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Research article

Global stability of a novel nonlinear diffusion online game addiction model with unsustainable control

  • Received: 19 June 2022 Revised: 31 August 2022 Accepted: 12 September 2022 Published: 26 September 2022
  • MSC : 35K57, 35B40, 34K09, 92D25

  • In this paper, we build a novel nonlinear diffusion online game addiction model with unsustainable control. The existence and boundedness of a solution are investigated by a C0-semigroup and differential inclusion. Simultaneously, we study the global asymptotic stability of steady states of the model. Finally, a concrete example is theoretically analyzed and numerically simulated.

    Citation: Kaihong Zhao. Global stability of a novel nonlinear diffusion online game addiction model with unsustainable control[J]. AIMS Mathematics, 2022, 7(12): 20752-20766. doi: 10.3934/math.20221137

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  • In this paper, we build a novel nonlinear diffusion online game addiction model with unsustainable control. The existence and boundedness of a solution are investigated by a C0-semigroup and differential inclusion. Simultaneously, we study the global asymptotic stability of steady states of the model. Finally, a concrete example is theoretically analyzed and numerically simulated.



    Based on the experimental data which were observed and summarized by Nicholson [1], Gurney et al. [2] presented a classic biological dynamical system model

    N(t)=δN(t)+pN(tτ)eaN(tτ). (1.1)

    Here, N(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, δ is the per capita daily adult death rate, and τ is the generation time. The research on the Nicholson's blowfly model and its modifications has realized a remarkable progress in the past fifty years and an abundance of results on the existence of positive solutions, persistence, oscillation, stability, periodic solutions, almost periodic solutions, pseudo almost periodic solutions, etc. (see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]) have been obtained. Furthermore, Berezansky et al. [22] systematically collected and compared the results in the above-mentioned studies and put forward several open problems that have been partially answered in recent works [23,24,25,26,27,28,29,30], such as Nicholson's blowfly model with impulsive perturbation, harvesting term and nonlinear density-dependent mortality term. On the other hand, Liu [31] considered the following cooperative Nicholson's blowfly equation with a patch structure:

    xi(t)=nj=1aijxj(t)+βxi(tri)exi(tri)dixi(t),i=1,2,,n. (1.2)

    Also, Berezansky et al. [32] studied the following cooperative Nicholson-type delay differential system

    {x1(t)=a1x1(t)+b1x2(t)+c1x1(tτ)ex1(tτ),x2(t)=a2x2(t)+b2x1(t)+c2x2(tτ)ex2(tτ). (1.3)

    In this paper, we will discuss the following competitive and cooperative Nicholson's blowfly system

    {x1(t)=δ1(t)x1(t)+a1(t)x2(t)+nj=1c1j(t)x1(tτ1j(t))eb1j(t)x1(tτ1j(t))k1(t)x1(t)x2(t),x2(t)=δ2(t)x2(t)+a2(t)x1(t)+nj=1c2j(t)x2(tτ2j(t))eb2j(t)x2(tτ2j(t))k2(t)x1(t)x2(t) (1.4)

    where δi,ai,bij,cij,τij,ki:R1[0,+) are almost periodic functions i=1,2,j=1,2,,n. x1(t),x2(t) denote the sizes of the different populations at time t, cij denotes the maximum per capita daily egg production of xi, 1bij represents the size at which the population xi reproduces at its maximum rate, δi is the per capita daily adult death rate of xi, and τij denotes the generation time of xi. a1(t) represents the rate at which x2 contributes to x1 and a2(t) represents the rate at which x1 contributes to x2 at time t. ki(t) denotes the death rate of xi due to the competition between x1 and x2 at time t. The cooperative terms a1(t)x2(t) and a2(t)x1(t) and the competitive terms k1(t)x1(t)x2(t) and k1(t)x1(t)x2(t) reflect the degree at which they cooperate and compete with each other, respectively.

    Recently, there have been wide-ranging results obtained on competitive and cooperative systems added the literatures [33,34,35,36,37,38] due to its extensive applicability. However, to the best of our knowledge, few results are presented in literatures about the existence of positive almost periodic solutions for competitive and cooperative Nicholson's blowfly system. In the real world, since the competition is inevitable and there exists an almost periodically changing environment, it is worth studying the positive almost periodic solution for competitive and cooperative Nicholson's blowfly system. Based on the above idea, we shall consider the existence and exponential convergence of positive almost periodic solutions of system (1.4) which possesses obvious dynamics significance.

    For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function g defined on R1, let g+ and g be defined as follows:

    g=inftR1g(t),g+=suptR1g(t).

    It will be assumed that

    δi>0,bij>0,ri=max1jn{τ+ij}>0i=1,2,j=1,2,,n. (1.5)

    Let Rn(Rn+) be the set of all (nonnegative) real vectors; we will use x=(x1,x2,,xn)TRn to denote a column vector, in which the symbol (T) denotes the transpose of a vector. we let |x| denote the absolute-value vector given by |x|=(|x1|,|x2|,,|xn|)T and define ||x||=max1in|xi|. For a matrix A=(aij)n×n, AT, A1, |A| and ρ(A) denote the transpose, inverse, the absolute-value matrix and the spectral radius of A respectively. A matrix or vector A0 means that all entries of A are greater than or equal to zero. A>0 can be defined similarly. For matrices or vectors A and B, AB (resp. A>B) means that AB0 (resp. AB>0). Denote C=2i=1C([ri,0],R1)  and  C+=2i=1C([ri,0],R1+) as Banach spaces equipped with the supremum norm defined by

    ||φ||=suprit0max1i2|φi(t)|   for all  φ(t)=(φ1(t),φ2(t))TC(orC+).

    If xi(t) is defined on [t0ri,ν) with t0,νR1 and i=1,2, then we define xtC as xt=(x1t,x2t)T where xit(θ)=xi(t+θ) for all θ[ri,0] and i=1,2.

    The initial conditions associated with system (1.4) are of the following form:

    xt0=φ, φ=(φ1,φ2)TC+. (1.6)

    We write xt(t0,φ)(x(t;t0,φ)) for a solution of the initial value problems (1.4) and (1.6). Also, let [t0,η(φ)) be the maximal right-interval of the existence of xt(t0,φ).

    The remaining part of this paper is organized as follows. In Section 2, we shall give some definitions and preliminary results. In Section 3, we shall derive sufficient conditions for checking the existence, uniqueness and exponential convergence of the positive almost periodic solution of (1.4). In Section 4, we shall give an example and numerical simulation to illustrate the results obtained in the previous section.

    In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.

    Definition 2.1. [39,40] Let u(t):R1Rn be continuous in t. u(t) is said to be almost periodic on R1, if for any ε>0, the set T(u,ε)={δ:|u(t+δ)u(t)|<ε  for all  tR1} is relatively dense, i.e., for any ε>0, it is possible to find a real number l=l(ε)>0, such that for any interval with length l(ε), there exists a number δ=δ(ε) in this interval such that |u(t+δ)u(t)|<ε,  for all  tR1.

    Definition 2.2. [39,40] Let xRn and Q(t) be an n×n continuous matrix defined on R1. The linear system

    x(t)=Q(t)x(t) (2.1)

    is said to admit an exponential dichotomy on R1 if there exist positive constants k,α, projection P and the fundamental solution matrix X(t) of (2.1) satisfying

    Definition 2.3. A real n\times n matrix K = (k_{ij}) is said to be an M-matrix if k_{ij}\leq0, i, j = 1, \ldots, n, i\neq j and K^{-1}\geq 0 .

    Set

    B = \{ \varphi|\varphi = (\varphi_{1}(t), \varphi_{2}(t) )^T \ \mbox{ is an almost periodic vector function on } R^{1} \}.

    For any \varphi \in B , we define an induced module \|\varphi\|_{B} = \sup\limits_{t\in R^{1}}\max\limits_{1\leq i\leq 2}|\varphi_i (t)| ; then, B is a Banach space.

