Main Times Series The positive almost periodic solution for Nicholsontype delay systems with linear harvesting terms

Applied Mathematical Modelling 36 (2012) 3289–3298 Contents lists available at SciVerse ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm The positive almost periodic solution for Nicholson-type delay systems with linear harvesting terms q Xingguo Liu a, Junxia Meng b,⇑ a b College of Business Administration, Hunan University, Changsha, Hunan 410082, PR China College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, PR China a r t i c l e i n f o Article history: Received 28 April 2011 Received in revised form 24 September 2011 Accepted 29 September 2011 Available online 18 October 2011 a b s t r a c t In this paper, we study the existence and exponential convergence of positive almost periodic solutions for a class of Nicholson-type delay system with linear harvesting terms Under appropriate conditions, we establish some criteria to ensure that the solutions of this system converge locally exponentially to a positive almost periodic solution Moreover, we give an example to illustrate our main results Ó 2011 Elsevier Inc All rights reserved Keywords: Positive almost periodic solution Exponential convergence Nicholson-type delay system Linear harvesting term Introduction In [1], to describe the models of Marine Protected Areas and B-cell Chronic Lymphocytic Leukemia dynamics that belong to the Nicholson-type delay differential systems, Berezansky et al [1] considered the dynamics of the following autonomous Nicholson-type delay systems: ( x01 tị ẳ a1 x1 tị ỵ b1 x2 tị ỵ c1 x1 t sịex1 tsị ; x02 tị ẳ a2 x2 tị ỵ b2 x1 tị ỵ c2 x2 t sịex2 tsị ; 1:1ị with initial conditions: xi sị ẳ ui sị; s ẵs; 0; ui 0ị > 0; ð1:2Þ where ui C([Às, 0], [0, +1)), ai, bi, ci and s are nonnegative constants, i = 1, Furthermore, Wang et al [2] showed the existence and exponential convergence of positive almost periodic solutions for the following non-autonomous Nicholson-type delay systems: q This work was supported by the Natural Scientific Research Fund of Zhejiang Provincial of PR China (Grant No Y6110436), the Natural Scientific Research Fund of Hunan Provincial of PR China (Grant No 11JJ6006), and the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grant Nos 11C0916, 11C0915, 11C1186) ⇑ Corresponding author Tel./fax: +86 057383643075 E-mail address: mengjunxia1968@yahoo.com.cn (J Meng) 0307-904X/$ - see front matter Ó 2011 Elsevier Inc All rights reserved doi:10.1016/j.apm.2011.09.087 3290 X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 m P > Àc1j ðtÞx1 ðtÀs1j ðtÞÞ > ; > < x1 tị ẳ a1 tịx1 tị ỵ b1 tịx2 tị ỵ c1j tịx1 t s1j tịịe jẳ1 m P > > c tịx ts tịị > : x2 tị ẳ a2 tịx2 