TN recent years, various multi-species logarithmic population

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1 Th Study on Almot Priodic Solution for A Nutral Multi-pci Logarithmic Population Modl With Fdback Control By Matrix, Spctral Thory Changjin Xu, and Yun Wu Abtract In thi papr, a nutral multi-pci logarithmic population modl i invtigatd. By applying th matrix, pctral thory which i diffrnt from th mthod mployd in th litratur, a t of ufficint condition ar obtaind for th xitnc and uniqun of almot priodic olution of th nutral multi-pci logarithmic population modl. Th obtaind ufficint condition ar givn in trm of pctral radiu of xplicit matric which ar much diffrnt from tho by th algbraic inqualiti. An xampl i givn to illutrat th faibility and ffctivn of th obtaind rult. Th rult of thi papr ar compltly nw and gnraliz tho of th prviou tudi. Indx Trm Nicholon-typ ytm, poitiv olution, xponntial tability, dlay, Lyapunov mthod. I. INTRODUCTION TN rcnt yar, variou multi-pci logarithmic population modl hav bn xtnivly invtigatd by many cholar du to thir thortical and practical ignificanc in biology. Gopalamy 1 and Kirlingr 2 propod th following ingl pci logarithmic modl = N(t)a b ln N(t) c ln N(t τ). (1) In 1997, Li 3 gnralizd ytm (1) to th following nonautonomou form = N(t)a(t)b(t) ln N(t)c(t) ln N(tτ(t)). (2) Applying th coincidnc dgr thory, Li 3 tablihd om ufficint condition for th xitnc of poitiv priodic olution of ytm (2). In 23, Chn t al. 4 gnralizd ytm (2) to th ytm with tat dpndnt dlay and invtigatd th xitnc of poitiv priodic olution of th ytm. In 5, Liu propod th following Manucript rcivd Novmbr 6, 214; rvid Jun 17, 215. Thi work wa upportd in part by th Thi work i upportd by National Natural Scinc Foundation of China(No , No ), Natural Scinc and Tchnology Foundation of Guizhou Provinc(J215225) and 125 Spcial Major Scinc and Tchnology of Dpartmnt of Education of Guizhou Provinc (21211). C. Xu i with th Dpartmnt of Guizhou Ky Laboratory of Economic Sytm Simulation, Guizhou Univrity of Financ and Economic, Guiyang 554, PR China -mail: xcj43@126.com. Y. Wu i with chool of Mathmatic and Statitic, Hnan Univrity of Scinc and Tchnology, Luoyang 47123, PR China, -mail: wuyun621@126.com multipci priodic logarithmic population modl = N i (t) r i (t) a ij (t) ln N j (t) b ij (t) ln N j (t τ ij (t)). (3) Applying th coincidnc dgr thory and contructing Lyapunov functional, a t of ufficint condition which guarant th xitnc, uniqun and tability of th poitiv priodic olution of ytm (3) ar tablihd. In 25, Chn 6 propod th following multipci logarithmic population modl = N i (t) r i (t) a ij (t) ln N j (t) b ij (t) ln N j (t τ ij (t)) t c ij (t) K ij (t ) ln N j ()d. (4) By uing th fixd point thory and contructing a uitabl Lyapunov functional, a t of aily applicabl critria ar obtaind for th xitnc, uniqun and global attractivity of poitiv priodic olution (poitiv almot priodic olution) of th modl (4). Gopalamy 1 pointd out that in om ca, th nutral dlay population modl ar mor ralitic. Thn Li 7 and Li t al. 8 conidrd th priodic olution or almot priodic olution of th following two ingl pci nutral Logarithmic modl and = N(t) r(t) a(t) ln N(t σ) d ln N(t τ) b(t) = N(t) r(t) a j (t) ln N(t σ j (t)) b j (t) d ln N(t τ j(t)) (5), (6) rpctivly. In 23, Yang and Cao 9 addrd th xitnc of poitiv priodic olution of th nutral logarithmic (Advanc onlin publication: 15 Fbruary 216)