    Lemma 2.1. [39,40] If the linear system (2.1) admits an exponential dichotomy, then the almost periodic system

    \begin{align} \begin{array}{rcl} x'(t) = Q(t)x+g(t) \end{array} \end{align} (2.2)

    has a unique almost periodic solution x(t) , and

    \begin{align} x(t) = \int_{-\infty}^{t}X(t)PX^{-1}(s)g(s)ds-\int^{+\infty}_{t}X(t)(I-P)X^{-1}(s)g(s)ds. \end{align} (2.3)

    Lemma 2.2. [39,40] Let c_{i}(t) be an almost periodic function on R^{1} and

    M[c_{i}] = \lim\limits_{T\rightarrow +\infty}\frac{1}{T}\int_{t}^{t+T}c_{i}(s)ds > 0, \quad i = 1, 2, \cdots, n.

    Then the linear system

    x'(t) = diag (-c_{1}(t), -c_{2}(t), \cdots, -c_{n}(t))x(t)

    admits an exponential dichotomy on R^{1} .

    Lemma 2.3. [41,42] Let A \geq 0 be an n\times n matrix and \rho(A) < 1 ; then, (I_n-A)^{-1}\geq0 , where I_n denotes the identity matrix of size n .

    Lemma 2.4. Suppose that there exist positive constants E_{i1} and E_{i2} such that

    \begin{align} E_{i1} > E_{i2} , \sum\limits_{j = 1}^n \frac{c^{+}_{1j} }{\delta_{1} ^{-}b_{1j}^{-}e}+\frac{a_1^+E_{21}}{\delta_{1} ^{-}} < E_{11}, \sum\limits_{j = 1}^n \frac{c^{+}_{2j} }{\delta_{2} ^{-}b_{2j}^{-}e}+\frac{a_2^+E_{11}}{\delta_{2} ^{-}} < E_{21}, \end{align} (2.4)
    \begin{align} \frac{a_{1}^{-}}{\delta_{1} ^{+}}E_{22}+\sum\limits_{j = 1}^n \frac{c^{-}_{1j}}{\delta_{1} ^{+}} E_{11} e^{- b_{1j}^{+}E_{11}}-\frac{k_{1}^+}{\delta_{1} ^{+}}E_{11}E_{21} > E_{12}\geq \frac{1}{\min\limits _{1\leq j\leq m}b_{1j}^{-}}\; , \end{align} (2.5)
    \begin{align} \frac{a_{2}^{-}}{\delta_{2} ^{+}}E_{12}+\sum\limits_{j = 1}^n \frac{c^{-}_{2j}}{\delta_{2} ^{+}} E_{21} e^{- b_{2j}^{+}E_{21}}-\frac{k_{2}^+}{\delta_{2} ^{+}}E_{11}E_{21} > E_{22}\geq \frac{1}{\min\limits _{1\leq j\leq m}b_{2j}^{-}}\; , \end{align} (2.6)

    where i = 1, 2 . Let

    C^{0}: = \{\varphi|\varphi \in C, E_{i2} < \varphi_{i}(t) < E_{i1}, \ \mbox{ for all } \ t\in [-r_{i}, 0], \ i = 1, 2\}.

    Moreover, assume that x(t; t_{0}, \varphi) is the solution of (1.4) with \varphi \in C^{0} . Then,

    \begin{align} E_{i2} < x_{i}(t; t_{0}, \varphi) < E_{i1}, \ \mbox{ for all } \ \ t\in [t_0, \eta(\varphi)), i = 1, 2, \end{align} (2.7)

    and \eta(\varphi) = +\infty .

    Proof. We rewrite the system (1.4) as

    x'(t) = f(t, x_t) ,

    where x(t) = (x_1(t), x_2(t))^T , f(t, \varphi) = (f_1(t, \varphi), f_2(t, \varphi))^T , f_1(t, \varphi) = -\delta_1(t) \varphi_{1}(0)+ a_{1}(t)\varphi_{2}(0)+\sum\limits_{j = 1}^nc_{1j}(t)\varphi_{1}(-\tau_{1j}(t))e^{-b_{1j}(t)\varphi_{1}(-\tau_{1j}(t))} -k_1(t)\varphi_1(0)\varphi_2(0) , f_2(t, \varphi) = -\delta_2(t) \varphi_{2}(0)+ a_{2}(t)\varphi_{1}(0)+\sum\limits_{j = 1}^nc_{2j}(t)\varphi_{2}(-\tau_{2j}(t))e^{-b_{2j}(t)\varphi_{2}(-\tau_{2j}(t))} -k_2(t)\varphi_1(0)\varphi_2(0) , \varphi(t) = (\varphi_1(t), \varphi_2(t))^T\in C^0 . It is obvious that f: R^1\times C^0\rightarrow R^2 is continuous and C^0\subset C is open. Let \phi, \psi \in C^0 ; then, considering that \sup\limits_{u\geq 0}|\frac{1-u}{e^{u}}| = 1 and the inequality

    \begin{align} &|xe^{-x}-ye^{-y}| = |\frac{1-(x+\theta (y-x))}{e^{x+\theta (y-x)}}|| x -y | \\ &\ \ \ \ \ \ \ \leq | x -y | \ \mbox{ where } \ x, y\in [0, +\infty) , 0 < \theta < 1, \end{align} (2.8)

    we obtain

    \begin{eqnarray*} |f_1(t, \phi)-f_1(t, \psi)| & \leq &\delta_1(t) |\phi_{1}(0)-\psi_{1}(0)|+ a_{1}(t)|\phi_{2}(0)-\psi_{2}(0)|+\sum\limits_{j = 1}^nc_{1j}(t)\times\\ && |\phi_{1}(-\tau_{1j}(t))e^{-b_{1j}(t)\phi_{1}(-\tau_{1j}(t))}-\psi_{1}(-\tau_{1j}(t))e^{-b_{1j}(t)\psi_{1}(-\tau_{1j}(t))}|\\ &&+k_1(t)|\phi_1(0)\phi_2(0)-\psi_1(0)\psi_2(0)|\\ & \leq & (\delta_1^++ a_{1}^+)||\phi-\psi||+\sum\limits_{j = 1}^n\frac{c_{1j}(t)}{b_{1j}(t)}\times|b_{1j}(t)\phi_{1}(-\tau_{1j}(t))e^{-b_{1j}(t)\phi_{1}(-\tau_{1j}(t))}\\ && -b_{1j}(t)\psi_{1}(-\tau_{1j}(t))e^{-b_{1j}(t)\psi_{1}(-\tau_{1j}(t))}|\\ &&+k_1^+(|\phi_1(0)||\phi_2(0)-\psi_2(0)|+|\psi_2(0)||\phi_1(0)-\psi_1(0)|)\\ & \leq & (\delta_1^++ a_{1}^+)||\phi-\psi||+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{b_{1j}(t)}\times\\ && |b_{1j}(t)\phi_{1}(-\tau_{1j}(t))-b_{1j}(t)\psi_{1}(-\tau_{1j}(t))|\\ &&+k_1^+(E_{11}||\phi-\psi||+E_{21}||\phi-\psi||)\\ & \leq & (\delta_1^++ a_{1}^++\sum\limits_{j = 1}^nc_{1j}^++k_1^+E_{11}+k_1^+E_{21})||\phi-\psi||. \end{eqnarray*} (2.9)

    In the same way, we also get

    \begin{align} |f_2(t, \phi)-f_2(t, \psi)| \leq (\delta_2^++ a_{2}^++\sum\limits_{j = 1}^nc_{2j}^++k_2^+E_{11}+k_2^+E_{21}))||\phi-\psi||. \end{align} (2.10)

    Then (2.9) and (2.10) imply that f satisfies the Lipschitz condition in its second argument on each compact subset of R^1\times C^0 . Moreover, since \varphi\in C_+, it is easy to get that x_t(t_0, \varphi)\in C_+ for all t\in [t_0, \eta(\varphi)) by using Theorem 5.2.1 from [43, p. 81]. Set x(t) = x(t; t_{0}, \varphi) \mbox{ for all } \ t\in [t_0, \eta(\varphi)) .

