Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 61, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTICALLY ALMOST PERIODICITY OF DELAYED NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE CHUANGXIA HUANG, JIAFU WANG, LIHONG HUANG Abstract. In this article we study a delayed Nicholson-type system involv- ing patch structure. We apply differential inequality techniques to establish a sufficient condition for the existence of positive asymptotically almost periodic solutions. By constructing suitable Lyapunov functions, we obtain a new cri- terion for the uniqueness and global attractivity of the asymptotically almost periodic solutions. 1. Introduction Several classes of differential equations models arising from biological mathemat- ics have been intensively investigated, the hot topics include: stability, limit cycles, bifurcation and periodic solutions [2, 5, 7, 9, 8, 10, 13, 14, 18]. Although period- icity is important in real surroundings and world, when adding the factors of the environmental vary, almost periodicity is always more accurate, more realistic and more general than periodicity. As given in [1, 3, 4, 23] in comparison with periodic effects, almost periodic effects are more frequent in lots of real world applications. In particular, the existence and global stability of almost periodic solutions for the famous scalar Nicholson’s blowflies equation x′(t) = −a(t)x(t) + m∑ j=1 βj(t)x(t− τj(t))e−γj(t)x(t−τj(t)), (1.1) and the Nicholson’s blowflies systems with patch structure x′i(t) = −aii(t)xi(t) + n∑ j=1,j 6=i aij(t)xj(t) + m∑ j=1 βij(t)xi(t− τij(t))e−γij(t)xi(t−τij(t)), (1.2) for i ∈ Q := {1, 2, . . . , n}, has been extensively investigated [12, 19, 20]. Here, the scalar Nicholson’s blowflies equation (1.1) is a special case of Nicholson’s blowflies system (1.2). xi(t) denotes the density of the ith population at time t, aij(t)(i 6= j) is the rate of the population moving from class j to class i at time t, aii(t) is the coefficient of instantaneous loss for class i at time t (which integrates both the death 2010 Mathematics Subject Classification. 34C25, 34K13, 34K25. Key words and phrases. Nicholson-type system; global attractivity; patch structure; asymptotically almost periodic solution. c©2020 Texas State University. Submitted July 19, 2019. Published June 16, 2020. 1 2 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 rate and the dispersal rates of the population in class i moving to the other classes), βij(t)xi(t− τij(t))e−γij(t)xi(t−τij(t)) is the birth function for class i at time t, βij(t) is the per capita daily adult death rate for the species in the patch i at time t and 1/γij(t) is the size at which the ith-population reproduces at its maximum rate in time t and τij(t) is the generation time of ith-population at time t and i ∈ Q. In particular, the results of [19] complement previously known results [12, 20]. It should be mentioned that Wang et al [19] established the existence and global convergence of almost periodic solutions for Nicholson’s blowflies systems (1.2) un- der the additional conditions that there exists a positive constant M > κ such that γij(t)M ≤ κ̃, for all t ∈ R, i ∈ Q, j ∈ I = {1, 2, . . . ,m}, (1.3) sup t∈R {−aii(t) + n∑ j=1,j 6=i aij(t) + 1 eM m∑ j=1 βij(t) γij(t) } < 0, i ∈ Q, (1.4) inf t∈R {−aii(t) + n∑ j=1,j 6=i aij(t) + m∑ j=1 βij(t) γij(t) e−κ} > 0, i ∈ Q, (1.5) inf t≥t0 m∑ j=1 βij(t) γij(t) > 0, i ∈ Q, j ∈ I, (1.6) where κ ∈ (0, 1), 1− κ eκ = 1 e2 , κ̃ ∈ (1,+∞), κe−κ = κ̃e−κ̃, κ ≈ 0.7215, κ̃ ≈ 1.3423 . (1.7) Obviously, (1.6) can