2 population modl with multipl dlay = N(t) a(t) β(t)n(t) (b i (t) i=1 N(t τ i (t))) c i (t) d ln N(t γ i(t)). (7) i=1 In 24, Lu and G 1 pointd out that th proof of Thorm 3.1 i incomplt and analyzd th xitnc of poitiv priodic olution for nutral logarithmic population modl with multipl dlay m = N(t) r(t) a i (t) ln N(t σ i (t)) b j (t) d ln N(t τ j(t)). (8) With th hlp of an abtract continuou thorm of k- t contractiv oprator, author obtaind om ufficint condition for th xitnc, global attractivity of poitiv priodic olution of (8). In 29, Wang t al. 11 focud on th xitnc and uniqun of poitiv priodic olution for a following nutral logarithmic population modl = N(t) r(t) a(t) ln N(t) b j (t) ln N(t τ j (t)) c j (t) k j (t ) ln N()d d j (t) d ln N(t η j(t)). (9) Applying an abtract continuou thorm of k-t contractiv oprator, author tablihd om ufficint condition for th xitnc, global attractivity of poitiv priodic olution of (9). In 21 and 211, Alzabut t al tudid th almot priodic olution for dlay logarithmic population modl. Rcntly, Chn 14 had invtigatd th priodic olution and almot priodic olution of th following nutral multi-pci logarithmic population modl = N i (t) r i (t) a ij (t) ln N j (t) b ij (t) ln N j (t τ ij (t)) c ij (t) K ij (t ) ln N j ()d d ij (t) d ln N j(t η ij (t), (1) whr i = 1, 2,,, n, a ij (t), b ij (t), c ij (t), d ij (t) C(R, (, )), τ ij (t), η ij (t) C(R, R ) ar all continuou function. K ij ()d = 1, K ij ()d <. Many cholar 15-18,22-24 argu that coytm in th ral world i continuouly ditributd by unprdictabl forc which can rult in chang in th biological paramtr uch a urvival rat. Of practical intrt in cology i th qution of whthr or not an coytm can withtand tho unprdictabl diturbanc which prit for a finit priod of tim. In th languag of control variabl, w call th diturbanc function a control variabl. Motivatd by th dicuion abov, w will invtigat th nutral multipci logarithmic population modl with fdback control a follow = N i (t) r i (t) a ij (t) ln N j (t) b ij (t) ln N j (t τ ij (t)) c ij (t) K ij (t ) ln N j ()d d ij (t) d ln N j(t η ij (t) i (t)u i (t) f i (t)u i (t σ i (t)), du i (t) = α i (t)u i (t) β i (t) ln N i (t) γ i (t) ln N i (t δ i (t)), (11) whr i = 1, 2,, n, u i (i = 1, 2,, n) dnot indirct fdback control variabl. Th main aim of thi articl i to tablih om ufficint condition for th xitnc and uniqun of almot priodic olution of (11). Our rult ar nw and complmnt tho of th prviou tudi in To th bt of our knowldg, it i th firt tim to invtigat th nutral multipci logarithmic population modl with fdback control by applying th matrix, pctral thory. So far, thr ar vry fw papr that dal with