    We claim that

    \begin{align} 0\leq x_{i}(t) < E_{i1} \ \mbox{ for all } \ t\in [t_0, \eta(\varphi)), i = 1, 2 . \end{align} (2.11)

    By contradiction, assume that (2.11) does not hold. Then, there exists t_1\in(t_0, \eta(\varphi)) such that one of the following two cases must occur:

    \begin{align} (1)\; \; x_1(t_1) = E_{11} , \; \; 0\leq x_i(t) < E_{i1}\; \; \mbox{for all}\; \; t\in[t_0-r_i, t_1), \; i = 1, 2; \end{align} (2.12)
    \begin{align} (2)\; \; x_2(t_1) = E_{21} , \; \; 0\leq x_i(t) < E_{i1}\; \; \mbox{for all}\; \; t\in[t_0-r_i, t_1), \; i = 1, 2. \end{align} (2.13)

    In the sequel, we consider two cases.

    Case (ⅰ). Suppose that (2.12) holds. Considering the derivative of x_1(t) , together with (2.4) and the fact that \sup\limits_{u\geq0}ue^{-u} = \frac{1}{e} , we have

    \begin{eqnarray*} 0 & \leq & x_{1}'(t_1)\\ & = & -\delta_1(t_1) x_{1}(t_1)+ a_{1}(t_1)x_{2}(t_1)+\sum\limits_{j = 1}^nc_{1j}(t_1)x_{1}(t_1-\tau_{1j}(t_1))e^{-b_{1j}(t_1)x_{1}(t_1-\tau_{1j}(t_1))}\\ &&-k_1(t_1)x_1(t_1)x_2(t_1)\\ & \leq & -\delta_1^- x_{1}(t_1)+ a_{1}^+x_{2}(t_1)+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{b_{1j}^-}\frac{1}{e}\\ & \leq&-\delta_1^- E_{11}+a_{1}^+E_{21}+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{b_{1j}^-}\frac{1}{e}\\ & = & \delta_1^-(- E_{11}+\sum\limits_{j = 1}^n \frac{c^{+}_{1j} }{\delta_{1} ^{-}b_{1j}^{-}e}+\frac{a_1^+E_{21}}{\delta_{1} ^{-}}) \\ & < &0, \end{eqnarray*}

    which is a contradiction.

    Case (ⅱ). Suppose that (2.13) holds. Considering the derivative of x_2(t), together with (2.4) and the fact that \sup\limits_{u\geq0}ue^{-u}=\frac{1}{e}, we have

    \begin{eqnarray*} 0 & \leq & x_{2}'(t_1)\\ & = & -\delta_2(t_1) x_{2}(t_1)+ a_{2}(t_1)x_{1}(t_1)+\sum\limits_{j = 1}^nc_{2j}(t_1)x_{2}(t_1-\tau_{1j}(t_1))e^{-b_{2j}(t_1)x_{2}(t_1-\tau_{1j}(t_1))}\\ &&-k_2(t_1)x_1(t_1)x_2(t_1)\\ &\leq& -\delta_2^- x_{2}(t_1)+ a_{2}^+x_{1}(t_1)+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{b_{2j}^-}\frac{1}{e}\\ &\leq &-\delta_2^- E_{21}+a_{2}^+E_{11}+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{b_{2j}^-}\frac{1}{e}\\ & = & \delta_2^-(- E_{21}+\sum\limits_{j = 1}^n \frac{c^{+}_{2j} }{\delta_{2} ^{-}b_{2j}^{-}e}+\frac{a_2^+E_{11}}{\delta_{2} ^{-}}) \\ & < &0, \end{eqnarray*}

    which is a contradiction. Together with Cases (ⅰ) and (ⅱ), (2.11) holds for t\in[t_0, \eta(\varphi)).

    We next show that

    \begin{align} x_{i}(t) > E_{i2} , \ \mbox{ for all } \ t\in (t_0, \eta(\varphi)), i = 1, 2. \end{align} (2.14)

    Suppose, for the sake of contradiction, that (2.14) does not hold. Then, there exists t_2\in(t_0, \eta(\varphi)) such that one of the following two cases must occur:

    \begin{align} (1)\; \; x_1(t_2) = E_{12} , \; \; E_{i2} < x_i(t) < E_{i1}\; \; \mbox{for all}\; \; t\in[t_0-r_i, t_2), \; i = 1, 2; \end{align} (2.15)
    \begin{align} (2)\; \; x_2(t_2) = E_{22} , \; \; E_{i2} < x_i(t) < E_{i1}\; \; \mbox{for all}\; \; t\in[t_0-r_i, t_2), \; i = 1, 2 . \end{align} (2.16)

    If (2.15) holds, from (2.5), (2.6), (2.11) and (2.15), we get

    \begin{align} E_{i2} < x_{i}(t) < E_{i1}, \ \ b_{ ij} ^{+} x _{i}(t)\geq b_{ ij} ^{+}E_{i2}\geq b_{ ij} ^{+}\frac{1}{\min\limits _{1\leq j \leq n}b_{ij}^{-}}\geq 1 , \end{align} (2.17)

    for all t\in [t_{0}-r_{i}, t_2), i = 1, 2, \ j = 1, 2\cdots, n. Calculating the derivative of x_{1}(t), together with (2.5) and the fact that \min\limits_{1\leq u\leq \kappa} ue^{- u} = \kappa e^{ - \kappa} , (1.4), (2.15) and (2.17) imply that

    \begin{eqnarray*} 0 & \geq & x_{1}'(t_2)\\ & = & -\delta_1(t_2) x_{1}(t_2)+ a_{1}(t_2)x_{2}(t_2)+\sum\limits_{j = 1}^nc_{1j}(t_2)x_{1}(t_2-\tau_{1j}(t_2))e^{-b_{1j}(t_2)x_{1}(t_2-\tau_{1j}(t_2))}\\ &&-k_1(t_2)x_1(t_2)x_2(t_2)\\ & \geq& -\delta_1^+ x_{1}(t_2)+ a_{1}^-x_{2}(t_2)+\sum\limits_{j = 1}^n\frac{c_{1j}(t_2)}{b_{1j}^+}b_{1j}^+x_{1}(t_2-\tau_{1j}(t_2))e^{-b_{1j}^+x_{1}(t_2-\tau_{1j}(t_2))}\\ &&-k_1^+x_1(t_2)x_2(t_2)\\ & > & -\delta_1^+ E_{12}+ a_{1}^- E_{22}+\sum\limits_{j = 1}^nc_{1j}^- E_{11}e^{-b_{1j}^+ E_{11}}-k_1^+ E_{11} E_{21}\\ & = &\delta_1^+(-E_{12}+ \frac{a_{1}^{-}}{\delta_{1} ^{+}}E_{22}+\sum\limits_{j = 1}^n \frac{c^{-}_{1j}}{\delta_{1} ^{+}} E_{11} e^{- b_{1j}^{+}E_{11}}-\frac{k_{1}^+}{\delta_{1} ^{+}}E_{11}E_{21})\\ & > &0, \end{eqnarray*}

    which is absurd and implies that (2.14) holds. If (2.16) holds, we can prove that (2.14) also holds in a similar way.

    It follows from (2.11) and (2.14) that (2.7) is true. From Theorem 2.3.1 in [44], we easily obtain \eta(\varphi) = +\infty. This completes the proof.

    Theorem 3.1. Let (2.4) – (2.6) hold. Moreover, suppose that

    \begin{align} \rho(A^{-1}B) < 1. \end{align} (3.1)

    where

    A = \left( \begin{array}{ccc} \delta_1^- & 0 \\ 0 & \delta_2^- \end{array} \right), B = \left( \begin{array}{ccc} \sum\limits_{j = 1}^n \frac{c_{1j}^+}{e^2}+k_1^+E_{21}& a_1^++k_1^+E_{11} \\ a_2^++k_2^+E_{21} & \sum\limits_{j = 1}^n \frac{c_{2j}^+}{e^2}+k_2^+E_{11} \end{array} \right) .

    Then, there exists a unique positive almost periodic solution of system (1.4) in the region B^{*} = \{\varphi|\varphi \in B, E_{i2}\leq \varphi_{i}(t) \leq E_{i1}, \ \mbox{ for all } \ t\in R^{1}, i = 1, 2, \cdots, n \} .