be obtained from (1.4) and (1.5). Unfortunately, the techni- cal conditions (1.3)–(1.5) on the coefficient functions are limited in the whole real axis, which are clearly not consistent with the biological background in the consid- ered systems. Obviously, according to the biological interpretation of Nicholson’s blowflies models in [22, 17], it is necessary to relax the above technical conditions as follows: M lim sup t→+∞ γij(t) ≤ κ̃, for all i ∈ Q, j ∈ I, (1.8) sup t∈[t0,+∞) {−aii(t) + n∑ j=1,j 6=i aij(t) + 1 eM m∑ j=1 βij(t) γij(t) } < 0, i ∈ Q, (1.9) lim inf t→+∞ {−aii(t) + n∑ j=1,j 6=i aij(t) + m∑ j=1 βij(t) γij(t) e−κ} > 0, i ∈ Q. (1.10) Motivated by the above discussions, in this paper, we apply a novel proof to establish the existence and global attractivity of positive asymptotically almost periodic solutions for system (1.2) with weaker conditions (1.8)–(1.10). This article is organized as follows: In Section 2, some necessary definitions, lem- mas, assumptions are presented. In Section 3, the existence and global attractivity of positive asymptotically almost periodic solutions are demonstrated by virtue of some differential inequalities and analytic techniques. To verify our theoretical re- sults, a numerical experiment is carried out in Section 4. Conclusions are drawn in Section 5. EJDE-2020/61 NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE 3 2. Preliminary results Throughout this paper, we assume that there exists t̃0 > t0 such that σi = max j∈I sup t∈R τij(t) > 0, inf t≥t̃0 γij(t) ≥ 1, i ∈ Q, j ∈ I, (2.1) which is a weaker condition than that inft∈R γij(t) ≥ 1 adopted in [18, 8, 7]. For x = (x1, . . . , xn) ∈ Rn, define |x| = (|x1|, . . . , |xn|) and ‖x‖ = maxi∈Q |xi|. Let R+ = [0,+∞), and C+ = ∏n i=1 C([−σi, 0],R+). For J, J1, J2 ⊆ R, denote W0(R+, J) = {ν : ν ∈ C(R+, J), lim t→+∞ ν(t) = 0}, and let BC(J1, J2) be the set of bounded and continuous functions from J1 to J2. Definition 2.1 ([13, 10]). A subset P of R is said to be relatively dense in R if there exists a number l > 0 such that [t, t+ l]∩P 6= ∅ (t ∈ R). u ∈ BC(R, J) is said to be almost periodic on R if, for any ε > 0, the set T (u, ε) = {δ : |u(t+δ)−u(t)| < ε, ∀t ∈ R} is relatively dense. Definition 2.2 ([23, 4]). u ∈ C(R+, J) is said to be asymptotically almost periodic if there exist an almost periodic function h and a continuous function g ∈W0(R+, J) such that u = h+ g. For J ⊆ R, we denote the set of the almost periodic functions from R to J by AP (R, J). The collection of the asymptotically almost periodic functions will be denoted by AAP (R, J). In addition, AP (R, J) is a proper subspace of AAP (R, J) [23, 4]. The decomposition u = h+ g given in Definition 2.2 is unique; see [23, Remark 5.16]. Hereafter, we assume that aii, γij ∈ AAP (R, (0,+∞)), aij(i 6= j), βij , τij ∈ AAP (R,R+) and aij = ahij + agij , βij = βhij + βgj , γij = γhj + γgj , τij = τhij + τgij , where ahii, γ h ij ∈ AP (R, (0,+∞)), ahij(i 6= j), βhij , τ h ij ∈ AP (R,R+), agij , β g ij , γ g ij , τ g ij ∈ W0(R+,R+), and i ∈ Q, j ∈ I. To proceed further, we need to introduce a nonlinear almost periodic differential system: x′i(t) = −ahii(t)xi(t) + n∑ j=1,j 6=i ahij(t)xj(t) + m∑ j=1 βhij(t)xi(t− τhij(t))e−γ h ij(t)xi(t−τh ij(t)), i ∈ Q, (2.2) We will consider the admissible initial conditions xi(t0 + θ) = ϕi(θ), θ ∈ [−σi, 0], ϕ = (ϕ1, . . . , ϕn) ∈ C+, ϕi(0) > 0, (2.3) for i ∈ Q. Lemma 2.3. Let x(t; t0, ϕ) be a solution of the initial value problem (2.2) and (2.3). Suppose that there exists a positive constant M > κ such that (1.8), (1.10) and