th almot priodic olution by applying th matrix, pctral thory. Th rmaindr of th papr i organizd a follow. In Sction II, w introduc om notation and aumption, which can b ud to chck th xitnc and uniqun of almot priodic olution of ytm (11). In Sction III, w prnt om ufficint condition for th xitnc and uniqun of almot priodic olution of (11). An xampl i givn to illutrat th ffctivn of th obtaind rult in Sction V. A brif concluion i drawn in Sction VI. II. NOTATIONS AND ASSUMPTIONS In thi ction, w would lik to introduc om notation and aumption which ar ud in what follow. Lt x = (x 1, x 2,, x n ) T R n dnot a column vctor, D = (d ij ) n n b an n n matrix, D T b th tranpo of D, and E n b th idntity matrix of iz n. A matrix or vctor D > man that all ntri of D ar gratr than zro, likwi for D. For matric or vctor D and E, D > E(D E) man that D E > (D E ). ρ(d) dnot th pctral radiu of th matrix D. If v = (v 1, v 2,, v n ) T R n, thn w dfin th com- (Advanc onlin publication: 15 Fbruary 216)

3 monly ud norm a follow v 1 = v i, v 2 = v i 2 1 2, v = max 1 i n v i. If A = (a ij ) n n, thn w dfin th norm of th matrix A a follow A = In particular, Av up = up Av = up Av. v R n,v v v =1 v 1 A 1 = max 1 j n i=1 A = max 1 i n n a ij. Lt m(f) = a ij, A 2 = λ max (A T A) 1 2, 1 T lim f(t) =, T T whr f(t) i almot priodic function. Throughout thi papr, w mak th following aumption. (H1) r i (t), a ij (t), b ij (t), c ij (t), d ij (t), α i (t), β i (t), γ i (t), i (t), f i (t), i, j = 1, 2,, n ar continuou ral-valud nonngativ almot priodic function on R. (H2) Th krnl K ij (.), i, j = 1, 2,, n ar nonngativ continuou function dfind on, ) atifying K ij ()d = 1. (H3) τ ij (t), σ i (t), δ i (t) and η ij (t) ar nonngativ, continuouly diffrntiabl and almot priodic function on t R. Morovr, τ ij (t), σ i (t), δ i (t) and η ij (t) ar all uniformly continuou on R with inf {1 τ ij (t)} >, inf {1 σ i (t)} >, inf {1 δ i (t)} >, inf {1 η ij (t)} >. Sytm (11) i upplmntd with th initial valu condition N i () = ϕ Ni (), Ṅ i () = ϕ Ni (), (,, ϕ Ni () >, up (, ϕ Ni () <, up (, ϕ Ni () <, u i () = ϕ ui (), ϕ ui () >, (,. (12) It i ay to that thr xit a poitiv olution y(t) = (N 1 (t), N 2 (t),, x n (t), u 1 (t), u 2 (t),, u n (t)) of ytm (11) atifying th initial valu condition (12). III. EXISTENCE AND UNIQUENESS OF ALMOST PERIODIC SOLUTION In thi ction, w will tablih ufficint condition on th xitnc and uniqun of almot priodic olution of (11). For convninc, w introduc om dfinition and lmma which will b ud in what follow. Dfinition Lt f(t) : R R n b continuou in t. f(t) i aid to almot priodic on R, if for any ε >, th t T (f, ε) = {δ : f(t δ) f(t) < ε, t R} i rlativly dn, i.., for ε >, it i poibl to find a ral numbr l = l(ε) >, for any intrval with lngth l(ε), thr xit a numbr δ = δ(ε) in thi intrval uch that f(t δ) f(t) < ε, for R. Dfinition 3.2 Lt z R n and Q(t) b a n n continuou matrix dfind on R. Th linar ytm dz = Q(t)z(t) (13) i aid to admit an xponntial dichotomy on R if thr xit contant k, λ >, projction P and th fundamntal