    Proof. For any \phi \in B , we consider the following auxiliary system

    \begin{align} \left\{ \begin{array}{rcl} x_{1} '(t)& = & -\delta_1(t) x_{1}(t)+ a_{1}(t)\phi_{2}(t)+\sum\limits_{j = 1}^nc_{1j}(t)\phi_{1}(t-\tau_{1j}(t))e^{-b_{1j}(t)\phi_{1}(t-\tau_{1j}(t))}\\ &&-k_1(t)\phi_1(t)\phi_2(t) \\ x_{2} '(t) & = & -\delta_2(t) x_{2}(t)+ a_{2}(t)\phi_{1}(t)+\sum\limits_{j = 1}^n c_{2j}(t)\phi_{2}(t-\tau_{2j}(t))e^{-b_{2j}(t)\phi_{2}(t-\tau_{2j}(t))}\\ &&-k_2(t)\phi_1(t)\phi_2(t). \end{array} \right. \end{align} (3.2)

    Since M[\delta_{i}] > 0 (i = 1, 2) , it follows from Lemma 2.2 that the linear system

    \begin{align} x_{i} '(t) = -\delta_{i}(t) x_{i}(t), \; \; \; \; \; i = 1, 2 \end{align} (3.3)

    admits an exponential dichotomy on R^1 . Thus, by Lemma 2.1, we obtain that the system (3.2) has exactly one almost periodic solution x^{\phi}(t) = (x_1^{\phi}(t), x_2^{\phi}(t))^T :

    \begin{align} \left\{ \begin{array}{rcl} x_1^{\phi}(t)& = & \int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}[a_1(s)\phi_{2}(s)+\sum\limits^n_{j = 1}c_{ 1j}(s)\phi_{1} (s-\tau_{1j}(s))e^{-b_{1j}(s)\phi_{1} (s-\tau_{1j}(s))}\\ &&-k_1(s)\phi_1(s)\phi_2(s)]ds \\ x_2^{\phi}(t)& = & \int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_2(s)\phi_{1}(s)+\sum\limits^n_{j = 1}c_{ 2j}(s)\phi_{2} (s-\tau_{2j}(s))e^{-b_{2j}(s)\phi_{2} (s-\tau_{2j}(s))}\\ &&-k_2(s)\phi_1(s)\phi_2(s)]ds. \end{array} \right. \end{align} (3.4)

    Define a mapping T:B\longrightarrow B by setting

    T(\phi(t)) = x^{\phi}(t), \ \forall \ \phi\in B.

    Since B^{*} = \{\varphi|\varphi \in B, E_{i2}\leq \varphi_{i}(t) \leq E_{i1}, \ \mbox{ for all } \ t\in R^{1}, i = 1, 2\} , it is obvious that B^{*} is a closed subset of B . For i = 1, 2 and any \phi \in B^{*} , from (2.4), (3.4) and the fact that \sup\limits_{u\geq 0} ue^{- u} = \frac{1}{ e} , we have

    \begin{eqnarray*} \ \ \ \ \ \ x_1^{\phi}(t) &\leq&\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}[a_1^+E_{21}+\sum\limits^n_{j = 1}c_{ 1j}(s)\frac{1}{b_{1j}(s)e}]ds\\ &\leq& \frac{1}{\delta_{1} ^{-}}[a_1^+E_{21}+\sum\limits_{j = 1}^n \frac{c^{+}_{1j} }{b_{1j}^{-}e}]\\ & = &\sum\limits_{j = 1}^n \frac{c^{+}_{1j} }{\delta_{1} ^{-}b_{1j}^{-}e}+\frac{a_1^+E_{21}}{\delta_{1} ^{-}} < E_{11} \ \ \ \ \ \mbox{ for all } \ t\in R^{1}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \ \ \ \ \ \ \end{eqnarray*} (3.5)

    and

    \begin{eqnarray*} \ \ \ \ \ \ x_2^{\phi}(t) &\leq&\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_2^+E_{11}+\sum\limits^n_{j = 1}c_{ 2j}(s)\frac{1}{b_{2j}(s)e}]ds\\ &\leq& \frac{1}{\delta_{2} ^{-}}[a_2^+E_{11}+\sum\limits_{j = 1}^n \frac{c^{+}_{2j} }{b_{2j}^{-}e}]\\ & = &\sum\limits_{j = 1}^n \frac{c^{+}_{2j} }{\delta_{2} ^{-}b_{2j}^{-}e}+\frac{a_2^+E_{11}}{\delta_{2} ^{-}} < E_{21} \ \ \ \ \ \mbox{ for all } \ t\in R^{1}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \ \ \ \ \ \ \end{eqnarray*} (3.6)

    In view of the fact that \min\limits_{1\leq u\leq \kappa} ue^{- u} = \kappa e^{ - \kappa} , from (2.5)–(2.7) and (3.4), we obtain

    \begin{eqnarray*} x_1^{\phi}(t) &\geq&\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}[a_1^-E_{22}+\sum\limits^n_{j = 1}c_{ 1j}(s)\frac{1}{b_{1j}^+}b_{1j}^+\phi_{1} (s-\tau_{1j}(s))e^{-b_{1j}^+\phi_{1} (s-\tau_{1j}(s))}\\ &&-k_1^+\phi_1(s)\phi_2(s)]ds\\ &\geq& \frac{1}{\delta_{1}^+}[a_1^-E_{22}+\sum\limits^n_{j = 1}c_{ 1j}^-E_{11}e^{-b_{1j}^+E_{11}}-k_1^+E_{11}E_{21}]\\ & = &\frac{a_{1}^{-}}{\delta_{1} ^{+}}E_{22}+\sum\limits_{j = 1}^n \frac{c^{-}_{1j}}{\delta_{1} ^{+}} E_{11} e^{- b_{1j}^{+}E_{11}}-\frac{k_{1}^+}{\delta_{1} ^{+}}E_{11}E_{21} > E_{12} \ \mbox{ for all } \ t\in R^{1}, \end{eqnarray*} (3.7)

    and

    \begin{eqnarray*} x_2^{\phi}(t) &\geq&\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_2^-E_{12}+\sum\limits^n_{j = 1}c_{ 2j}(s)\frac{1}{b_{2j}^+}b_{2j}^+\phi_{2} (s-\tau_{2j}(s))e^{-b_{2j}^+\phi_{1} (s-\tau_{2j}(s))}\\ &&-k_2^+\phi_1(s)\phi_2(s)]ds\\ &\geq& \frac{1}{\delta_{2}^+}[a_2^-E_{12}+\sum\limits^n_{j = 1}c_{ 2j}^-E_{21}e^{-b_{2j}^+E_{21}}-k_2^+E_{11}E_{21}]\\ & = &\frac{a_{2}^{-}}{\delta_{2} ^{+}}E_{12}+\sum\limits_{j = 1}^n \frac{c^{-}_{2j}}{\delta_{2} ^{+}} E_{21} e^{- b_{2j}^{+}E_{21}}-\frac{k_{2}^+}{\delta_{2} ^{+}}E_{11}E_{21} > E_{22} \ \mbox{ for all } \ t\in R^{1}. \end{eqnarray*} (3.8)

    Therefore, (3.5)–(3.8) show that the mapping T is a self-mapping from B^{*} to B^{*} .