sup t∈[t0,+∞) { − ahii(t) + n∑ j=1,j 6=i ahij(t) + 1 eM m∑ j=1 βhij(t) γhij(t) } < 0, i ∈ Q. (2.4) 4 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 hold. Then, x(t) = x(t; t0, ϕ) exists on [t0,+∞), and there is tϕ ∈ [t0,+∞) such that κ < xi(t) < M for all t ∈ [tϕ,+∞), i ∈ Q. (2.5) Proof. First, we claim that xi(t) > 0 for all t ∈ [t0, η(ϕ)), i ∈ Q, (2.6) where [t0, η(ϕ)) is the maximal right existence interval of x(t). Otherwise, we can find i0 ∈ Q and t̄i0 ∈ (t0, η(ϕ)) that satisfy xi0(t̄i0) = 0, xj(t) > 0 for all t ∈ [t0, t̄i0), j ∈ Q. From the facts that xi0(t0) = ϕi0(0) > 0 and x′i0(t) ≥ −ahi0i0(t)xi0(t) + m∑ j=1 βhi0j(t)xi0(t− τhi0j(t))e −γh i0j(t)xi0 (t−τh i0j(t)), for t ∈ [t0, t̄i0), we obtain 0 = xi0(t̄i0) ≥ e− ∫ t̄i0 t0 ahi0i0 (u)duxi0(t0) + e− ∫ t̄i0 t0 ahi0i0 (u)du ∫ t̄i0 t0 e ∫ s t0 ahi0i0 (v)dv × m∑ j=1 βhi0j(s)xi0(s− τhi0j(s))e −γh i0j(s)xi0 (s−τh i0j(s))ds > 0, which is a contradiction. Now, we demonstrate that x(t) is bounded on [t0, η(ϕ)). For t ∈ [t0 − σi, η(ϕ)) and i ∈ Q, we define Mi(t) = max{ξ : ξ ≤ t, xi(ξ) = max t0−σi≤s≤t xi(s)}. Suppose that x(t) is unbounded on [t0, η(ϕ)). Then, we can choose i∗ ∈ Q and a strictly monotone increasing sequence {ζn}+∞n=1 such that xi∗(Mi∗(ζn)) = max j∈Q {xj(Mj(ζn))}, lim n→+∞ xi∗(Mi∗(ζn)) = +∞, lim n→+∞ ζn = η(ϕ), and lim n→+∞ Mi∗(ζn) = η(ϕ). (2.7) It follows that there exists n∗ > 0 satisfying Mi∗(ζn) > t0, xi∗(Mi∗(ζn)) > M for all n > n∗. From supu≥0 ue −u = 1 e , (2.2), (2.4) and (2.7) it follows that, for all n > n∗, 0 ≤ x′i∗(Mi∗(ζn)) = −ahi∗i∗(Mi∗(ζn))xi∗(Mi∗(ζn)) + n∑ j=1,j 6=i ahi∗j(Mi∗(ζn))xj(Mi∗(ζn)) + m∑ j=1 βhi∗j(Mi∗(ζn)) γhi∗j(Mi∗(ζn)) γhi∗j(Mi∗(ζn))xi∗(Mi∗(ζn)− τhi∗j(Mi∗(ζn))) EJDE-2020/61 NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE 5 × e−γ h i∗j(Mi∗ (ζn))xi∗ (Mi∗ (ζn)−τh i∗j(Mi∗ (ζn))) ≤ [−ahi∗i∗(Mi∗(ζn)) + n∑ j=1,j 6=i ahi∗j(Mi∗(ζn))]xi∗(Mi∗(ζn)) + m∑ j=1 βhi∗j(Mi∗(ζn)) γhi∗j(Mi∗(ζn)) 1 e ≤ − 1 eM m∑ j=1 βhi∗j(Mi∗(ζn)) γhi∗j(Mi∗(ζn)) xi∗(Mi∗(ζn)) + m∑ j=1 βhi∗j(Mi∗(ζn)) γhi∗j(Mi∗(ζn)) 1 e < 0, which is absurd and suggests that x(t) is bounded on [t0, η(ϕ)). By [6, Theorem 2.3.1], we easily show η(ϕ) = +∞. � Hereafter, we assume that (2.5) is true. Designate il, iL ∈ Q such that l = lim inf t→+∞ xil(t) = min i∈Q lim inf t→+∞ xi(t), L = lim sup t→+∞ xiL(t) = max i∈Q lim sup t→+∞ xi(t). By the fluctuation lemma (see [15, Lemma A.1]), we can select a sequence {t∗k} +∞ k=1 satisfying lim k→+∞ t∗k = +∞, lim k→+∞ xiL(t∗k) = L = lim sup t→+∞ xiL(t), lim k→+∞ x′iL(t∗k) = 0. (2.8) From the almost periodicity of (2.2), we can select a subsequence of {k}k≥1, still de- noted by {k}k≥1, such that limk→+∞ ahiLj(t ∗ k), limk→+∞ bhiLj(t ∗ k), limk→+∞ βhiLq(t ∗ k), limk→+∞ γhiLq(t ∗ k), limk→+∞ xj(t ∗ k) and limk→+∞ xiL(t∗k − τhiLq(t ∗ k)) exist for all j ∈ Q and all q ∈ I. Furthermore, by taking limits, we have from (2.2) and (2.8) that 0 = lim k→+∞ x′iL(t∗k) = − lim k→+∞ ahiLiL(t∗k)L+ n∑ j=1,j 6=iL lim k→+∞ ahiLj(t ∗ k) lim k→+∞ xj(t ∗ k) + m∑ j=1 lim k→+∞ βhiLj(t ∗ k) γh iLj (t∗k) lim k→+∞ γhiLj(t ∗ k)xiL(t∗k − τhiLj(t ∗ k)) × e− limk→+∞ γh iLj (t∗k) limk→+∞ xiL (t∗k−τ h iLj (t∗k)) ≤ − lim k→+∞ ahiLiL(t∗k)L+ n∑ j=1,j 6=iL lim k→+∞ ahiLj(t ∗ k)L+ m∑ j=1 lim k→+∞ βhiLj(t ∗ k) γh iLj (t∗k) 1 e = lim k→+∞ L[−ahiLiL(t∗k) + n∑ j=1,j 6=iL ahiLj(t ∗ k) + m∑ j=1 βhiLj(t ∗ k) γh iLj (t∗k) 1 eL ] ≤ sup t∈[t0,+∞) { − ahiLiL(t)L+ n∑ j=1,j 6=iL ahiLj(t)L+ m∑ j=1 βhiLj(t) γh iLj (t) 1 e } , which, together with (2.4), entails that sup t∈[t0, +∞) { − ahiLiL(t)M + n∑ j=1,j 6=iL ahiLj(t)M + m∑ j=1 βhiLj(t) γh iLj (t) 1 e } 6 