matrix Z(t) of (13) atifying Z(t)P Z 1 () k λ(t), for t, Z(t)(I P )Z 1 () k λ(t), for t. Lmma If th linar ytm (13) admit an xponntial dichotomy, thn almot priodic ytm dz = Q(t)z(t) g(t) (14) ha a uniqu almot priodic olution z(t) and z(t) = t Z(t)P Z 1 ()g()d Z(t)(I P )Z 1 ()g()d. Lmma Lt a i (t) b an almot priodic function on R and a i (t) >. Thn th ytm dz = diag(a 1(t), a 2 (t),, a n (t))z(t) (15) admit an xponntial dichotomy. Rmark 3.1 It follow from Lmma 3.2 that ytm (15) ha a uniqu almot priodic olution z(t) which tak th form z(t) = ( = Z(t)Z 1 ()g()d a1(u)du g 1 ()d,, an(u)du g n ()d ). Lmma Aum that v(t), η(t) ar all continuouly diffrntiabl T -priodic function, a(t), b(t) ar all nonngativ continuou T -priodic function uch that T a(t) >, thn a(τ)dτ b()v ()( η())d = c(t)v(t η(t)) whr c() = a(τ)dτ (a()c() c ())(v( η())d, b() 1η (). Lmma 3.4 Lt m b a poitiv intgr and B b an Banach pac. If th mapping Γ : B B i a contraction mapping, thn Γ : B B ha xactly on fixd point in B, whr Γ m = Γ(Γ m1 ). By (H1), m(α i ) >. In viw of Lmma 3.1, w hav th following rult. (Advanc onlin publication: 15 Fbruary 216)

4 Lmma 3.5 (N 1 (t), N 2 (t),, N n (t), u 1 (t), u 2 (t),, u n (t)) T i an almot priodic olution of ytm (11) if and only if it i an almot priodic olution of u i (t) = = N i (t) r i (t) a ij (t) ln N j (t) b ij (t) ln N j (t τ ij (t)) c ij (t) K ij (t ) ln N j ()d d ij (t) d ln N j(t η ij (t) i (t)u i (t) f i (t)u i (t σ i (t)), αi(ζ)dζ β i () ln N i () γ i () ln N i ( δ i ())d, (16) whr i = 1, 2,, n. Obviouly, (16) i quivalnt to th following ytm = N i (t) r i (t) a ij (t) ln N j (t) b ij (t) ln N j (t τ ij (t)) c ij (t) K ij (t ) ln N j ()d d ij (t) d ln N j(t η ij (t) i (t) αi(ζ)dζ (β i () ln N i () γ i () ln N i ( δ i ()))d σi(t) f i (t) σ i (t) α i(ζ)dζ (β i () ln N i () γ i () ln N i ( δ i ()))d. (17) Now w ar in a poition to tat our main rult on th xitnc and uniqun of almot priodic olution for ytm (11). Thorm 3.1 In addition to (H1) (H3), if th following condition (H4) ρ(λ) < 1, whr Λ = (Λ ij ) n n and Λ ii = d ii (t) Λ ij = d ij (t) aii(ζ)dζ Θ ii ()d, Θ ii () = a ii () b ii () c ii () aii(ζ)dζ Θ ij ()d, i j, d ii ()(a ii ()d ii () d ii() ) K ii (θ)dθ ( i () f i ()) ξ αi(ζ)dζ (β i (ξ) γ i (ξ))dξ, Θ ij () = (a ij () b ij () c ij () d ij ()(a ii ()d ij () d ij() ), i j. K ij (θ)dθ whr i = 1, 2,, n. Thn ytm (11) ha a uniqu poitiv almot priodic olution. Proof Lt N i (t) = xi(t), thn (17) tak th form dx i (t) = r i (t) a ij (t)x j (t) b ij (t)x j (t τ ij (t)) c ij (t) K ij (t )x j ()d d ij (t)ẋ j (t η ij (t)(1 η ij (t)) i (t) αi(ζ)dζ (β i ()x i () γ i ()x i ( δ i ()))d f i (t) σi(t) γ i ()x i ( δ i ()))d = a ii (t)x i (t) σ i (t) α i(ζ)dζ (β i ()x i (),j i b ij (t)x j (t τ ij (t)) a ij (t)x j (t) c ij (t) K ij (t )x j ()d d ij (t)ẋ j (t η ij (t)(1 η ij (t)) i (t) αi(ζ)dζ (β i ()x i () γ i ()x i ( δ i ()))d f i (t) σi(t) σ i (t) α i(ζ)dζ (β i ()x i () γ i ()x i ( δ i ()))d r i (t). (18) Clarly, if ytm (18) ha an almot priodic olution (x 1(t), x 2(t),, x n(t)) T, thn (N1 (t), N2 (t),, Nn(t)) T = ( x 1 (t), x 1 (t),, x n (t) ) T i an almot priodic olution of (17). In viw of Lmma 3.5, w can conclud that ( x 1 (t), x 1 (t),, x n (t), u 1(t), u 1(t),, u n(t)) T i an almot priodic olution of (11), whr u i (t) = αi(ζ)dζ β i ()x i ()γ i ()x i (δ i ())d, whr i = 1, 2,, n. Now w will how that (18) ha a uniqu almot almot priodic olution. Firtly, w dfin B = {ψ(t) = (ψ 1 (t), ψ 2 (t),, ψ n (t)) T ψ(t) i a continuou almot priodic function}. Obviouly, B i a Banach pac with th norm ψ = max 1 i n up x i (t). (Advanc onlin publication: 15 Fbruary 216)

5 For any ψ(t) = (ψ 1 (t), ψ 2 (t),, ψ n (t)) T B, w conidr th following almot priodic ytm dx i (t) = a ii (t)x i (t),j i b ij (t)ψ j (t τ ij (t)) a ij (t)ψ j (t) c ij (t) K ij (t )ψ j ()d d ij (t) ψ j (t η ij (t)(1 η ij (t)) i (t) αi(ζ)dζ (β i ()ψ i () γ i ()ψ i ( δ i ()))d f i (t) σi(t) σ i (t) α i(ζ)dζ (β i ()ψ i () γ i ()ψ i ( δ i ()))d r i (t). (19) By (H1), w know that m(a ii >. In viw of Lmma 3.2, th linar ytm dx i (t) = a ii (t)x i (t), i = 1, 2,, n, (2) admit an xponntial dichotomy on T. Thn ytm (2) ha xactly on almot priodic olution a follow whr x ψ i (t) = (xψ 1 (t), xψ 2 (t),, xψ n(t)) T t a11(ζ)dζ h ψ 1 ()d = t a22(ζ)dζ h ψ 2 ()d, (21) t ann(ζ)dζ h ψ n()d h ψ i () = n,j i c ij () a ij ()ψ j () b ij ()ψ j ( τ ij ()) K ij ( θ)ψ j (θ)dθ d ij () ψ j ( η ij ()(1 η ij ()) i () ξ αi(ζ)dζ (β i (ξ)ψ i (ξ) γ i (ξ)ψ i (ξ δ i (ξ)))dξ f i () σi() σi () ξ α i(ζ)dζ (β i (ξ)ψ i (ξ) γ i (ξ)ψ i (ξ δ i (ξ)))d r i (). (22) In viw of Lmma 3.2, x ψ i (t) can b xprd a x ψ i (t) = (xψ 1 (t), xψ 2 (t),, xψ n(t)) T = (A 1, A 2,, A n ) T, (23) whr and A 1 = A 2 = d 1j (t)ψ j (t η 1j (t)) a11(ζ)dζ l ψ 1 ()d, d 2j (t)ψ j (t η 2j (t)) a22(ζ)dζ l ψ 2 ()d, n A n d nj (t)ψ j (t η nj (t)) l ψ i () = n,j i a ij ()ψ j () b ij ()ψ j ( τ ij ()) ann(ζ)dζ l ψ n ()d c ij () K ij ( θ)ψ j (θ)dθ d ij ()(a ii ()d ij () d ij())ψ j ( η ij ()) i () ξ αi(ζ)dζ (β i (ξ)ψ i (ξ) γ i (ξ)ψ i (ξ δ i (ξ)))dξ f i () σi() σi () ξ α i(ζ)dζ (β i (ξ)ψ i (ξ) γ i (ξ)ψ i (ξ δ i (ξ)))dξ r i (). (24) Dfin a mapping F : B B a follow F ψ(t) = Z ψ (t), for any ψ B. (25) For any φ, ψ B, w hav (F (φ) F (ψ)) = ( (F (φ(t)) F (ψ(t))) 1, (F (φ(t)) F (ψ(t))) 2,, (F (φ(t)) F (ψ(t))) n ) T n d 1j(t) φ j (t η 1j (t)) ψ j (t η 1j (t)) t a11(ζ)dζ l φ 1 () lψ 1 () d n d 2j(t) φ j (t η 2j (t)) ψ j (t η 2j (t)) t a22(ζ)dζ l φ 2 () lψ 2 () d. n d nj(t) φ j (t η nj (t)) ψ j (t η nj (t)) t t ann(ζ)dζ ln() φ ln ψ () d (26) On th othr hand, by (24), w gt l φ i () lψ i () = a ij () φ j () ψ j (),j i b ij () φ j ( τ ij ()) ψ j ( τ ij ()) c ij () K ij ( θ) φ j (θ) ψ j (θ) dθ d ij ()(a ii ()d ij () d ij() ) φ j ( η ij ()) (Advanc onlin publication: 15 Fbruary 216)