    Let \varphi, \psi \in B^{*} ; for i = 1, 2 , we get

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{1} | \\ & = & \sup\limits_{t \in R^{1}}|\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}[a_{ 1}(s)(\varphi_{2} (s)-\psi_{2} (s)) \\ & &+\sum^n_{j = 1}c_{ 1j}(s)(\varphi_{1} (s-\tau_{1j}(s))e^{-b_{ 1j}(s)\varphi_{1} (s-\tau_{1j}(s))}-\psi_{1} (s-\tau_{1j}(s))e^{-b_{1j}(s)\psi_{1} (s-\tau_{1j}(s))}) \\ & &- k_{1}(s)(\varphi_{1}(s)\varphi_{2}(s)-\psi_{1}(s)\psi_{2}(s))]ds| \\ & = & \sup\limits_{t \in R^{1}}|\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}[a_{ 1}(s)(\varphi_{2} (s)-\psi_{2} (s))+\sum^n_{j = 1}\frac{c_{ 1j}(s)}{b_{ 1j}(s)}\times \\ & &(b_{ 1j}(s)\varphi_{1} (s-\tau_{1j}(s))e^{-b_{ 1j}(s)\varphi_{1} (s-\tau_{1j}(s))}-b_{ 1j}(s)\psi_{1} (s-\tau_{1j}(s))e^{-b_{1j}(s)\psi_{1} (s-\tau_{1j}(s))}) \\ & &- k_{1}(s)(\varphi_{1}(s)\varphi_{2}(s)-\varphi_{1}(s)\psi_{2}(s)+\varphi_{1}(s)\psi_{2}(s)-\psi_{1}(s)\psi_{2}(s))]ds| , \end{eqnarray} (3.9)

    and

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{2} | \\ & = & \sup\limits_{t \in R^{1}}|\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_{ 2}(s)(\varphi_{1} (s)-\psi_{1} (s)) \\ & &+\sum^n_{j = 1}c_{ 2j}(s)(\varphi_{2} (s-\tau_{2j}(s))e^{-b_{ 2j}(s)\varphi_{2} (s-\tau_{2j}(s))}-\psi_{2} (s-\tau_{2j}(s))e^{-b_{2j}(s)\psi_{2} (s-\tau_{2j}(s))}) \\ & &- k_{2}(s)(\varphi_{1}(s)\varphi_{2}(s)-\psi_{1}(s)\psi_{2}(s))]ds| \\ & = & \sup\limits_{t \in R^{1}}|\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{2}(u)du}[a_{ 2}(s)(\varphi_{1} (s)-\psi_{1} (s))+\sum^n_{j = 1}\frac{c_{ 2j}(s)}{b_{ 2j}(s)}\times \\ & &(b_{ 2j}(s)\varphi_{2} (s-\tau_{2j}(s))e^{-b_{ 2j}(s)\varphi_{2} (s-\tau_{2j}(s))}-b_{ 2j}(s)\psi_{2} (s-\tau_{2j}(s))e^{-b_{2j}(s)\psi_{2} (s-\tau_{2j}(s))}) \\ & &- k_{2}(s)(\varphi_{1}(s)\varphi_{2}(s)-\varphi_{1}(s)\psi_{2}(s)+\varphi_{1}(s)\psi_{2}(s)-\psi_{1}(s)\psi_{2}(s))]ds| . \end{eqnarray} (3.10)

    Since

    b_{ij}(s)\varphi_{i} (s-\tau_{ ij}(s))\geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq n}b_{ij}^{-}}\geq 1,\ \mbox{ for all } \ s\in R^{1}, i = 1, 2, j = 1, 2, \cdots, n,

    and

    \ b_{ij}(s)\psi_{i} (s-\tau_{i j}(s))\geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq n}b_{ij}^{-}}\geq 1, \ \mbox{ for all }\ s\in R^{1}, i = 1, 2, j = 1, 2, \cdots, n.

    According to (1.4), (2.5), (3.5), (3.7) and (3.9), together with \sup\limits_{u\geq 1}|\frac{1-u}{e^{u}}| = \frac{1}{e^{2}} and the inequality

    \begin{align} &|xe^{-x}-ye^{-y}| = |\frac{1-(x+\theta (y-x))}{e^{x+\theta (y-x)}}|| x -y | \\&\ \ \ \ \ \ \ \ \\ \ \leq\frac{1}{e^{2}}| x -y | \ \mbox{ where } \ x, y\in [1, +\infty) , 0 < \theta < 1, \end{align} (3.11)

    we have

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{1} | \\ &\leq & \frac{a^{+}_{1}}{ \delta^{-}_{1}}\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)| \\ & &+\sup\limits_{t \in R^{1}}\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}\sum^n_{j = 1}c^{+}_{ 1j}\frac{1}{e^{2}} |\varphi_{1} (s-\tau_{1j}(s)) - \psi _{1}(s-\tau_{1j}(s))| ds \\ & &+\sup\limits_{t \in R^{1}}\int_{-\infty}^{t}e^{-\int_{s}^{t}\delta_{1}(u)du}k_1(s)( |\varphi_{1} (s)||\varphi_2(s)-\psi_2(s)|+|\psi_{2} (s)||\varphi_1(s)-\psi_1(s)|) ds \\ &\leq&\frac{a^{+}_{1}}{\delta^{-}_{1}}\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|+\sum^n_{j = 1}\frac{c^{+}_{ 1j}}{\delta^{-}_1e^{2}}\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|\\ &&+\frac{k_1^+}{\delta^{-}_{1}}E_{11}\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)| +\frac{k_1^+}{\delta^{-}_{1}}E_{21}\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|, \\ & = &(\sum^n_{j = 1}\frac{c^{+}_{ 1j}}{\delta^{-}_1e^{2}}+\frac{k_1^+}{\delta^{-}_{1}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|+ (\frac{a^{+}_{1}}{\delta^{-}_{1}}+\frac{k_1^+}{\delta^{-}_{1}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|. \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \end{eqnarray} (3.12)

    Similarly, we also get

    \begin{eqnarray} && \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{2} | \\ &\leq & (\sum^n_{j = 1}\frac{c^{+}_{ 2j}}{\delta^{-}_2e^{2}}+\frac{k_2^+}{\delta^{-}_{2}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|+ (\frac{a^{+}_{2}}{\delta^{-}_{2}}+\frac{k_2^+}{\delta^{-}_{2}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|. \end{eqnarray} (3.13)

    Hence

    \begin{eqnarray} &&( \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{1}|, \sup\limits_{t \in R^{1}} |(T(\varphi(t))-T(\psi(t)))_{2}|)^T \\ &\leq & ( (\sum\limits_{j = 1}^n\frac{c^{+}_{1j}}{ \delta^{-}_{1}e^2}+\frac{k_1^+}{\delta^{-}_{1}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|+ (\frac{a^{+}_{1}}{ \delta^{-}_{1}}+\frac{k_1^+}{\delta^{-}_{1}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|, \\ & &(\sum\limits_{j = 1}^n\frac{c^{+}_{2j}}{ \delta^{-}_{2}e^2}+\frac{k_2^+}{\delta^{-}_{2}}E_{11})\sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)|+ (\frac{a^{+}_{2}}{ \delta^{-}_{2}}+\frac{k_2^+}{\delta^{-}_{2}}E_{21})\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|)^T \\ & = & F(\sup\limits_{t\in R^{1}}|\varphi_1(t)-\psi_1(t)|, \sup\limits_{t\in R^{1}}|\varphi_2(t)-\psi_2(t)| )^T \\ & = & F(\sup\limits_{t\in R^{1}}|(\varphi(t)-\psi(t))_1|, \sup\limits_{t\in R^{1}}|(\varphi(t)-\psi(t))_2| )^T, \end{eqnarray} (3.14)

    where F = A^{-1}B. Let \mu be a positive integer. Then, from (3.14) we get

    \begin{eqnarray} &&( \sup\limits_{t \in R^{1}} |(T^\mu(\varphi(t))-T^\mu(\psi(t)))_{1}|, \sup\limits_{t \in R^{1}} |(T^\mu(\varphi(t))-T^\mu(\psi(t)))_{2}|)^T \\ & = & ( \sup\limits_{t \in R^{1}} |(T(T^{\mu-1}(\varphi(t)))-T(T^{\mu-1}(\psi(t))))_{1}|, \\ &\ &\sup\limits_{t \in R^{1}}|(T(T^{\mu-1}(\varphi(t)))-T(T^{\mu-1}(\psi(t))))_{2}|)^T \\ &\leq& F( \sup\limits_{t \in R^{1}} |(T^{\mu-1}(\varphi(t))-T^{\mu-1}(\psi(t)))_{1}|, \sup\limits_{t \in R^{1}}|(T^{\mu-1}(\varphi(t))-T^{\mu-1}(\psi(t)))_{2}|)^T\\ &\ & \vdots\\ &\leq& F^\mu( \sup\limits_{t \in R^{1}} |(\varphi(t)-\psi(t))_{1}|, \sup\limits_{t \in R^{1}}|(\varphi(t)-\psi(t))_{2}|)^T\\ & = & F^\mu( \sup\limits_{t \in R^{1}} |\varphi_{1}(t)-\psi_{1}(t)|, \sup\limits_{t \in R^{1}}|\varphi_{2}(t)-\psi_{2}(t)|)^T. \end{eqnarray} (3.15)