tị þ b2 ðtÞx1 ðtÞ þ c2j ðtÞx2 ðt À s2j tịịe 2j 2j ; 1:3ị jẳ1 where ai, bi, cij, cij, sij : R ? (0, +1) are almost periodic functions, and i = 1, 2, j = 1, 2, , m Recently, assuming that a harvesting function is a function of the delayed estimate of the true population, Berezansky et al [3] proposed the Nicholson’s blowflies model with a linear harvesting term: x0 ðtÞ ẳ dxtị ỵ pxt sịeaxtsị Hxt rị; d; p; s; a; H; r 0; ỵ1ị; 1:4ị where Hx(t À r) is a linear harvesting term, x(t) is the size of the population at time t, P is the maximum per capita daily egg production, 1a is the size at which the population reproduces at its maximum rate, d is the per capita daily adult death rate, and s is the generation time Moreover, Berezansky et al [3] pointed out an open problem: How about the dynamic behaviors of the Nicholson’s blowflies model with a linear harvesting term Now, motivated by Berezansky et al [1], Wang et al [2], Berezansky et al [3] a corresponding question arises: How about the existence and convergence of positive almost periodic solutions of Nicholson-type delay differential systems with linear harvesting terms The main purpose of this paper is to give the conditions to ensure the existence and convergence of positive almost periodic solutions of the following non-autonomous Nicholson-type delay systems with linear harvesting terms: m P > Àc1j ðtÞx1 ðtÀs1j tịị > > > x1 tị ẳ a1 tịx1 tị ỵ b1 tịx2 tị ỵ c1j tịx1 t s1j tịịe > jẳ1 > > > < H1 tịx1 ðt À r1 ðtÞÞ; m P > > > x2 tị ẳ a2 tịx2 tị ỵ b2 tịx1 tị þ c2j ðtÞx2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ > > > jẳ1 > > : H2 tịx2 t À r2 ðtÞÞ; ð1:5Þ where ai, bi, Hi, ri, cij, cij, sij : R1 ? [0, +1) are almost periodic functions, and i = 1, 2, j = 1, 2, , m For convenience, we introduce some notations Throughout this paper, given a bounded continuous function g defined on R1, let g+ and gÀ be defined as g ẳ inf gtị; t2R g ỵ ẳ sup gðtÞ: t2R It will be assumed that & n o ' aÀi > 0; bÀi > 0; cÀij > 0; ri ẳ max max sỵij ; rỵi > 0; i ¼ 1; 2: 16j6m ð1:6Þ À Á Denote by Rn Rnỵ the set of all (nonnegative) real vectors Let     C ẳ Cẵr ; 0; R1 Þ Â Cð½Àr ; 0Š; R1 Þ and C ỵ ẳ C ẵr1 ; 0; R1ỵ C ẵr ; 0; R1ỵ : If xi(t) is defined on [t0 À ri, r) with t0, r R1 and i = 1, 2, then we define xt C as xt ¼ x1t ; x2t where xit hị ẳ xi t ỵ hị for all h [Àri, 0] and i = 1, A matrix or vector A P means that all entries of A are greater than or equal to zero A > can be defined similarly For matrices or vectors A and B, A P B (resp A > B) means that A À B P (resp A À B > 0) For vector X = (x1, x2) R2, we let jXj denote the absolute-value vector given by jXj = (jx1j, jx2j), and define jXk = max16i62jxij The initial conditions associated with system (1.5) are of the form: xt0 ¼ u; u ẳ u1 ; u2 ị C ỵ and ui 0ị > 0; i ẳ 1; 2: 1:7ị We write xt(t0, u)(x(t; t0, u)) for a solution of the initial value problem (1.5) and (1.7) Also, let [t0, g(u)) be the maximal rightinterval of existence of xt(t0, u) The remaining part of this paper is organized as follows In Section 2, we shall give some notations and preliminary results In Section 3, we shall derive new sufficient conditions for checking the existence, uniqueness and local exponential convergence of the positive almost periodic solution of (1.5) In Section 4, we shall give some example and remark to illustrate our results obtained in the previous sections Preliminary results In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 3291 Definition 2.1 (See [4,5]) Let u(t) : R1 ? Rn be continuous in t u(t) is said to be almost periodic on R1, if for any e > 0, the set T(u,e) = {d:ju(t + d) À u(t)j < e for all t R1} is relatively dense, i.e., for any e > 0, it is possible to find a real number l = l(e) > 0, such that for any interval with length l(e), there exists a number d = d(e) in this interval such that ju(t + d) À u(t)j < e, for all t R1 Definition 2.2 (See [4,5]) Let x Rn and Q(t) be a n  n continuous matrix defined on R1 The linear system x0 tị ẳ Q tịxtị ð2:1Þ is said to admit an exponential dichotomy on R if there exist positive constants k, a, projection P and the fundamental solution matrix X(t) of (2.1) satisfying ÀaðtÀsÞ kXðtÞPX À1 ðsÞk ke for all t P s; ÀaðsÀtÞ kXðtÞðI À PÞX À1 ðsÞk ke for all t s: Set B ẳ fuju ẳ u1 tị; u2 ðtÞÞ is an almost periodic function on R1 g: For any u B, we define induced module kukB ¼ supt2R1 kuðtÞk, then B is a Banach space Lemma 2.1 (See [4,5]) If the linear system (2.1) admits an exponential dichotomy, then almost periodic system x0 tị ẳ Q tịx ỵ gtị 2:2ị has a unique almost periodic solution x(t), and xtị ẳ Z t XtịPX sịgsịds Z ỵ1 XtịI PịX sịgsị ds: 2:3ị t À1 Lemma 2.2 (See [4,5]) Let ci(t) be an almost periodic function on R1 and Mẵci ẳ lim T!ỵ1 Z T tỵT ci sịds > 0; i ẳ 1; 2; ; n: t Then the linear system x0 tị ẳ diag c1 tị; c2 tị; ; Àcn ðtÞÞxðtÞ admits an exponential dichotomy on R1 Lemma 2.3 Suppose that there exist two positive constants Ei1 and Ei2 such that Ei1 > Ei2 ; b1 E22 ỵ a b2 E12 ỵ a þ a À m X cÀ1j j¼1 þ bþ1 E21 ỵ a m c X 2j jẳ1 ỵ a m X cỵ1j < E11 ; a1 c1j e jẳ1 ỵ cỵ E 1j 11 E11 e cỵ E 2j 21 E21 e Hỵ1 E11 aỵ1 Hỵ2 E21 aỵ2 bỵ2 E11 a > E12 P ỵ m X cỵ2j < E21 ; a2 c2j e jẳ1 2:4ị ; cÀ1j ð2:5Þ ; cÀ2j ð2:6Þ 16j6m > E22 P 16j6m where i = 1, Let C :ẳ fuju C; Ei2 < ui tị < Ei1 ; for all t ½Àr i ; 0Š; i ¼ 1; 2g: Moreover, assume that x(t; t0, u) is the solution of (1.5) with u C0 Then, Ei2 < xi ðt; t ; uÞ < Ei1 ; for all t ẵt ; guịị; and