6 ψ j ( η ij ()) i () ξ αi(ζ)dζ (β i (ξ) φ i (ξ) ψ i (ξ) γ i (ξ) φ i (ξ δ i (ξ)) ψ i (ξ δ i (ξ)) )dξ f i () σi() σi () ξ α i(ζ)dζ (β i (ξ) φ i (ξ) ψ i (ξ) γ i (ξ) φ i (ξ δ i (ξ)) ψ i (ξ δ i (ξ)) )dξ a ij () up φ j (t) ψ j (t),j i b ij () up φ j (t) ψ j (t) c ij () K ij ( θ) up φ j (t) ψ j (t) dθ d ij ()(a ii ()d ij () d ij() ) up φ j (t) ψ j (t) i () (β i (ξ) up φ i (t) ψ i (t) γ i (ξ) up φ i (t) ψ i (t) )dξ f i () σi() σi () ξ ξ αi(ζ)dζ α i(ζ)dζ (β i (ξ) up φ i (t) ψ i (t) γ i (ξ) up φ i (t) ψ i (t) )dξ = a ij () b ij (),j i c ij () K ij (θ)dθ d ij ()(a ii ()d ij () d ij() ) up φ j (t) ψ j (t) ( i () f i ()) ξ αi(ζ)dζ (β i (ξ) γ i (ξ))dξ up φ i (t) ψ i (t) = a ii () b ii () c ii () d ii ()(a ii ()d ii () d ii() ) K ii (θ)dθ ( i () f i ()) ξ αi(ζ)dζ (β i (ξ) γ i (ξ))dξ up φ i (t) ψ i (t) { n (a ij () b ij () c ij () K ij (θ)dθ,j i d ij ()(a ii ()d ij () d ij() ) } up φ j (t) ψ j (t). (27) Lt Θ ii () = a ii () b ii () c ii () d ii ()(a ii ()d ii () d ii() ) ( i () f i ()) Θ ij () = (a ij () b ij () c ij () K ii (θ)dθ ξ αi(ζ)dζ (β i (ξ) γ i (ξ))dξ, d ij ()(a ii ()d ij () d ij() ), i j. whr i = 1, 2,, n. It follow from (27) that K ij (θ)dθ l φ i () lψ i () Θ ii() up φ i (t) ψ i (t),j i whr i = 1, 2,, n. Thn whr Θ ij () up φ j (t) ψ j (t), (28) d ij (t) φ j (t η ij (t)) ψ j (t η ij (t)) aii(ζ)dζ l φ i () lψ i () d d ij (t) up φ j (t) ψ j (t),j i aii(ζ)dζ Θ ii ()d up φ i (t) ψ i (t) up φ j (t) ψ j (t) = Λ ii up φ i (t) ψ i (t),j i Λ ii = d ii (t) Λ ij = d ij (t) aii(ζ)dζ Θ ij ()d Λ ij up φ j (t) ψ j (t), (29) aii(ζ)dζ Θ ii ()d, aii(ζ)dζ Θ ij ()d, i j, whr i = 1, 2,, n. It follow from (26) and (29) that (F (φ) F (ψ)) = ( (F (φ(t)) F (ψ(t))) 1, (F (φ(t)) F (ψ(t))) 2,, (F (φ(t)) F (ψ(t))) n ) T (3) (Advanc onlin publication: 15 Fbruary 216)

7 = = Λ(up Λ 11 up φ 1 (t) ψ 1 (t) n,j 1 Λ 1j up φ j (t) ψ j (t) Λ 22 up φ 2 (t) ψ 2 (t) n,j 2 Λ 2j up φ j (t) ψ j (t) Λ nn up φ n (t) ψ n (t) n,j n Λ nj up φ j (t) ψ j (t) Λ 11 Λ 11 Λ 1n Λ 21 Λ 22 Λ 2n Λ 21 Λ n2 Λ nn up φ 1 (t) ψ 1 (t) up φ 2 (t) ψ 2 (t) up φ n (t) ψ n (t) φ 1 (t) ψ 1 (t), up, up φ n (t) ψ n (t) ) T = Λ(up Thn w gt n n n 1 φ 2 (t) ψ 2 (t), (φ(t) ψ(t) 1, up (φ(t) ψ(t)) 2,, up (φ(t) ψ n (t)) n ) T. (31) up (F (φ(t)) F (ψ(t))) 1 up (F (φ(t)) F (ψ(t))) 2 up (F (φ(t)) F (ψ(t))) n up (φ(t) ψ(t)) 1 up (φ(t) ψ(t)) 2 up (φ(t) ψ(t)) n For any poitiv intgr m, by (31), w hav up (F m (φ(t)) F m (ψ(t))) 1 up (F m (φ(t)) F m (ψ(t))) 2 = up (F m (φ(t)) F m (ψ(t))) n. (32) up (F (F m1 (φ(t))) F (F m1 (ψ(t)))) 1 up (F (F m1 (φ(t))) F (F m1 (ψ(t)))) 2 up (F (F m1 (φ(t))) F (F m1 (ψ(t)))) n Λ Λ m up (F m1 (φ(t)) F m1 (ψ(t))) 1 up (F m1 (φ(t)) F m1 (ψ(t))) 2 up (F m1 (φ(t)) F (F m1 (ψ(t))) n By (H4), w gt up (φ(t) ψ(t)) 1 up (φ(t) ψ(t)) 2 up (φ(t) ψ(t)) n (33) lim m Λm =, (34) which impli that thr xit a poitiv intgr N and a poitiv contant µ < 1 uch that Λ N = (κ ij ) n n and κ ij µ, i = 1, 2,, n. (35) It follow form (32) and (34) that Thu (F N (φ) F N (ψ)) i κ ij up φ(t) ψ(t) max φ(t) ψ(t) up 1 i n j1 κ ij µ φ ψ, i = 1, 2,, n. (36) F N (φ) F N (ψ) = max (F N (φ) F N (ψ)) i µ φ ψ, (37) 1 i n which impli that th mapping F N : B B i a contraction mapping. In viw of Lmma 3.4, F ha a uniqu a fixd point x (t) in B. Thu ytm (18) ha a uniqu almot priodic olution x (t) = (x 1(t), x 2(t),, x n(t)) T, thn (N 1 (t), N 2 (t),, N n(t)) T = ( x 1 (t), x 2 (t),, x n (t) ) T i th uniqu almot priodic