    Since \rho(F)<1, we obtain

    \lim\limits_{\mu\rightarrow +\infty}F^\mu = 0,

    which implies that there exist a positive integer N and a positive constant r<1 such that

    \begin{align} F^N = (A^{-1}B)^N = (g_{ij})_{2\times 2} \; \; \mbox{and}\; \; \sum\limits_{j = 1}^2 g_{ij}\leq r, i = 1, 2. \end{align} (3.16)

    In view of (3.15) and (3.16), we have

    \begin{eqnarray} |(T^N(\varphi(t))-T^N(\psi(t)))_{i}|&\leq& \sup\limits_{t\in R^{1}}|(T^N(\varphi(t))-T^N(\psi(t)))_{i}| \\ &\leq& \sum\limits_{j = 1}^2 g_{ij} \sup\limits_{t \in R^{1}} |\varphi_{j}(t)-\psi_{j}(t)|\\ &\leq& \sup\limits_{t \in R^{1}} \max\limits_{1\leq j\leq 2}|\varphi_{j}(t)-\psi_{j}(t)|\sum\limits_{j = 1}^2 g_{ij}\\ &\leq& r \| \varphi(t)-\psi(t)\|_B, \end{eqnarray}

    for all t\in R, i = 1, 2. It follows that

    \begin{align} \|T^N(\varphi(t))-T^N(\psi(t))\|_B = \sup\limits_{t\in R^{1}} \max\limits_{1\leq i\leq 2}|(T^N(\varphi(t))-T^N(\psi(t)))_i|\leq r\|\varphi(t)-\psi(t)\|_B. \end{align} (3.17)

    This implies that the mapping T^N:B^*\rightarrow B^* is a contraction mapping.

    By the fixed point theorem for Banach space, T possesses a unique fixed point \varphi^{*}\in B^{*} such that T\varphi^{*} = \varphi^{*}. By (3.2), \varphi^{*} satisfies (1.4). So \varphi^{*} is an almost periodic solution of (1.4) in B^{*}. The proof of Theorem 3.1 is now completed.

    Theorem 3.2. Let x^{*}(t) be the positive almost periodic solution of system (1.4) in the region B^{*} . Suppose that (2.4)–(2.6) and (3.1) hold. Then, the solution x(t; t_{0}, \varphi) of (1.4) with \varphi \in C^{0} converges exponentially to x ^{* }(t) as t\rightarrow +\infty.

    Proof. Since \rho(A^{-1}B) < 1, it follows from Theorem 3.1 that system (1.4) has a unique almost periodic solution x^*(t) = (x^*_1(t), x^*_2(t))^T in the region B^{*} . Set x(t) = x(t; t_{0}, \varphi), x^*(t) = x^*(t; t_{0}, \varphi^*) and y_{i}(t) = x_{i}(t)-x_{i}^{*}(t), where \varphi, \varphi^*\in C^{0}, t\in [t_{0}-r_{i}, +\infty), \ i = 1, 2. Then, we have the following:

    \begin{align} \left\{ \begin{array}{rcl} y_{1} '(t)& = & -\delta_1(t) y_{1}(t)+ a_{1}(t)y_{2}(t)+\sum\limits_{j = 1}^nc_{1j}(t)(x_{1}(t-\tau_{1j}(t))e^{-b_{1j}(t)x_{1}(t-\tau_{1j}(t))}\\ &&-x_{1}^*(t-\tau_{1j}(t))e^{-b_{1j}(t)x_{1}^*(t-\tau_{1j}(t))})-k_1(t)(x_1(t)x_2(t)-x_1^*(t)x_2^*(t)), \\ y_{2} '(t) & = & -\delta_2(t) y_{2}(t)+ a_{2}(t)y_{1}(t)+\sum\limits_{j = 1}^n c_{2j}(t)(x_{2}(t-\tau_{2j}(t))e^{-b_{2j}(t)x_{2}(t-\tau_{2j}(t))}\\ &&-x_{2}^*(t-\tau_{2j}(t))e^{-b_{2j}(t)x_{2}^*(t-\tau_{2j}(t))})-k_2(t)(x_1(t)x_2(t)-x_1^*(t)x_2^*(t)). \end{array} \right. \end{align} (3.18)

    Again from \rho(A^{-1}B) < 1, it follows from Lemma 2.3 that I_2-A^{-1}B is an M-matrix; we obtain that there exists a constant \bar{\mu}>0 and a vector \xi = (\xi_1, \xi_2)^T>(0, 0)^T such that

    (I_2-A^{-1}B)\xi > (\bar{\mu}, \bar{\mu})^T.

    Therefore,

    \left\{ \begin{array}{rcl} && \xi_1-(\sum\limits_{j = 1}^n\frac{c_{1j}^+}{\delta_1^-e^2}+\frac{k_1^+}{\delta_1^-}E_{21}) \xi_1-(\frac{a_1^+}{\delta_1^-}+\frac{k_1^+}{\delta_1^-}E_{11}) \xi_2 > \overline{\mu}, \\ && \xi_2-(\sum\limits_{j = 1}^n\frac{c_{2j}^+}{\delta_2^-e^2}+\frac{k_2^+}{\delta_2^-}E_{11}) \xi_2-(\frac{a_2^+}{\delta_2^-}+\frac{k_2^+}{\delta_2^-}E_{21}) \xi_1 > \overline{\mu}, \\ \end{array} \right.

    which implies that

    \begin{align} \left\{ \begin{array}{rcl} && (-\delta_1^-+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}+k_1^+E_{21}) \xi_1+(a_1^++k_1^+E_{11}) \xi_2 < -\delta_1^-\overline{\mu}, \\ && (-\delta_2^-+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}+k_2^+E_{11}) \xi_2+(a_2^++k_2^+E_{21}) \xi_1 < -\delta_2^-\overline{\mu}. \\ \end{array} \right. \end{align} (3.19)

    We can choose a positive constant \eta<1 such that

    \begin{align} \left\{ \begin{array}{rcl} && \eta\xi_1+(-\delta_1^-+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}e^{\eta r_1}+k_1^+E_{21}) \xi_1+(a_1^++k_1^+E_{11}) \xi_2 < p > 0, \\ && \eta\xi_2+(-\delta_2^-+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}e^{\eta r_2}+k_2^+E_{11}) \xi_2+(a_2^++k_2^+E_{21}) \xi_1 < p > 0. \\ \end{array} \right. \end{align} (3.20)

    In the sequel, we consider the following Lyapunov function:

    \begin{align} V_i(t) = |y_i(t)|e^{\eta(t-t_0)}, \; \; i = 1, 2. \end{align} (3.21)