g(u) = +1 i ẳ 1; 2:7ị 3292 X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 Proof Set x(t) = x(t; t0, u) for all t [t0, g(u)) Let [t0, T) # [t0, g(u)) be an interval such that < xi ðtÞ for all t ẵt0 ; Tị; i ẳ 1; 2; ð2:8Þ we claim that < xi ðtÞ < Ei1 for all t ẵt0 ; Tị; i ẳ 1; 2: ð2:9Þ Assume, by way of contradiction, that (2.9) does not hold Then, one of the following cases must occur Case i: There exists t1 (t0, T) such that x1 t1 ị ẳ E11 and < xi tị < Ei1 for all t ½t À r i ; t1 ị; i ẳ 1; 2: 2:10ị i ¼ 1; 2: ð2:11Þ Case ii: There exists t2 (t0, T) such that x2 t2 ị ẳ E21 and < xi ðtÞ < Ei1 for all t ½t À r i ; t2 Þ; If Case i holds, calculating the derivative of x1(t), together with (2.4) and the fact that supuP0 ueÀu ¼ 1e , (1.5) and (2.10) imply that x01 ðt Þ ẳ a1 t ịx1 t1 ị ỵ b1 t1 ịx2 t ị ỵ m X c1j t Þx1 ðt À s1j ðt ÞÞeÀc1j ðt1 Þx1 ðt1 Às1j ðt1 ÞÞ À H1 ðt Þx1 ðt r1 t1 ịị jẳ1 x1 t ị a ỵ bỵ1 E21 m cỵ m X cỵ1j bỵ E21 X 1j ẳ a1 E11 ỵ ỵ ỵ c1j e a1 a1 c1j e j¼1 j¼1 ! < 0; which is a contradiction and implies that (2.9) holds If Case ii holds, calculating the derivative of x2(t), together with (2.4) and the fact that supuP0 ueÀu ¼ 1e , (1.5) and (2.11) imply that x02 ðt ị ẳ a2 t ịx2 t2 ị ỵ b2 t2 ịx1 t ị ỵ m X c2j t Þx2 ðt À s2j ðt ÞÞeÀc2j ðt2 Þx2 ðt2 Às2j ðt2 ÞÞ À H2 ðt Þx2 t r2 t2 ịị jẳ1 a2 x2 t ị ỵ bỵ2 E11 ỵ m cỵ m X cỵ2j bỵ E11 X 2j ẳ a2 E21 ỵ ỵ c2j e a2 a2 c2j e j¼1 j¼1 ! < 0; which is a contradiction and implies that (2.9) holds We next show that xi ðtÞ > Ei2 ; for all t t0 ; guịị; i ẳ 1; 2: 2:12ị Suppose, for the sake of contradiction, that (2.12) does not hold Then, one of the following cases must occur Case I: There exists t3 (t0, g(u)) such that x1 ðt3 Þ ¼ E12 and xi ðtÞ > Ei2 for all t ẵt0 r i ; t ị; i ẳ 1; 2: 2:13ị Case II: There exists t4 (t0, g(u)) such that x2 t4 ị ẳ E22 and xi tị > Ei2 for all t ẵt r i ; t4 ị; i ẳ 1; 2: ð2:14Þ If Case I holds Then, from (2.5), (2.6), (2.9) and (2.13), we get Ei2 < xi ðtÞ < Ei1 ; cỵij xi tị P cỵij Ei2 P cỵij P 1; cÀij ð2:15Þ 16j6m for all t [t0 À ri, t3), i = 1, 2, j = 1, 2, , m Calculating the derivative of x1(t), together with (2.5) and the fact that min16u6jueÀu = jeÀj, (1.5), (2.13) and (2.15) imply that P x01 t3 ị ẳ a1 t ịx1 t ị ỵ b1 t3 ịx2 t3 ị ỵ m X c1j ðt3 Þx1 ðt À s1j ðt ÞÞeÀc1j ðt3 Þx1 ðt3 Às1j ðt3 ÞÞ À H1 ðt ịx1 t r1 t3 ịị jẳ1 ẳ a1 t ịx1 t ị ỵ b1 t ịx2 t ị ỵ m X c1j t3 ị jẳ1 cỵ1j cỵ1j x1 t3 s1j t3 ịịec1j ðt3 Þx1 ðt3 Às1j ðt3 ÞÞ À H1 ðt3 Þx1 ðt3 À r1 ðt3 ÞÞ 3293 X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 P Àa1 ðt ịx1 t ị ỵ b1 E22 ỵ m c X 1j jẳ1 ỵ x1 t ị P a ỵ b1 E22 ỵ m X c1j ỵ 1j jẳ1 c ỵ ỵ 1j c cỵ1j x1 t3 À s1j ðt3 ÞÞeÀc1j x1 ðt3 Às1j ðt3 ÞÞ À Hỵ1 E11 cỵ E ỵ 1j 11 1j E11 e c Hỵ1 E11 ỵ ẳa E12 ỵ b1 E22 aỵ1 ỵ m X