olution of (17). Thu, by Lmma 3.5, ( x 1 (t), x 2 (t),, x n (t), u 1(t), u 2(t),, u n(t)) T i th uniqu almot priodic olution of (11). Th proof of Thorm 3.1 i compltd. IV. NUMERICAL EXAMPLE In thi ction, w will giv an xampl to illutrat th faibility and ffctivn of our main rult obtaind in prviou ction. Conidring th following nutral multipci logarithmic population modl with fdback control dn 1 (t) 2 = N 1 (t) r 1 (t) a 1j (t) ln N j (t) 2 b 1j (t) ln N j (t τ 1j (t)) 2 c 1j (t) K 1j (t ) ln N j ()d 2 d 1j (t) d ln N j(t η 1j (t) 1 (t)u 1 (t) f 1 (t)u 1 (t σ 1 (t)), du 1 (t) = α 1 (t)u 1 (t) β 1 (t) ln N 1 (t) γ 1 (t) ln N 1 (t δ 1 (t)), (38) whr k ij =, r 1 (t) = 1 in t, a 11 = 1 in t, a 12 = 1 co t, b 11 =.3 in t, b 12 =.2 co t, c 11 =.1 in t, c 12 =.3 co t, d 11 =.4 in t, d 12 =.5 co t, τ 11 =.2.4 in t, τ 12 =.3.1 co t, η 11 =.3.2 in t, η 12 =.2.1 co t, 1 (t) =.2 in t, f 1 (t) =.2 co t, σ 1 (t) =.4.2 in t, δ 1 (t) =.3.3 in t, α 1 (t) =.4 co t, β 1 (t) =.3 in, γ 1 (t) =.5 co t. Thn by Matlab oftwar, w hav K ij ()d = 1, ρ(λ).3472 < 1. Thu all aumption in Thorm 3.1 ar fulfilld. Thu w can conclud that (37) ha a uniqu poitiv priodic olution. Th rult ar vrifid by th numrical imulation in Fig. 1. (Advanc onlin publication: 15 Fbruary 216)

8 Fig. 1. N 1 (t), u 1 (t) N 1 (t) u 1 (t) t Tim rpon of tat variabl N 1 (t) and u 1 (t). V. CONCLUSIONS In thi papr, w tudy a nutral multi-pci logarithmic population modl. Applying th matrix, pctral thory, w tablih om ufficint condition for th xitnc and uniqun of almot priodic olution of th nutral multi-pci logarithmic population modl. Th obtaind ufficint condition ar givn in trm of pctral radiu of xplicit matric which ar much diffrnt from tho by th algbraic inqualiti. An xampl i givn to illutrat th faibility and ffctivn of th obtaind rult. Th rult of thi papr ar compltly nw and gnraliz tho of th prviou tudi in Rcntly, th almot priodic olution of dicrt nutral multi-pci logarithmic population modl ha alo paid mor attntion by numrou rarchr. Howvr, thr ar vry fw rult on th almot priodic olution of dicrt nutral multi-pci logarithmic population modl, which might b our futur rarch topic. 11 Q. Wang, Y. Wang, and B.X. Dai, Exitnc and uniqun of poitiv priodic olution for a nutral logarithmic population modl, Applid Mathmatic and Computation, vol. 213, no. 1, pp: , J.O. Alzabut, G.T. Stamov, and E. Srmutlu, Poitiv almot priodic olution for a dlay logarithmic population modl, Mathmatical and Computr Modlling, vol. 53, no. 1-2, pp: , J.O. Alzabut, G.T. Stamov, and E. Srmutlu, On almot priodic olution for an impuliv dlay logarithmic population modl, Mathmatical and Computr Modlling, vol. 51, no. 5-6, pp: , F.D. Chn, Priodic olution and almot priodic olution of a nutral multipci Logarithmic population modl, Applid Mathmatic and Computation, vol. 176, no. 2, pp: , Y.H. Xia, Almot priodic olution of a population modl: via pctral radiu of matrix, Bulltin of th Malayian Mathmatical Scinc Socity, vol. 37, no. 1, pp: , L.J. Chn, and X.D. Xi, Prmannc of an N-pci