    In view of (2.5)–(2.7), for i\in \{1, 2\} and j\in \{1, 2, \cdots, n\}, we obtain

    b_{ij}(t)x_i (t-\tau_{ij}(t))\geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq m}b_{ij}^{-}}\geq 1, \ \mbox{ for all } \ t\in [t_{0}-r_{i}, +\infty),

    and

    b_{ij}(t)x_i^*(t-\tau_{ij}(t)) \geq b^{-}_{ij}E_{i2}\geq b^{-}_{ij} \frac{1}{\min\limits _{1\leq j\leq m}b_{ij}^{-}}\geq 1,\ \mbox{ for all } \ t\in R^{1},

    which, together with (3.11) and (3.18), imply that

    \begin{array}{rcl} D^-V_1(t) &\leq & -\delta_1(t)|y_1(t)|e^{\eta(t-t_0)}+a_{1}(t)|y_2(t)|e^{\eta(t-t_0)}+\sum\limits_{j = 1}^nc _{1j}(t)e^{\eta(t-t_0)}\times\\ &&|x_1(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1(t-\tau_{1j}(t))}-x_1^*(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1^*(t-\tau_{1j}(t))}|\\ &&+ k_1(t)e^{\eta(t-t_0)}|x_1(t)x_2(t)-x_1^*(t)x_2^*(t)|+\eta|y_1(t)|e^{\eta(t-t_0)} \\ &\leq& (\eta-\delta_1(t))V_1(t)+a_1^+V_2(t)+\sum\limits_{j = 1}^n\frac{c _{1j}(t)}{b_{1j}(t)}e^{\eta(t-t_0)}\times\\ &&|b_{1j}(t)x_1(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1(t-\tau_{1j}(t))}\\ & &-b_{1j}(t)x_1^*(t-\tau_{1j}(t))e^{-b_{1j}(t)x_1^*(t-\tau_{1j}(t))}|\\ &&+ k_1(t)|x_1(t)||x_2(t)-x_2^*(t)|e^{\eta(t-t_0)} +k_1(t)|x_2^*(t)||x_1(t)-x_1^*(t)|e^{\eta(t-t_0)}\\ &\leq& (\eta-\delta_1^-)V_1(t)+a_1^+V_2(t)+\sum\limits_{j = 1}^n\frac{c _{1j}(t)}{e^2}|y_1(t-\tau_{1j}(t))|e^{\eta(t-t_0)}\\ &&+ k_1^+E_{11}V_2(t)+k_1^+E_{21}V_1(t)\\ &\leq&(\eta-\delta_1^-+k_1^+E_{21})V_1(t)+\sum\limits_{j = 1}^n\frac{c _{1j}^+}{e^2}e^{\eta r_1}V_1(t-\tau_{1j}(t))\\ &&+ (a_1^++k_1^+E_{11})V_2(t), \end{array} (3.22)

    and

    \begin{array}{rcl} D^-V_2(t) &\leq & -\delta_2(t)|y_2(t)|e^{\eta(t-t_0)}+a_{2}(t)|y_1(t)|e^{\eta(t-t_0)}+\sum\limits_{j = 1}^nc _{2j}(t)e^{\eta(t-t_0)}\times\\ &&|x_2(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2(t-\tau_{2j}(t))}-x_2^*(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2^*(t-\tau_{2j}(t))}|\\ &&+ k_2(t)e^{\eta(t-t_0)}|x_1(t)x_2(t)-x_1^*(t)x_2^*(t)|+\eta|y_2(t)|e^{\eta(t-t_0)} \\ &\leq& (\eta-\delta_2(t))V_2(t)+a_2^+V_1(t)+\sum\limits_{j = 1}^n\frac{c _{2j}(t)}{b_{2j}(t)}e^{\eta(t-t_0)}\times\\ &&|b_{2j}(t)x_2(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2(t-\tau_{2j}(t))}\\ & &-b_{2j}(t)x_2^*(t-\tau_{2j}(t))e^{-b_{2j}(t)x_2^*(t-\tau_{2j}(t))}|\\ &&+ k_2(t)|x_1(t)||x_2(t)-x_2^*(t)|e^{\eta(t-t_0)} +k_2(t)|x_2^*(t)||x_1(t)-x_1^*(t)|e^{\eta(t-t_0)}\\ &\leq& (\eta-\delta_2^-)V_2(t)+a_2^+V_1(t)+\sum\limits_{j = 1}^n\frac{c _{2j}(t)}{e^2}|y_2(t-\tau_{2j}(t))|e^{\eta(t-t_0)}\\ &&+ k_2^+E_{11}V_2(t)+k_2^+E_{21}V_1(t)\\ &\leq&(\eta-\delta_2^-+k_2^+E_{11})V_2(t)+\sum\limits_{j = 1}^n\frac{c _{2j}^+}{e^2}e^{\eta r_2}V_2(t-\tau_{2j}(t))\\ &&+ (a_2^++k_2^+E_{21})V_1(t). \end{array} (3.23)

    Let \varsigma>1 denote an arbitrary real number such that

    \varsigma\xi_i > ||\varphi-\varphi^*|| = \sup\limits_{-r_i\leq s\leq0}\max\limits_{1\leq i\leq 2}|\varphi_i(s)-\varphi^*_i(s)| > 0 , \; i = 1, 2.

    It follows from (3.21) that

    V_i(t) = |y_i(t)|e^{\eta(t-t_0)} < \varsigma\xi_i , \; \; \mbox{for all}\; t\in[t_0-r_i, t_0], \; i = 1, 2.

    We claim that

    \begin{align} V_i(t) = |y_i(t)|e^{\eta(t-t_0)} < \varsigma\xi_i , \; \; \mbox{for all}\; t > t_0, \; i = 1, 2. \end{align} (3.24)

    In contrast, there must exist i\in \{1, 2\} and r^*>t_0 such that

    \begin{align} V_i(r^*) = \varsigma\xi_i, \; \mbox{and}\; V_j(t) < \varsigma\xi_j, \; \; \mbox{for all}\; t\in[t_0-r_j, r^*), \; j = 1, 2. \end{align} (3.25)

    Thus,

    \begin{align} V_1(r^*)-\varsigma\xi_1 = 0, \; \mbox{and}\; V_j(t)-\varsigma\xi_j < 0, \; \; \mbox{for all}\; t\in[t_0-r_j, r^*), \; j = 1, 2, \end{align} (3.26)

    or

    \begin{align} V_2(r^*)-\varsigma\xi_2 = 0, \; \mbox{and}\; V_j(t)-\varsigma\xi_j < 0, \; \; \mbox{for all}\; t\in[t_0-r_j, r^*), \; j = 1, 2. \end{align} (3.27)

    Together with (3.20), (3.22), (3.23), (3.26) and (3.27), we obtain

    \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \begin{array}{rcl} 0&\leq&D^-(V_1(r^*)-\varsigma\xi_1) \\ & = &D^-(V_1(r^*))\\ &\leq&(\eta-\delta_1^-+k_1^+E_{21})V_1(r^*)+\sum\limits_{j = 1}^n\frac{c _{1j}^+}{e^2}e^{\eta r_1}V_1(r^*-\tau_{1j}(r^*))\\ &&+ (a_1^++k_1^+E_{11})V_2(r^*)\\ &\leq&\varsigma[\eta\xi_1+(-\delta_1^-+\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}e^{\eta r_1}+k_1^+E_{21}) \xi_1+(a_1^++k_1^+E_{11}) \xi_2]\\ & < &0, \end{array} (3.28)

    or

    \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \begin{array}{rcl} 0&\leq&D^-(V_2(r^*)-\varsigma\xi_2) \\ & = &D^-(V_2(r^*))\\ &\leq&(\eta-\delta_2^-+k_2^+E_{11})V_2(r^*)+\sum\limits_{j = 1}^n\frac{c _{2j}^+}{e^2}e^{\eta r_2}V_2(r^*-\tau_{2j}(r^*))\\ &&+ (a_2^++k_2^+E_{21})V_1(r^*)\\ &\leq&\varsigma[\eta\xi_2+(-\delta_2^-+\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}e^{\eta r_2}+k_2^+E_{11}) \xi_2+(a_2^++k_2^+E_{21}) \xi_1]\\ & < &0, \end{array} (3.29)

    which are both contradictory. Hence, (3.24) holds. Let M>1 such that

    \begin{align} \varsigma\xi_i\leq M||\varphi-\varphi^*||, \; i = 1, 2. \end{align} (3.30)

    In view of (3.24) and (3.30), we get

    |x_i(t)-x_i^*(t)| = |y_i(t)|\leq \varsigma\xi_ie^{-\eta (t-t_0)}\leq M||\varphi-\varphi^* || e^{-\eta (t-t_0)}, \; \; \mbox{for all }\; \; t > t_0\; \; i = 1, 2.

    This completes the proof.

    Corollary 3.1. Let (2.4)–(2.6) hold. Suppose that I_2-A^{-1}B is an M-matrix. Then system (1.4) has exactly one almost periodic solution x^*(t). Moreover, the solution x(t; t_{0}, \varphi) of (1.4) with \varphi \in C^{0} converges exponentially to x ^{* }(t) as t\rightarrow +\infty.