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 = M sup t∈[t0,+∞) {−ahiLiL(t) + n∑ j=1,j 6=iL ahiLj(t) + m∑ j=1 βhiLj(t) γh iLj (t) 1 Me } < 0 ≤ sup t∈[t0,+∞) { − ahiLiL(t)L+ n∑ j=1,j 6=iL ahiLj(t)L+ m∑ j=1 βhiLj(t) γh iLj (t) 1 e } , and then L < M . Consequently, there exists t∗0 ≥ t0 such that xi(t) < M, for all t ≥ t∗0, i ∈ Q. Next, we show that l > 0. By way of contradiction, we assume that lim inf t→+∞ xil(t) = min i∈Q lim inf t→+∞ xi(t) = 0. (2.9) Let ωi(t) = max{ξ : ξ ≤ t, xi(ξ) = mint0≤s≤t xi(s)} for each t ≥ t0. From (2.9), we can choose i∗∗ ∈ Q and a strictly monotone increasing sequence {ξn}+∞n=1 such that xi∗∗(ωi∗∗(ξn)) = min j∈Q {xj(ωj(ξn))}, lim n→+∞ xi∗∗(ωi∗∗(ξn)) = 0, lim n→+∞ ξn = +∞, and then lim n→+∞ ωi∗∗(ξn) = +∞. According to (1.8), (2.1), (2) and L < M , one can find there exists n∗∗ > 0 such that, for n > n∗∗ and j ∈ I, ωi∗∗(ξn) > t∗0 + σi∗∗ , xi∗∗(ωi∗∗(ξn)) < κ, quadγhi∗∗j(ωi∗∗(ξn)) ≥ 1, (2.10) xi∗∗(ωi∗∗(ξn)) ≤ γhi∗∗j(ωi∗∗(ξn))xi∗∗(ωi∗∗(ξn)− τhi∗∗j(ωi∗∗(ξn))) ≤ κ̃. (2.11) It follows from (1.2), (2.10) and (2.11) that 0 ≥ x′i∗∗(ωi∗∗(ξn)) = −ahi∗∗i∗∗(ωi∗∗(ξn))xi∗∗(ωi∗∗(ξn)) + n∑ j=1,j 6=i∗∗ ahi∗∗j(ωi∗∗(ξn))xj(ωi∗∗(ξn)) + m∑ j=1 βhi∗∗j(ωi∗∗(ξn)) γhi∗∗j(ωi∗∗(ξn)) γhi∗∗j(ωi∗∗(ξn))xi∗∗(ωi∗∗(ξn)− τhi∗∗j(ωi∗∗(ξn))) × e−γ h i∗∗j(ωi∗∗ (ξn))xi∗∗ (ωi∗∗ (ξn)−τh i∗∗j(ωi∗∗ (ξn))) ≥ −ahi∗∗i∗∗(ωi∗∗(ξn))xi∗∗(ωi∗∗(ξn)) + n∑ j=1,j 6=i∗∗ ahi∗∗j(ωi∗∗(ξn))xj(ωi∗∗(ξn)) + m∑ j=1 βhi∗∗j(ωi∗∗(ξn)) γhi∗∗j(ωi∗∗(ξn)) xi∗∗(ωi∗∗(ξn))e−xi∗∗ (ωi∗∗ (ξn)), n > n∗∗, and ahi∗∗i∗∗(ωi∗∗(ξn)) ≥ n∑ j=1,j 6=i∗∗ ahi∗∗j(ωi∗∗(ξn)) xj(ωi∗∗(ξn)) xi∗∗(ωi∗∗(ξn)) + m∑ j=1 βhi∗∗j(ωi∗∗(ξn)) γhi∗∗j(ωi∗∗(ξn)) e−xi∗∗ (ωi∗∗ (ξn)) ≥ n∑ j=1,j 6=i∗∗ ahi∗∗j(ωi∗∗(ξn)) + m∑ j=1 βhi∗∗j(ωi∗∗(ξn)) γhi∗∗j(ωi∗∗(ξn)) e−xi∗∗ (ωi∗∗ (ξn)), n > n∗∗, EJDE-2020/61 NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE 7 This inequality and (1.10), yield 0 ≥ lim inf n→+∞ {−ahi∗∗i∗∗(ωi∗∗(ξn)) + n∑ j=1,j 6=i∗∗ ahi∗∗j(ωi∗∗(ξn)) + m∑ j=1 βhi∗∗j(ωi∗∗(ξn)) γhi∗∗j(ωi∗∗(ξn)) e−xi∗∗ (ωi∗∗ (ξn)) ≥ lim inf t→+∞ {−ahi∗∗i∗∗(t) + n∑ j=1,j 6=i∗∗ ahi∗∗j(t) + m∑ j=1 βhi∗∗j(t) γhi∗∗j(t) } ≥ lim inf t→+∞ {−ahi∗∗i∗∗(t) + n∑ j=1,j 6=i∗∗ ahi∗∗j(t) + m∑ j=1 βhi∗∗j(t) γhi∗∗j(t) e−κ} = lim inf t→+∞ {−ai∗∗i∗∗(t) + n∑ j=1,j 6=i∗∗ ai∗∗j(t) + m∑ j=1 βi∗∗j(t) γi∗∗j(t) e−κ} > 0. This is a clear contradiction and proves that l > 0. Finally, we show that l > κ. Again from the fluctuation lemma [15, Lemma A.1] and the almost periodicity of (2.2), we can pick a sequence {t∗∗k } +∞ k=1 such that lim k→+∞ t∗∗k = +∞, lim k→+∞ xil(t ∗∗ k ) = l = lim inf t→+∞ xil(t), lim k→+∞ x′il(t ∗∗ k ) = 0 (2.12) and limk→+∞ ahilj(t ∗∗ k ), limk→+∞ bhilj(t ∗∗ k ), limk→+∞ βhilq(t ∗∗ k ), limk→+∞ γhilq(t ∗∗ k ), limk→+∞ xj(t ∗∗ k ), limk→+∞ xil(t ∗∗ k − τhilq(t ∗∗ k )) exist for all j ∈ Q and all q ∈ I. Furthermore, l ≤ lim k→+∞ xj(t ∗∗ k ) ≤ L < M, l ≤ lim k→+∞ γhilq(t ∗∗ k ) lim k→+∞ xil(t ∗∗ k − τhilq(t ∗∗ k )) ≤ κ̃, ∀j ∈ Q, q ∈ I. (2.13) By way of contradiction, we assume that 0 < l ≤ κ. With the help of (1.10), (2.12) and (2.13), we have 0 = lim k→+∞ x′il(t ∗∗ k ) ≥ − lim k→+∞ ahilil(t ∗∗ k )l + n∑ j=1,j 6=il lim k→+∞ ahilj(t ∗∗ k )l + m∑ j=1 limk→+∞ βhilj(t ∗∗ k ) limk→+∞ γh ilj (t∗∗k ) lim k→+∞ γhilj(t ∗∗ k )xil(t ∗∗ k − τhilj(t ∗∗ k )) × e− limk→+∞ γh ilj (t∗∗k )x il (t∗∗k −τ h ilj (t∗∗k )) ≥ − lim k→+∞ ahilil(t ∗∗ k )l + n∑ j=1,j 6=il lim k→+∞ ahilj(t ∗∗ k )l + m∑ j=1 limk→+∞ βhilj(t ∗∗ k ) limk→+∞ γh ilj (t∗∗k ) le−l ≥ l lim inf t→+∞ {−ahilil(t) + n∑ j=1,j 6=il ahilj(t) + m∑ j=1 βhilj(t) γh ilj (t) e−l} ≥ l lim inf t→+∞ {−ahilil(t) + n∑ j=1,j 6=il ahilj(t) + m∑ j=1 βhilj(t) γh ilj (t) e−κ} 8 