coopration ytm with continuou tim dlay and fdback control, Nonlinar Analyi: Ral World Application, vol. 12, no. 1, pp: 34-38, Y.K. Li, and T.W. Zhang, Prmannc of a dicrt n-pci coopration ytm with tim-varying dlay and fdback control, Mathmatical and Computr Modlling, vol. 53, no. 5-6, pp: , Y.K. Li, and T.W. Zhang, Prmannc and almot priodic qunc olution for a dicrt dlay logitic quation with fdback control, Nonlinar Analyi: Ral World Application, vol. 12, no. 3, pp: , Z. Huang, S. Mohamad, X. Wang, and C. Fng, Convrgnc analyi of gnral nural ntwork undr almot priodic timli, Intrnational Journal of Circuit Thory and Application, vol. 37, no. 6, pp: , F.X. Lin, Exponntial Dichotomi of Linar Sytm, Anhui Univrity Pr, Hfi, 1999 (in Chin). 21 W.A. Coppl, Dichotomi in Stability Thory, Lctur Not in Mathmatic, vol. 629, Springr, Brlin, M. Kolar, M. Bn, D. Svcovic, and J. Kratochvil, Mathmatical modl and computational tudi of dicrt dilocation dynamic, IAENG Intrnational Journal of Applid Mathmatic, vol. 45, no. 3, pp: , L.Y. Pang, and T.W. Zhang, Almot priodic Ocillation in a watttyp prdator-pry modl with diffuion and tim dlay, IAENG Intrnational Journal of Applid Mathmatic, vol. 45, no. 2, pp: 92-11, Z.J. Gng, and M. Liu, Analyi of tochatic Gilpin-Ayala modl in pollutd nvironmnt, IAENG Intrnational Journal of Applid Mathmatic, vol. 45, no. 2, pp: , 215. REFERENCES 1 K. Gopalamy, Stability and Ocillation in Dlay Diffrntial Equation of Population DynamicMathmatic and it Application, vol. 74, Kluwr Acadmic Publihr Group, Dordrcht, G. Kirlingr, Prmannc in Lotka-Voltrra quation linkd pryprdator ytm, Mathmatical Biocinc, vol. 82, no. 2, pp: , Y.K. Li, Attractivity of a poitiv priodic olution for all othr poitiv olution in a dlay population modl, Mathmatica Applicata, vol. 12, no. 3, pp: , (in Chin). 4 F.D. Chn, X.X. Chn, F.X.Lin, and J.L. Shi, Poitiv priodic olution of tat-dpndnt dlay logarithm population modl, Journal of Fuzhou Univrity, vol. 31, no. 3, pp: 1-4, 23. (in Chin). 5 Z.J. Liu, Poitiv priodic olution for dlay multipci logrithmic population modl, Journal of Enginring Mathmatic, vol. 19, no. 2, pp: 11-16, 22. (in Chin). 6 F.D. Chn, Priodic olution and almot priodic olution for a dlay multipci logarithmic population modl, Applid Mathmatic and Computuation, vol. 171, no. 2, pp: 76-77, Y.K. Li, On a priodic nutral dlay logarithmic population modl, Journal of Sytm Scinc and Mathmatical Scinc, vol. 19, no. 1, pp: 34-38, S.P. Lu, and W.G. G, Exitnc of poitiv priodic olution for nutral logarithmic population modl with multipl dlay, Journal of Computational and Applid Mathmatic, vol. 166, no. 2, pp: , Z.H. Yang, and J.D Cao, Sufficint conditon for th xitnc of poitiv priodic olutio of a cla of nutral dlay modl, Applid Mathmatic and Computation, vol. 142, no. 1, pp: , S.P. Lu, and W.G. G, Exitnc of poitiv priodic olution for nutral logarithmic population modl with multipl dlay, Journal of Computational and Applid Mathmatic, vol. 166, no. 2, pp: , 24. (Advanc onlin publication: 15 Fbruary 216)