    Proof. Since I_2-A^{-1}B is an M-matrix, it follows that there exists a vector d = (d_1, d_2)^T>(0, 0)^T such that

    \begin{align} (I_2-A^{-1}B)d > 0, \end{align} (3.31)

    hence

    \begin{align} \left\{ \begin{array}{rcl} && -\delta_1^-d_1+(\sum\limits_{j = 1}^n\frac{c_{1j}^+}{e^2}+k_1^+E_{21}) d_1+(a_1^++k_1^+E_{11}) d_2 < 0 \\ && -\delta_2^-d_2+(\sum\limits_{j = 1}^n\frac{c_{2j}^+}{e^2}+k_2^+E_{11}) d_2+(a_2^++k_2^+E_{21}) d_1 < 0. \\ \end{array} \right. \end{align} (3.32)

    For any matrix norm ||\cdot|| and nonsingular matrix D, ||A||_D = ||D^{-1}AD|| also defines a matrix norm. Let D = diag(d_1, d_2). Then (3.32) implies that the row norm of matrix D^{-1}A^{-1}BD is less than 1. Hence \rho(A^{-1}B) < 1. Corollary 3.1 follows immediately from Theorems 3.1 and 3.2.

    In this section, we give an example and present a numerical simulation to demonstrate the results obtained in previous sections.

    Example 4.1. Consider the following competitive and cooperative Nicholson's blowfly system:

    \begin{align} \left\{ \begin{array}{rcl} x_{1} '(t) & = & -(18+\cos^{2}\sqrt{5}t) x_{1} (t)+(1+0.7\sin ^{2} t)e^{e-2} x_{2} (t)\\&\; &+ e^{e-1}(9.5+0.005|\sin \sqrt{2}t|) x_{1} (t-e^{|\sin t|+|\sin \sqrt{2}t|})e^{- x _{1} (t-e^{|\sin t|+|\sin\sqrt{2} t|})} , \ \ \\&\; &+ e^{e-1}(9.5+0.005|\sin \sqrt{5}t|) x _{1} (t-e^{|\cos \sqrt{3} t|+|\cos t|})e^{- x _{1} (t-e^{|\cos \sqrt{3}t|+|\cos t|})}\\&\; &- 0.1e^{-2}\cos ^{2} t x _{1} (t)x _{2} (t) \\ x_{2} '(t) & = & -(18+\sin^{2}\sqrt{5}t) x_{2} (t)+(1+0.7\cos ^{2} t)e^{e-2} x_{1} (t)\\&\; &+ e^{e-1}(9.5+0.005|\cos \sqrt{2}t|) x_{2} (t-e^{|\cos t|+|\cos \sqrt{7} t|})e^{- x _{2} (t-e^{|\cos t|+|\cos \sqrt{7} t|})} , \ \ \\&\; &+ e^{e-1}(9.5+0.005|\sin \sqrt{6}t|) x _{2} (t-e^{|\cos \sqrt{7}t|+|\cos \sqrt{3} t|})e^{- x _{2} (t-e^{|\cos \sqrt{7}t|+|\cos \sqrt{3} t|})}\\&\; &- 0.1e^{-2}\sin ^{2} t x _{1} (t)x _{2} (t) . \end{array} \right. \end{align} (4.1)

    Obviously, \delta_{i}^{-} = 18, \delta_{i}^{+} = 19, a_i^- = e^{e-2}, a_i^+ = 1.7e^{e-2}, b_{ij}^{-} = b_{ij} ^{+} = 1, \ c_{ij}^{-} = 9.5e^{e-1}, \ c_{ij}^{+} = 9.505e^{e-1}, k_i^+ = 0.1e^{-2} and r_i = e^2 (i, j = 1, 2). Let E_{i1} = e and E_{i2} = 1 for i = 1, 2; we obtain

    A = \left( \begin{array}{ccc} 18 & 0 \\ 0 & 18 \end{array} \right)\; \; , B = \left( \begin{array}{ccc} 19.01e^{e-3}+0.1e^{-1} & 1.7e^{e-2}+0.1e^{-1} \\ 1.7e^{e-2}+0.1e^{-1} & 19.01e^{e-3}+0.1e^{-1} \end{array} \right)
    \begin{align} \sum\limits_{j = 1}^2 \frac{c^{+}_{1j} }{\delta_{1} ^{-}b_{1j}^{-}e}+\frac{a_1^+E_{21}}{\delta_{1} ^{-}} = \frac{19.01e^{e-2}+1.7e^{e-1}}{18} < e \end{align} (4.2)
    \begin{align} \sum\limits_{j = 1}^n \frac{c^{+}_{2j} }{\delta_{2} ^{-}b_{2j}^{-}e}+\frac{a_2^+E_{11}}{\delta_{2} ^{-}} = \frac{19.01e^{e-2}+1.7e^{e-1}}{18} < e \end{align} (4.3)
    \begin{align} \frac{a_{1}^{-}}{\delta_{1} ^{+}}E_{22}+\sum\limits_{j = 1}^n \frac{c^{-}_{1j}}{\delta_{1} ^{+}} E_{11} e^{- b_{1j}^{+}E_{11}}-\frac{k_{1}^+}{\delta_{1} ^{+}}E_{11}E_{21} = \frac{19+e^{e-2}-0.1}{19} > 1 \end{align} (4.4)
    \begin{align} \frac{a_{2}^{-}}{\delta_{2} ^{+}}E_{12}+\sum\limits_{j = 1}^n \frac{c^{-}_{2j}}{\delta_{2} ^{+}} E_{21} e^{- b_{2j}^{+}E_{21}}-\frac{k_{2}^+}{\delta_{2} ^{+}}E_{11}E_{21} = \frac{19+e^{e-2}-0.1}{19} > 1 \end{align} (4.5)
    \begin{align} \rho(A^{-1}B)\approx 0.9946 < 1. \end{align} (4.6)

    Then (4.2) - (4.6) imply that the competitive and cooperative Nicholson's blowfly system (4.1) satisfies (2.4)–(2.6) and (3.1). Hence, from Theorems 3.1 and 3.2, system (4.1) has a positive almost periodic solution

    x^{*}(t )\in B^{*} = \{\varphi|\varphi \in B, 1\leq \varphi_{i}(t) \leq e, \ \mbox{ for all } \ t\in R, i = 1, 2 \} .

    Moreover, if \varphi\in C^{0} = \{\varphi|\varphi \in C, 1 < \varphi_{i}(t) < e, \ for all \ t\in [-e^{2}, 0], i = 1, 2 \}, then x(t; t_{0}, \varphi) converges exponentially to x^{*}(t) as t\rightarrow +\infty. The fact is verified by the numerical simulation illustrated in Figure 1.

    Figure 1.  Numerical solution x(t) = (x_1(t), x_2(t))^T of system (4.1) for the initial value \varphi(t)\equiv (1.5, 2.5)^T.

    Remark 4.1. To the best of our knowledge, few authors have considered the problems related to positive almost periodic solutions of competitive and cooperative Nicholson's blowfly systems. Therefore, the main results in [31,32] and the references therein can not be appled to prove that all solutions of (4.1) with initial the value \varphi\in C^{0} converge exponentially to the positive almost periodic solution. This implies that the results in this paper are new and this complements previously obtained results.

    This article investigated a class of competitive and cooperative Nicholson's blowfly system. Unlike what has been done for some known cooperative Nicholson's blowfly systems [31,32], we have introduced the competitive terms to describe two distinct blowfly populations that compete with each other. By constructing invariant sets and applying the fixed point theorem, we derived some sufficient conditions to ensure that the addressed system has a unique exponential stable positive almost periodic solution. Inspired by the latest Nicholson's blowfly models [3,4,5,6,7,8,9,10], our future works will be devoted to competitive and cooperative Nicholson's blowfly systems involving distinct delays, distributed delays and mixed delays.

    The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

    This work was supported by Natural Scientific Research Fund of Zhejiang Provincial of China (grant nos. LY18A010019, LY16A010018).

    The author declares no conflicts of interest.



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