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 = l lim inf t→+∞ {−ailil(t) + n∑ j=1,j 6=il ailj(t) + m∑ j=1 βilj(t) γilj(t) e−κ} > 0, which results in a contradiction. This proves that l > κ. Hence, there exits tϕ > t0 such that κ < xi(t; t0, ϕ) < M for all t ≥ tϕ, i ∈ Q. The proof is now complete. By using a similar argument as in Lemma 2.3, we can show the following result. Lemma 2.4. Let x(t) = x(t; t0, ϕ) be a solution of the initial value problem (1.2) and (2.3). Suppose that there exists a positive constant M > κ such that (1.8), (1.9) and (1.10) hold. Then, x(t) exists on [t0,+∞), κ < min i∈Q lim inf t→+∞ xi(t) ≤ max i∈Q lim sup t→+∞ xi(t) < M, and there is t∗ϕ ∈ [t0,+∞) such that κ < xi(t) < M for all t ∈ [t∗ϕ,+∞), i ∈ Q. (2.14) Lemma 2.5. Suppose that there exists a positive constant M > κ such that (1.8), (1.10) and (2.4) hold. Moreover, assume that x(t) = x(t; t0, ϕ) is a solution of equation (2.2) and (2.3). Then, for any ε > 0, we can choose a relatively dense subset Pε of R with the property that, for each δ ∈ Pε, there exists T = T (δ) > 0 satisfying ‖x(t+ δ)− x(t)‖ < ε 2 , for all t > T. Proof. With the help of Lemma 2.3, (1.8), (2.1) and (2.4), we can choose positive constants T1 > max{0, tϕ} and ζ such that for all t ≥ T1 and i ∈ Q, γhij(t)xi(t− τhij(t)) > κ, 1 ≤ κ̃ Mγhij(t) , 1 ≤ κ̃ Mγij(t) , κ̃ e < 1, and − ahii(t) + n∑ j=1,j 6=i ahij(t) + 1 e2 m∑ j=1 βhij(t) ≤ −ahii(t) + n∑ j=1,j 6=i ahij(t) + 1 eM m∑ j=1 βhij(t) γhij(t) κ̃ e ≤ −ahii(t) + n∑ j=1,j 6=i ahij(t) + 1 eM m∑ j=1 βhij(t) γhij(t) < −ζ. Then there exist two constants η > 0 and λ ∈ (0, 1] such that, for i ∈ Q, sup t∈[T1, +∞) { − [ahii(t)− λ] + n∑ j=1,j 6=i ahij(t) + m∑ j=1 βhij(t) 1 e2 eλσi } < −η. (2.15) Define xi(t) ≡ xi(t0 − σi), for all t ∈ (−∞, t0 − σi], i ∈ Q, (2.16) EJDE-2020/61 NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE 9 and Ai(δ, t) = −[ahii(t+ δ)− ahii(t)]xi(t+ δ) + n∑ j=1,j 6=i [ahij(t+ δ)− ahij(t)]xj(t+ δ) + m∑ j=1 [βhij(t+ δ)− βhij(t)]xi(t+ δ − τhij(t+ δ))e−γ h ij(t+δ)xi(t+δ−τh ij(t+δ)) + m∑ j=1 βhij(t)[xi(t+ δ − τhij(t+ δ))e−γ h ij(t+δ)xi(t+δ−τh ij(t+δ)) − xi(t− τhij(t) + δ)e−γ h ij(t+δ)xi(t−τh ij(t)+δ)] + m∑ j=1 βhij(t)[xi(t− τhij(t) + δ)e−γ h ij(t+δ)xi(t−τh ij(t)+δ) − xi(t− τhij(t) + δ)e−γ h ij(t)xi(t−τh ij(t)+δ)], for all t ∈ R, i ∈ Q. From Lemma 2.3, one can see that x(t) is bounded and the right-hand side of (2.2) is also bounded. It follows from (2.16) that x(t) is uniformly continuous on R. Therefore, for any ε > 0, we can choose a sufficiently small constant ε∗ > 0 such that |ahij(t)− ahij(t+ δ)| < ε∗, |βhij(t)− βhij(t+ δ)| < ε∗, |γhij(t)− γhij(t+ δ)| < ε∗, |τhij(t)− τhij(t+ δ)| < ε∗ . It follows that |Ai(δ, t)| < 1 2 ηε, (2.17) where t ∈ R, i ∈ Q and j ∈ I. Furthermore, for ε∗ > 0, from the uniformly almost periodic family theory in [4, p. 19, Corollary 2.3], one can choose a relatively dense subset Pε∗ of R such that |ahij(t)− ahij(t+ δ)| < ε∗, |βhij(t)− βhij(t+ δ)| < ε∗, |γhij(t)− γhij(t+ δ)| < ε∗, |τhij(t)− τhij(t+ δ)| < ε∗, (2.18) δ ∈ Pε∗ , t ∈ R, i ∈ Q, and j ∈ I. Let Pε = Pε∗ . Then for any δ ∈ Pε, from (2.17) and (2.18), we have |Ai(δ, t)| < 1 2 ηε, for all t ∈ R, i ∈ Q. (2.19) Let Λ0 ≥ max { |t0|+ T1 + max i∈Q σi, |t0|+ T1 + max i∈Q σi − δ } . For t ∈ R, denote u(t) = (u1(t), u2(t), . . . , un(t)), ui(t) = xi(t+ δ)− xi(t), U(t) = (U1(t), U2(t), . . . , Un(t)), Ui(t) = eλtui(t), where i ∈ Q. Let it be an index such that |Uit(t)| = ‖U(t)‖. (2.20) 10 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 Then, for all t ≥ Λ0, we have u′i(t) = −ahii(t)[xi(t+ δ)− xi(t)] + n∑ j=1,j 6=i ahij(t)[xj(t+ δ)− xj(t)] + m∑ j=1 βhij(t)[xi(t− τhij(t) + δ)e−γ h ij(t)xi(t−τh ij(t)+δ) − xi(t− τhij(t))e−γ h ij(t)xi(t−τh ij(t))] +Ai(δ, t). From the above equality, (2.4) and |αe−α − βe−β | ≤ 1 e2 |α− β| where α, β ∈ [κ,+∞), (2.21) we obtain D−(|Uis(s)|)|s=t ≤ λeλt|uit(t)|+ eλt { − ahitit(t)[xit(t+ δ)− xit(t)] sgn(xit(t+ δ)− xit(t)) + n∑ j=1,j 6=it ahitj(t)|xj(t+ δ)− xj(t)|+ m∑ j=1 βhitj(t) ∣∣xit(t− τhitj(t) + δ) × e−γ h itj (t)xit (t−τh itj (t)+δ) − xit(t− τhitj(t))e −γh itj (t)xit (t−τh itj (t)) ∣∣ + |Ait(δ, t)| } = λeλt|uit(t)|+ eλt { − ahitit(t)[xit(t+ δ)− xit(t)] sgn(xit(t+ δ)− xit(t)) + n∑ j=1,j 6=it ahitj(t)|xj(t+ δ)− xj(t)|+ m∑ j=1 βhitj(t) γhitj(t) × |γhitj(t)xit(t− τ h itj(t) + δ)e−γ h itj (t)xit (t−τh itj (t)+δ) − γhitj(t)xit(t− τ h itj(t))e −γh itj (t)xit (t−τh itj (t))|+ |Ait(δ, t)| } ≤ λeλt|uit(t)|+ eλt { − ahitit(t)|uit(t)|+ n∑ j=1,j 6=it ahitj(t)|uj(t)| + m∑ j=1 βhitj(t) 1 e2 |uit(t− τhitj(t))|+ |Ait(δ, t)| } = −[ahitit(t)− λ]|Uit(t)|+ n∑ j=1,j 6=it ahitj(t)|Uj(t)| + m∑ j=1 βhitj(t) 1 e2 eλτ h itj (t)|Uit(t− τhitj(t))|+ eλt|Ait(δ, t)| for all t ≥ Λ0. (2.22) Let E(t) = sup −∞ eλt‖u(t)‖ for all t ≥ Λ0, we assert that E(t) ≡ ‖U(Λ0)‖ for all t ≥ Λ0. (2.23) In the opposite case, one can pick Λ1 > Λ0 such that E(Λ1) > E(Λ0). Because eλt‖u(t)‖ ≤ E(Λ0) for all t ≤ Λ0, there must exist β∗ ∈ (Λ0,Λ1) such that eλβ ∗ ‖u(β∗)‖ = E(Λ1) ≥ E(β∗), which contradicts that E(β∗) > eλβ ∗‖u(β∗)‖ and proves the above assertion. Then, we can select Λ2 > Λ0 satisfying ‖u(t)‖ ≤ e−λtE(t) = e−λtE(Λ0) < ε 2 for all t ≥ Λ2. (2.24) Step 2. If there exists ς ≥ Λ0 such that E(ς) = eλς‖u(ς)‖, we can have from (2.22) and the definition of E(t) that 0 ≤ D−(|Uis(s)|)|s=ς ≤ −[ahiς iς (t)− λ]|Uiς (ς)|+ n∑ j=1,j 6=iς ahiςj(t)|Uj(ς)| + m∑ j=1 βhiςj(ς) 1 e2 eλτ h iςj(ς)|Uiς (ς − τhiςj(ς))|+ eλς |Aiς (δ, ς)| ≤ { − [ahiς iς (t)− λ] + n∑ j=1,j 6=iς ahiςj(t) + m∑ j=1 βhiςj(ς) 1 e2 eλτ h iςj(ς) } E(ς) + 1 2 ηεeλς < −ηE(ς) + 1 2 ηεeλς , which leads to eλς‖u(ς)‖ = E(ς) < ε 2 eλς , and ‖u(ς)‖ < ε 2 . (2.25) For any t > ς satisfying E(t) = eλt‖u(t)‖, by the same method as that in the derivation of (2.25), we can show eλt‖u(t)‖ < ε 2 eλt, and ‖u(t)‖ < ε 2 . (2.26) Furthermore, if E(t) > eλt‖u(t)‖ and t > ς, one can pick Λ3 ∈ [ς, t) such that E(Λ3) = eλΛ3 |u(Λ3)‖, E(s) > eλs‖u(s)‖ for all s ∈ (Λ3, t], which, together with (2.25) and (2.26), suggest that ‖u(Λ3)‖ < ε 2 . (2.27) With a similar reasoning as that in the proof of Step one, we can entail that E(s) ≡ E(Λ3) is a constant for all s ∈ (Λ3, t], which, together with (2.27), follows that ‖u(t)‖ < e−λtE(t) = e−λtE(Λ3) = ‖u(Λ3)‖e−λ(t−Λ3) < ε 2 . 12 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 Finally, the above discussion infers that there exists Λ̂ > max{ς,Λ0, Lambda2} satisfying ‖u(t)‖ ≤ ε 2 < ε for all t > Λ̂, which completes the proof of Lemma 2.5. � 3. Main results The main result in this article reads as follows. Theorem 3.1. Assume that there exists a positive constant M > κ such that (1.7), (1.8), (1.9), (1.10) and (2.4) hold. Then system (2.2) has exactly one positive almost periodic solution x∗(t), and every solution of (1.2) with initial condition (2.3) is asymptotically almost periodic on R+, and converges to x∗(t) as t→ +∞. Proof. Let v(t) be a solution of system (2.2) with initial function ϕ satisfying (2.3), and vi(t) ≡ vi(t0 − σi), for all t ∈ (−∞, t0 − σi], i ∈ Q. Also we define Bi(q, t) = −[ahii(t+ tq)− ahii(t)]vi(t+ tq) + n∑ j=1,j 6=i [ahij(t+ tq)− ahij(t)]vj(t+ tq) + m∑ j=1 [βhij(t+ tq)− βhij(t)]vi(t+ tq − τhij(t+ tq))e −γh ij(t+tq)vi(t+tq−τh ij(t+tq)) + m∑ j=1 