c1j jẳ1 aỵ1 E11 e cỵ E 1j 11 Hỵ1 E11 ! aỵ1 > 0; which is a contradiction and implies that (2.12) holds If Case II holds, we can show that (2.15) holds for all t [t0 À ri, t4),i = 1, 2, j = 1, 2, , m Calculating the derivative of x2(t), together with (2.6) and the fact that min16u6jueÀu = jeÀj, (1.5), (2.14) and (2.15) imply that P x02 t4 ị ẳ a2 t ịx2 t ị ỵ b2 t4 ịx1 t4 ị þ m X c2j ðt4 Þx2 ðt4 À s2j ðt ÞÞeÀc2j ðt4 Þx2 ðt4 Às2j ðt4 ÞÞ À H2 t ịx2 t r2 t4 ịị jẳ1 ẳ a2 t4 ịx2 t4 ị ỵ b2 t ịx1 t ị ỵ m X c2j t4 ị cỵ2j jẳ1 P a2 t ịx2 t ị þ bÀ2 E12 þ m cÀ X 2j cþ2j j¼1 þ x2 ðt Þ P Àa þ bÀ2 E12 m c X 2j ỵ jẳ1 cỵ2j cỵ2j x2 ðt4 À s2j ðt4 ÞÞeÀc2j ðt4 Þx2 ðt4 Às2j ðt4 ÞÞ À H2 ðt4 Þx2 ðt4 À r2 ðt4 ÞÞ þ cþ2j x2 ðt4 À s2j ðt4 ÞÞeÀc2j x2 ðt4 s2j t4 ịị Hỵ2 E21 cỵ E ỵ 2j 21 2j E21 e c Hỵ2 E21 ỵ ẳa E22 ỵ b2 E12 aỵ2 ỵ m c X 2j jẳ1 aỵ2 E21 e cỵ E 2j 21 Hỵ2 E21 aỵ2 ! > 0; which is a contradiction and implies that (2.12) holds It follows from (2.9) and (2.12) that (2.7) is true From Theorem 2.3.1 in [6], we easily obtain g(u) = +1 This ends the proof of Lemma 2.3 h Main results Theorem 3.1 Let (2.4)(2.6) hold Moreover, suppose that ( max bỵ1 a1 ỵ m X cỵ1j jẳ1 a1 e2 Hỵ1 ỵ ; bỵ2 a1 a2 ỵ m X cỵ2j jẳ1 a2 e2 ỵ Hỵ2 ) a2 < 1: 3:1ị Then, there exists a unique positive almost periodic solution of system (1.5) in the region B⁄ = {uju B, Ei2 ui(t) Ei1, for all t R1, i = 1, 2} Proof For any / B, we consider an auxiliary system m P > > x01 tị ẳ a1 tịx1 tị ỵ b1 tị/2 tị ỵ c1j tị/1 t s1j tịịec1j tị/1 ts1j tịị > > > jẳ1 > > > > > < ÀH1 ðtÞ/1 ðt À r1 ðtÞÞ; ð3:2Þ m > P > > > x02 tị ẳ a2 tịx2 tị ỵ b2 tị/1 tị ỵ c2j ðtÞ/2 ðt À s2j ðtÞÞeÀc2j ðtÞ/ðtÀs2j ðtÞÞ > > > jẳ1 > > > : H2 tị/2 t r2 ðtÞÞ; Notice that M[ai] > 0(i = 1, 2), it follows from Lemma 2.2 that the linear system ( x01 tị ẳ a1 tịx1 tị; 3:3ị x02 tị ẳ Àa2 ðtÞx2 ðtÞ; admits an exponential dichotomy on R Thus, by Lemma 2.1, we obtain that the system (3.2) has exactly one almost periodic solution: / x tị ẳ x/1 tị; x/2 tị Z ẳ t e Rt s a1 uịdu b1 sị/2 sị ỵ m X À1 Z t À1 e À j¼1 Rt s a2 uịdu b2 sị/1 sị ỵ m X jẳ1 ! c1j ðsÞ/1 ðsÀs1j ðsÞÞ c1j ðsÞ/1 ðs À s1j ðsÞÞe À H1 ðsÞ/1 ðs À r1 ðsÞÞ ds; ! ! c2j ðsÞ/2 ðs À s2j ðsÞÞeÀc2j ðsÞ/2 ðsÀs2j ðsÞÞ À H2 ðsÞ/2 ðs À r2 ðsÞÞ ds : ð3:4Þ 3294 X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 Define a mapping T : B ? B by setting T/tịị ẳ x/ tị; 8/ B: Since B = {uju B, Ei2 ui(t) Ei1, for all t R1, i = 1, 2}, it is easy to see that B⁄ is a closed subset of B For any u B⁄, from (2.4) and (3.4) and the fact that supuP0 ueÀu ¼ 1e, we have ⁄ / x ðtÞ 6 ! Z ! ! Rt m t X 1 a2 uịdu ỵ e s ỵ e s b2 E11 ỵ c sị ds; c ðsÞ ds c1j ðsÞe 1j c2j ðsÞe 2j À1 jẳ1 jẳ1 ! m m cỵ1j bỵ2 E11 X cỵ2j bỵ1 E21 X < E11 ; E21 ị; for all t R1 : ỵ ỵ ; a1 a1 cÀ1j e aÀ2 aÀ2 