Tawe Zhag, Permaece ad almost perodc sequece soluto for a dscrete delay logstc equato wth feedback cotrol, Nolear Aalyss: Real World Applcato, 12(2011) [8] Yogku L, Tawe Zhag, Yua Ye, O the exstece ad stablty of a uque almost perodc sequece soluto dscrete predator-prey models wth tme delays, Appled Mathematcal Modelg, 35(2011) [9] Tawe Zhag, Yogku L, Yua Ye, Persstece ad almost perodc solutos for a dscrete fshg model wth feedback cotrol, Commu Nolear Sc Numer Smulat, 16(2011) [10] Yje Wag, Perodc ad almost perodc solutos of a olear sgle speces dscrete model wth feedback cotrol, Appled Mathematcs ad Computato, 219(2013) [11] Tawe Zhag, Xaorog Ga, Almost perodc solutos for a dscrete fshg model wth feedback cotrol ad tme delays, Commu Nolear Sc Numer Smulat, 19(2014) [12] Hu Zhag, B Jg, Ygq L, Xaofeg Fag, Global aalyss of almost perodc soluto of a dscrete multspeces mutualsm system, Joural of Appled Mathematcs, Volume 2014, Artcle ID , 12 pages. 33