βhij(t)[vi(t+ tq − τhij(t+ tq))e −γh ij(t+tq)vi(t+tq−τh ij(t+tq)) − vi(t− τhij(t) + tq)e −γh ij(t+tq)vi(t−τh ij(t)+tq)] + m∑ j=1 βhij(t)[vi(t− τhij(t) + tq)e −γh ij(t+tq)vi(t−τh ij(t)+tq) − vi(t− τhij(t) + tq)e −γh ij(t)vi(t−τh ij(t)+tq)], for all t ∈ R, i ∈ Q. where {tq}q≥1 ⊆ R is a sequence. Then v′i(t+ tq) = −ahii(t)vi(t+ tq) + n∑ j=1,j 6=i ahij(t)vj(t+ tq) + m∑ j=1 βhij(t)vi(t− τhij(t) + tq)e −γh ij(t)vi(t−τh ij(t)+tq) +Bi(q, t), (3.1) for all t+ tq ≥ t0, i ∈ Q. By using a similar proof as in Lemma 2.5, we can choose {tq}q≥1 such that |Bi(q, t)| < 1 q for all i, q, t. (3.2) By Arzala-Ascoli Lemma and the fact that the function sequence {v(t + tq)}q≥1 is uniformly bounded and equiuniformly continuous, we can choose a subsequence {tqj}j≥1 of {tq}q≥1, such that {v(t+ tqj )}j≥1 converges uniformly to a continuous function x∗(t) = (x∗1(t), x∗2(t), . . . , x∗n(t)) on any compact set of R (for convenience, we denote this subsequence by {v(t+ tq)}q≥1). EJDE-2020/61 NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE 13 Then, from Lemma 2.3, we have κ < min i∈Q lim inf t→+∞ vi(t) ≤ x∗i (t) ≤ max i∈Q lim sup t→+∞ vi(t) < M for all t ∈ R, i ∈ Q, (3.3) and −ahii(t)vi(t+ tq)⇒ −ahii(t)x∗i (t), i ∈ Q, n∑ j=1,j 6=i ahij(t)vj(t+ tq)⇒ n∑ j=1,j 6=i ahij(t)x ∗ j (t), i ∈ Q, m∑ j=1 βhij(t)vi(t− τhij(t) + tq)e −γh ij(t)vi(t−τh ij(t)+tq) ⇒ m∑ j=1 βhij(t)x ∗ i (t− τhij(t))e−γ h ij(t)x∗(t−τh ij(t)), i ∈ Q, (3.4) as q → +∞, on any compact set of R, where ⇒ denotes uniformly converge. Thus, for i ∈ Q, (3.1), (3.2) and (3.4) produce that {v′i(t+ tq)}q≥1 converges uniformly to −ahii(t)x∗i (t) + n∑ j=1,j 6=i ahij(t)x ∗ j (t) + m∑ j=1 βhij(t)x ∗ i (t− τhij(t))e−γ h ij(t)x∗(t−τh ij(t)) on any compact subset of R. According to the properties of the uniform convergence function sequence, we obtain that x∗(t) is a solution of (2.2) and (x∗i (t)) ′ = −ahii(t)x∗i (t) + n∑ j=1,j 6=i ahij(t)x ∗ j (t) + m∑ j=1 βhij(t)x ∗ i (t− τhij(t))e−γ h ij(t)x∗(t−τh ij(t)), for all t ∈ R, i ∈ Q. Now, from Lemma 2.5, for any ε > 0, we can choose a relatively dense subset Pε of R with the property that, for each δ ∈ Pε, there exists T = T (δ) > 0 satisfying ‖v(s+ tq + δ)− v(s+ tq)‖ < ε 2 , for all s+ tq > T, lim q→+∞ ‖v(s+ tq + τ)− v(s+ tq)‖ = ‖x∗(s+ δ)− x∗(s)‖ ≤ ε 2 < ε for all s ∈ R, which implies that x∗(t) is a positive almost periodic solution of (2.2). Next, we show that all solutions of (1.2) converge to x∗(t) as t → +∞. Let x(t) be an arbitrary solution of system (1.2) with initial value ϕ satisfies (2.3). Define y(t) = x(t) − x∗(t), add the definition of xi(t) with xi(t) ≡ xi(t0 − σi) for all t ∈ (−∞, t0 − σi], and let Fi(t) = −[(ahii(t) + agii(t))xi(t)− a h ii(t)xi(t)] + n∑ j=1,j 6=i [(ahij(t) + agij(t))xj(t)− a h ij(t)xj(t)] + m∑ j=1 [(βhij(t) + βgij(t))xi(t− (τhij(t) + τgij(t))) × e−(γh ij(t)+γg ij(t))xi(t−(τh ij(t)+τg ij(t))) − βhij(t)xi(t− τhij(t))e−γ h ij(t)xi(t−τh ij(t))]. 14 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 Then y′i(t) = −ahii(t)yi(t) + n∑ j=1,j 6=i ahij(t)yj(t) + m∑ j=1 βhij(t)[xi(t− τhij(t))e−γ h ij(t)xi(t−τh ij(t)) − x∗i (t− τhij(t))e−γ h ij(t)x∗i (t−τh ij(t))] + Fi(t), for all t ≥ t0, i ∈ Q. (3.5) For any ε > 0, in view of the global existence and uniform continuity of x and the fact that agij , β g ij , γ g ij , τ g ij ∈ W0(R+,R+), we can choose a constant T ∗∗ϕ > max{T1, t ∗ ϕ} such that |Fi(t)| < η ε 2 , for all t > T ∗∗ϕ . (3.6) Set G(t) = sup −∞ T ∗∗ϕ such that κ < xi(t), x ∗ i (t), γ h ij(t)xi(t− τhij(t)) ≤ κ̃ for all t > Tϕ,x∗ , i ∈ Q. (3.7) In view of (2.21), (3.5) and (3.7), we have D−(eλs|yis(s)|)|s=t ≤ −[ahitit(t)− λ]eλt|yit(t)|+ n∑ j=1,j 6=it ahitj(t)e λt|yj(t)|+ m∑ j=1 βhitj(t) × 1 e2 eλτ h itj (t)eλ(t−τh itj (t))|yit(t− τhitj(t))|+ eλt|Fit(t)| (3.8) for all t ≥ Tϕ,x∗ and i ∈ Q. Then, from (2.15), (3.6) and (3.8), by employing the argument of Lemma 2.5, we know that there is a constant T̃ ≥ Tϕ,x∗ such that ‖y(t)‖ < ε 2 for all t ≥ T̃ , which yields lim t→+∞ x(t) = x∗(t), and x(t) ∈ AAP (R,Rn). It follows from the uniqueness of the limit function that (2.2) has exactly one positive almost periodic solution x∗(t). The proof is complete. � Remark 3.2. Under the conditions in Lemma 2.5, according to Lemma 2.3 and Lemma 2.5, by applying a similar way as in [19, Theorem 3.2], one can show that the solution x(t; t0, ϕ) of (2.2) converges exponentially to x∗(t) as t→ +∞. Since all conditions in (1.7)–(1.10) are weaker than those in (1.3)–(1.5), one can easily see that all results on almost periodicity of (2.2) in [12, 19, 20] are special cases of Theorem 3.1 in this article. EJDE-2020/61 NICHOLSON-TYPE SYSTEM INVOLVING PATCH STRUCTURE 15 0 20 40 60 80 100 120 140 160 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 t x i( t) ,i = 1 ,2 x1(t) x2(t) Figure 1. Numerical solutions of (4.1) for different initial values: (0.5,0,6), (1,1.1), (0,2,0.1). 4. Numerical simulation We consider the delayed Nicholson-type system with patch structure x′1(t) = −(1.85 + 0.1| sin √ 2t|+ 100t 1 + t2 )x1(t) + (1.5 + 0.1| sin t|+ 1 800 + t2 )x2(t) + (e0.4 + 1 100 + t2 )x1(t− 2 cos2 √ 3t− 2)e −(1.01− 2 cos t 100+t2 )x1(t−2 cos2 √ 3t−2) + (0.4 + 1 100 + t2 )x1(t− 2 sin2 √ 3t− 2)e −(1.01− 2 cos t 100+t4 )x1(t−2 sin2 √ 3t−2) , x′2(t) = −(2.85 + 0.3| cos t|+ 100t 2 + t2 )x2(t) + (2.5 + 0.3 sin2 √ 2t+ 1 100 + t4 )x1(t) + (0.4 + 1 100 + t4 )x2(t− cos t− 5)e −(1.01− 2 cos t 100+t4 )x2(t−cos t−5) + (0.4 + 1 100 + t4 )x2(t− cos t− 15)e −(1.01− 2 cos t 100+t4 )x2(t−cos t−15) , (4.1) where t0 = 0. Take M = 1.301, we can find that (1.8), (1.9), (1.10), (2.1) and (2.4) are satisfied. By Theorem 3.1, all solutions of (4.1) are asymptotically almost periodic functions on R+, and converge to a same almost periodic function as t→ +∞. This fact can be presented in the Figure 1. In system (4.1), { − aii(t) + 2∑ j=1,j 6=i aij(t) + 1 eM 2∑ j=1 βij(t) γij(t) }∣∣∣ t=−1,M>κ > 10, i ∈ Q = {1, 2}, 16 C. HUANG, J. WANG, L. HUANG EJDE-2020/61 and { − aii(t) + n∑ j=1,j 6=i aij(t) + m∑ j=1 βij(t) γij(t) e−κ} ∣∣∣ t=1 < −12, i ∈ Q = {1, 2}, imply that (4.1) does not satisfy conditions (1.4) and (1.5) in [12, 19, 20]. In addition, the asymptotically almost periodic dynamics of delayed Nicholson-type system with patch structure was not studied in [16, 17, 21, 22]. Hence, it is not hard to see that all results in the references [12, 17, 19, 20, 22] and [16, 11, 21] cannot be applied to conclude that all solutions of (4.1) converge globally are almost periodic solutions. Conclusions. 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Kluwer Academic/Science Press, Beijing, 2003 Chuangxia Huang School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineer- ing, Changsha, Hunan 410114, China Email address: cxiahuang@amss.ac.cn Jiafu Wang School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineer- ing, Changsha, Hunan 410114, China Email address: jfwang@csust.edu.cn Lihong Huang (corresponding author) School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineer- ing, Changsha, Hunan 410114, China Email address: lhhuang@csust.edu.cn 1. Introduction 2. Preliminary results 3. Main results 4. Numerical simulation Conclusions Acknowledgments References