cÀ2j e j¼1 j¼1 Z t Rt a1 uịdu m X bỵ1 E21 In view of the fact that min16u6jueÀu = jeÀj, from (2.5), (2.6) and (3.4), we obtain / x ðtÞ P Z t e À Rt s a1 ðuÞdu À1 Z t e Rt s ỵ m X c1j sị jẳ1 a2 uịdu P b1 E22 b2 E12 m X ỵ c2j sị jẳ1 b1 E22 aỵ1 ỵ m c X 1j cỵ E 1j 11 ỵ E11 e jẳ1 a1 cỵ2j cỵ1j ỵ 1j /1 s c þ 2j /2 ðs À c Hþ1 E11 aþ1 s1j sịịe s2j sịịe cỵ / ss1j sịị 1j cỵ / ss2j sịị 2j ; fracb2 E12 aỵ2 ỵ m c X 2j jẳ1 ! H1 sị/1 ðs À r1 ðsÞÞ ds; ! ! À H2 ðsÞ/2 s r2 sịị ds cỵ E 2j 21 ỵ E21 e a2 3:5ị Hỵ2 E21 ! > E12 ; E22 ị; for all t R1 : aỵ2 ð3:6Þ This implies that the mapping T is a self-mapping from B⁄ to B⁄ Now, we prove that the mapping T is a contraction mapping on B⁄ In fact, for u, w B⁄, we get    Z sup jðTðuÞðtÞ À TðwÞðtÞÞ1 j; sup jðTðuÞðtÞ À TðwÞðtÞÞ2 j ẳ sup j t2R t2R t2R ỵ m X t À e Rt s a1 ðuÞdu À1 ðb1 ðsÞðu2 ðsÞ À w2 ðsÞÞ c1j ðsÞðu1 ðs À s1j ðsÞÞeÀc1j ðsÞu1 ss1j sịị jẳ1 w1 s s1j sịịec1j sịw1 ss1j ðsÞÞ Þ À H1 ðsÞðu1 ðs À r1 ðsÞÞ Z t Rt À a ðuÞdu e s ðb2 ðsÞðu1 ðsÞ À w1 ðsÞÞ Àw1 ðs À r1 ðsÞÞÞÞdsj; sup j t2R ỵ m X c2j sịu2 s s2j sịịec2j sịu2 ss2j sịị jẳ1 w2 s s2j ðsÞÞeÀc2j ðsÞw2 ðsÀs2j ðsÞÞ Þ À H2 ðsÞðu2 ðs À r2 ðsÞÞ Àw2 ðs À r2 ðsÞÞÞÞdsjÞ  Z t Rt À a ðuÞdu e s ðb1 ðsÞðu2 ðsÞ w2 sịị ẳ sup j t2R m X c1j sị ỵ c1j sịu1 s s1j sịịec1j sịu1 ss1j sịị c 1j sị jẳ1 c1j sịw1 s s1j ðsÞÞeÀc1j ðsÞw1 ðsÀs1j ðsÞÞ Þ À H1 ðsÞðu1 ðs À r1 ðsÞÞ Z t Rt À a ðuÞdu Àw1 ðs À r1 ðsÞÞÞÞdsj; supt2R j e s ðb2 ðsÞðu1 ðsÞ À1 m X c2j ðsÞ ðc ðsÞu2 ðs s2j sịịec2j sịu2 ss2j sịị w1 sịị ỵ c2j sị 2j jẳ1 c2j sịw2 s s2j sịịec2j sịw2 ðsÀs2j ðsÞÞ Þ À H2 ðsÞðu2 ðs À r2 ðsÞÞ Àw2 ðs À r2 ðsÞÞÞÞdsjÞ: In view of (1.5), (2.5), (2.6), (3.5), (3.6) and (3.7), from supuP1 1Àu  u¼ e e2 and the inequality   1 À x ỵ hy xịị jx yj jx yj where x; y ẵ1; ỵ1ị; jxex yey j ẳ   exỵhyxị e2 we have ð3:7Þ < h < 1; ð3:8Þ X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 ðsup jðTðuÞðtÞ À TðwÞðtÞÞ1 j; sup jðTðuÞðtÞ À TðwÞðtÞÞ2 jÞ t2R t2R Z t Rt m X bỵ1 Hỵ bỵ a uịdu ku wkB ỵ sup e s cỵ1j ju1 s s1j sịị w1 s s1j sịịjds ỵ ku wkB ; 2À ku À wkB À e a1 a1 a2 t2R jẳ1 ! Z t Rt ỵ m X H a uịdu e s cỵ2j ju2 ðs À s2j ðsÞÞ À w2 ðs À s2j ðsÞÞjds þ À2 ku À wkB þ sup e a2 t2R jẳ1 ! ! ! ỵ ỵ ỵ ỵ m m ỵ ỵ b1 X c1j H1 b2 X c2j H2 ku wkB ; ku wkB : ỵ þ þ þ aÀ1 j¼1 aÀ1 e2 aÀ1 aÀ2 jẳ1 a2 e2 a2 Hence ( kTuị TwịkB max bỵ1 a1 max bỵ1 a1 ỵ ) m cỵ2j Hỵ1 bỵ2 X Hỵ2 ku wkB : ỵ ; þ þ aÀ1 e2 aÀ1 aÀ2 j¼1 aÀ2 e2 aÀ2 ỵ jẳ1 m X cỵ1j jẳ1 3:9ị m X cỵ1j Noting that ( 3295 m X cỵ2j Hỵ bỵ Hỵ þ À1 ; 2À þ þ À2 À À a1 e a1 a2 a2 e a2 j¼1 ) < 1; it is clear that the mapping T is a contraction on B⁄ Using Theorem 0.3.1 of [7], we obtain that the mapping T possesses a unique fixed point u⁄ B⁄, Tu⁄ = u⁄ By (3.2), u⁄ satisfies (1.5) So u⁄ is an almost periodic solution of (1.5) in B⁄ The proof of Theorem 3.1 is now complete h Theorem 3.2 Let x⁄(t) be the positive almost periodic solution of Eq (1.5) in the region B⁄ Suppose that (2.4)–(2.6) and (3.1) hold Then, the solution x(t; t0, u) of (1.5) with u C0 converges exponentially to x⁄(t) as t ? +1 Proof Set x(t) = x(t; t0, u) and yi tị ẳ xi tị xi tị, where t [t0 À ri, +1), i = 1, Then m P > > > y01 ðtÞ ẳ a1 tịy1 tị ỵ b1 tịy2 tị ỵ c1j ðtÞðx1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ > > > j¼1 > > > à < ÀxÃ1 ðt À s1j ðtÞÞeÀc1j ðtÞx1 ðtÀs1j ðtÞÞ Þ À H1 ðtÞy1 ðt À r1 ðtÞÞ m P > > > y2 tị ẳ a2 tịy2 tị ỵ b2 tịy1 tị ỵ c2j ðtÞðx2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ > > > j¼1 > > > à : ÀxÃ2 ðt À s2j ðtÞÞeÀc2j ðtÞx2 ðtÀs2j ðtÞÞ Þ À H2 ðtÞy2 ðt À r2 ðtÞÞ; ð3:10Þ Set À Á Ci ðuÞ ẳ u ỵ bỵi ỵ m X cỵij jẳ1 uri e ỵ Hỵi euri ; e2 u ẵ0; 1; i ẳ 1; 2: 3:11ị Clearly, Ci(u), i = 1, 2, are continuous functions on [0, 1] In view of (3.1), we obtain Ci 0ị ẳ ỵ bỵi ỵ m X jẳ1 cỵij ỵ Hỵi < 0; e2 i ẳ 1; 2; we can choose two constants g > and k (0, 1] such that Ci kị ẳ k ỵ bỵi ỵ m X jẳ1 cỵij kri e ỵ Hỵi ekri < g < 0; e2 i ¼ 1; 2: ð3:12Þ We consider the Lyapunov functional V tị ẳ jy1 tịjekt ; V tị ẳ jy2 ðtÞjekt : ð3:13Þ Calculating the upper right derivative of Vi(t)(i = 1, 2) along the solution y(t) of (3.10), we have Dỵ V tịị a1 tịjy1 tịjekt þ b1 ðtÞjy2 ðtÞjekt þ m X c1j ðtÞjx1 ðt s1j tịịec1j tịx1 ts1j tịị jẳ1 x1 t s1j tịịec1j tịx1 ts1j tịị jekt ỵ H1 tịjy1 t r1 tịịjekt ỵ kjy1 tịjekt " m X ẳ k a1 tịịjy1 tịj ỵ b1 tịjy2 tịj ỵ c1j tịjx1 t s1j tịịec1j tịx1 ts1j tịị jẳ1 x1 t s1j tịịec1j tịx1 ts1j tịị j ỵ H1 tịjy1 t r1 tịịj ekt ; for all t > t0 ; ð3:14Þ 3296 X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 32893298 and Dỵ V tịị a2 tịjy2 tịjekt þ b2 ðtÞjy1 ðtÞjekt þ m X c2j ðtÞjx2 ðt s2j tịịec2j tịx2 ts2j tịị jẳ1 x2 t s2j tịịec2j tịx2 ts2j tịị jekt ỵ H2 tịjy2 t r2 tịịjekt ỵ kjy2 tịjekt " m X ẳ k a2 tịịjy2 tịj ỵ b2 tịjy1 tịj ỵ c2j tịjx2 t s2j tịịec2j tịx2 ts2j tịị jẳ1 x2 t s2j tịịec2j tịx2 ts2j tịị j ỵ H2 tịjy1 t r2 tịịj ekt ; for all t > t : ð3:15Þ Let maxiẳ1;2 fekt0 maxt2ẵt0 ri ;t0 jui tị xi tịj ỵ 1ịg :ẳ M We claim that V i tị ẳ jyi tịjekt < M for all t > t ; i ẳ 1; 2: 3:16ị Otherwise, one of the following cases must occur Case 1: There exists T1 > t0 such that V ðT Þ ¼ M and V i ðtÞ < M for all t ½t À r i ; T ị; i ẳ 1; 2: 3:17ị i ẳ 1; 2: ð3:18Þ Case 2: There exists T2 > t0 such that V T ị ẳ M and V i tị < M for all t ẵt À r i ; T Þ; If Case holds, together with (2.7), (3.8) and (3.14), (3.17) implies that Dỵ V T ị Mị ẳ Dỵ V T ịị ẵk a1 T ịịjy1 T ịj ỵ b1 T ịjy2 T ịj ỵ " m X c1j ðT Þx1 ðT Às1j ðT ÞÞ c1j ðT Þjx1 ðT À s1j ðT ÞÞe À xÃ1 ðT Àc1j ðT ÞxÃ1 ðT Às1j ðT ÞÞ À s1j ðT ÞÞe # j ỵ H1 T ịjy1 T r1 T ịịj ekT jẳ1 m X c1j T Þ jc1j ðT Þx1 ðT À s1j ðT ÞÞeÀc1j ðT ÞxðT Às1j ðT ÞÞ c ðT Þ 1j j¼1 i à Àc1j ðT ÞxÃ1 ðT À s1j ðT ÞÞeÀc1j ðT Þx ðT Às1j ðT ÞÞ j þ H1 ðT Þjy1 ðT À r1 ðT ịịj ekT ẳ k a1 T ịịjy1 T ịj ỵ b1 T ịjy2 T ịj ỵ m X jy T s1j ðT ÞÞjekðT Às1j ðT ÞÞ eks1j ðT Þ e2 " # m X À k a1 ỵ bỵ1 ỵ cỵ1j ekr1 ỵ Hỵ1 ekr1 M: e jẳ1 k a1 T ịịjy1 T ịjekT ỵ b1 T ịjy2 T ịjekT ỵ c1j T ị jẳ1 ỵH1 T ịjy1 T À r1 ðT ÞÞjekðT Àr1 ðT ÞÞ ekr1 ðT Þ ð3:19Þ Thus, m X À Á k a1 ỵ bỵ1 ỵ cỵ1j ekr1 ỵ Hỵ1 ekr1 ; e jẳ1 which contradicts with (3.12) Hence, (3.16) holds If Case holds, together with (2.7), (3.8) and (3.15), (3.18) implies that Dỵ V T ị Mị ẳ Dỵ V T ịị ẵk a2 T ịịjy2 T ịj ỵ b2 T ịjy1 T ịj ỵ " m X c2j T Þx2 ðT Às2j ðT ÞÞ c2j ðT Þjx2 ðT À s2j ðT ÞÞe À xÃ2 ðT Àc2j ðT Þxà ðT Às2j ðT ịị s2j T ịịe # j ỵ H2 ðT Þjy2 ðT À r2 ðT ÞÞj ekT j¼1 m X c2j ðT Þ jc2j ðT Þx2 ðT À s2j ðT ÞÞeÀc2j ðT Þx2 ðT Às2j ðT ÞÞ c T ị 2j jẳ1 i c2j T ÞxÃ2 ðT À s2j ðT ÞÞeÀc2j ðT ịx2 T s2j T ịị j ỵ H2 ðT Þjy2 ðT À r2 ðT ÞÞj ekT ¼ ðk À a2 ðT ÞÞjy2 T ịj ỵ b2 T ịjy1 T ịj ỵ ỵH2 T ịjy2 T r2 ðT ÞÞjekðT Àr2 ðT ÞÞ ekr2 ðT Þ Thus, m X jy2 ðT À s2j ðT ÞÞjekðT Às2j ðT ÞÞ eks2j T ị e jẳ1 " # m X k a2 ỵ bỵ2 ỵ cỵ2j ekr2 ỵ Hỵ2 ekr2 M: e jẳ1 ðk À a2 ðT ÞÞjy2 ðT ÞjekT þ b2 ðT Þjy1 ðT ÞjekT þ c2j ðT Þ ð3:20Þ X Liu, J Meng / Applied Mathematical Modelling 36 (2012) 3289–3298 3297 m X À k a2 ỵ bỵ2 ỵ cỵ2j ekr2 ỵ Hỵ2 ekr2 ; e jẳ1 which contradicts with (3.12) Hence, (3.16) holds It follows that jyi ðtÞj < MeÀkt for all t > t ; i ẳ 1; 2: 3:21ị This completes the proof h Example and remark In this section, we give an example to demonstrate the results obtained in previous sections Example 4.1 Consider the following Nicholson-type delay system with linear harvesting terms:   18 ỵ cos2 t x1 tị ỵ 0:00001 ỵ 0:000005 sin t ee3 x2 ðtÞ  pffiffiffi  À Á 2j sin tj ị; ỵ ee1 9:5 ỵ 0:005j sin 2tj x1 t À e2j sin tj eÀx1 ðtÀe pffi     p p 2j cos 3tj ị ỵ ee1 9:5 þ 0:005j sin 5tj x1 t À e2j cos 3tj eÀx1 ðtÀe  pffiffi  À ð0:000001 cos2 tÞeeÀ3 x1 t À e2j cos 3tj   À Á > > x02 tị ẳ 18 ỵ sin t x2 tị ỵ 0:00001 ỵ 0:000005 cos2 t ee3 x1 ðtÞ > > > >  > pffiffiffi  À 2j cos tj > ị; > ỵ ee1 9:5 þ 0:005j cos 2tj x2 t À e2j cos tj eÀx2 ðtÀe > > > > pffi    > pffiffi  p ffiffiffi > 2j cos 7tj > ị > ỵ ee1 9:5 ỵ 0:005j sin 6tj x2 t À e2j cos 7tj eÀx2 ðtÀe > > > >   pffiffi > À Á > : À 0:000001 cos4 t eeÀ3 x2 t À e2j cos 3tj ; > x01 tị ẳ > > > > > > > > > > > > > > > > > > > > > > < ð4:1Þ ỵ e3 e3 e1 e1 Obviously, ẳ n18; aỵi ẳ 19; ; bỵ ; c ; cỵ ; Hỵ n coij ẳ co i ẳ i ẳ 0:000015e ij ¼ 1; bi ¼ 0:00001e ij ¼ 9:5e ij ẳ 9:505e ỵ ; ẳ e 0:000001ee3 ; r i ẳ max max16j6m sỵ r ; ij i þ bÀi þ þ þ i a j¼1 bþi e a c À e1Àcij e X ij À i ỵ Hỵi e ỵ i a ẳ 19 þ 0:00001eeÀ3 À 0:000001eeÀ2 > 1; 19 2 cþ X cỵij bỵi e X 0:000015ee2 ỵ 19:01ee2 ij ẳ ỵ ẳ