Global exponential stability of positive almost periodic solutions for a class of two-layer Gilpin–Ayala predator–prey model with time delays

Abstract

This paper mainly considers a class of two-layer Gilpin–Ayala predator–prey models with time delays. By means of Mawhin’s continuation theorem of coincidence degree theory, some new sufficient criteria for the existence of positive almost periodic solutions have been established. We also obtain the global exponential stability of the positive almost periodic solution for this system by constructing appropriate Lyapunov functionals and smart transformations. As an application, an example is given to illustrate the validity of our main results.

1 Introduction

In 1973, Ayala and Gilpin et al. [1] proposed the following model for studying the dynamics of competition in the fruit fly:

$$\begin{aligned} \textstyle\begin{cases} x'(t)=r_{1}x(t) [ 1- ( \frac{x(t)}{K_{1}} ) ^{\theta _{1}}-a_{12}\frac{y(t)}{K_{2}} ] , & \\ y'(t)=r_{2}y(t) [ 1-a_{21}\frac{x(t)}{K_{1}}- ( \frac{y(t)}{K_{2}} ) ^{\theta _{2}} ] , & \end{cases}\displaystyle \end{aligned}$$

(1.1)

where $r_{i}$ is the intrinsic rate of growth of species, $K_{i}$ is the environment carrying capacity of species i in the absence of competition, $\theta _{i}$ provides a nonlinear measure of interspecific interference, and $a_{ij}$ provides a measure of interspecific interference. In the same year, Gilpin and Ayala presented a more realistic and complicated competition model in the literature [2] as follows:

$$\begin{aligned} x'_{i}(t)=r_{i}x_{i}(t) \Biggl[ 1- \biggl( \frac{x_{i}(t)}{K_{i}} \biggr) ^{\theta _{i}}-\sum _{j=1,j\neq i}^{n}a_{ij}(t)\frac{x_{j}(t)}{K_{j}} \Biggr] , \quad i=1,2,\ldots,n, \end{aligned}$$

(1.2)

where $x_{i}(t)$ is the ith-species population density of at time t, $r_{i}$ is the ith-species intrinsic exponential growth rate, $K_{i}$ is the ith-species environment carrying capacity in the absence of competition, $\theta _{i}$ provides a nonlinear measure of intraspecific interference, and $a_{ij}(t)$ ($i\neq j$) is the interspecific competition rate between the ith species and the jth species at time t.

It is generally called to the Gilpin–Ayala population dynamics model such as (1.1) and (1.2). Compared with the Lotka–Volterra population model, the Gilpin–Ayala population model is important and essential due to its wide applications and advantages. In fact, it is easy to see that the rate of change in the size of each species is a nonlinear function of the sizes of the interacting species in the Gilpin–Ayala population model. However, the rate of change in the size of each species is a linear function of the sizes of the interacting species in the Lotka–Volterra population model. In addition, when the value of all nonlinear measure of interspecific interference is equal to 1, the Gilpin–Ayala population model is changed into the Lotka–Volterra population model. Therefore, as soon as it was put forward, the Gilpin–Ayala models have been widely focused and deeply studied by many scholars. For example, in [3], the author proved that system (1.1) is globally stable while $\theta _{i}\geq 1$ and $\theta _{i}<1$ ($i=1,2$). In [4], the authors discussed the structure and global stability of equilibria for system (1.1) with infinite delay. In addition, Li and Lu [5] studied the following Gilpin–Ayala population model, more complicated than (1.2):

$$\begin{aligned} \textstyle\begin{cases} x'_{i}(t)=x_{i}(t) [ r_{i}(t)- \sum_{j=1}^{n}a_{ij}(t)x_{j}^{\alpha _{ij}}(t) ] ,\quad i=1,2,\ldots,m, \\ x'_{i}(t)=x_{i}(t) [ -r_{i}(t)+ \sum_{j=1}^{m}a_{ij}(t)x_{j}^{\alpha _{ij}}(t) - \sum_{j=m+1}^{n}a_{ij}(t)x_{j}^{\alpha _{ij}}(t) ] ,\\ \quad i=m+1,\ldots,n. \end{cases}\displaystyle \end{aligned}$$

(1.3)

They obtained sufficient conditions for the existence of a unique globally attractive periodic solution of system (1.3). For more work on the Gilpin–Ayala population model, one could refer to [6–16] and the references cited therein.

Motivated by the above discussion, in this paper, we mainly study the two-layer Gilpin–Ayala predator–prey model with time delays described by

$$\begin{aligned} \textstyle\begin{cases} x'_{i}(t)=x_{i}(t) [r_{i}(t)- \sum_{k=1}^{n}a_{ik}(t)x_{k}^{\alpha _{ik}}(t)-\sum_{l=1}^{m}b_{il}(t)x_{n+l}^{\beta _{il}}(t) \\ \hphantom{x'_{i}(t)=}{}-\sum_{k=1}^{n}c_{ik}(t)x_{k}^{\gamma _{ik}}(t-\tau _{ik}(t)) -\sum_{l=1}^{m}d_{il}(t)x_{n+l}^{\delta _{il}}(t-\sigma _{il}(t)) ], \\ x'_{n+j}(t)=x_{n+j}(t) [-\hat{r}_{j}(t)+ \sum_{k=1}^{n}\hat{a}_{kj}(t)x_{k}^{\hat{\alpha }_{kj}}(t)-\sum_{l=1}^{m}\hat{b}_{lj}(t)x_{n+l}^{\hat{\beta }_{lj}}(t) \\ \hphantom{x'_{n+j}(t)=}{} +\sum_{k=1}^{n}\hat{c}_{kj}(t)x_{k}^{\hat{\gamma }_{kj}}(t-\hat{\tau }_{kj}(t)) -\sum_{l=1}^{m}\hat{d}_{lj}(t)x_{n+l}^{\hat{\delta }_{lj}}(t-\hat{\sigma }_{lj}(t)) ], \end{cases}\displaystyle \end{aligned}$$

(1.4)

where $i=1,2,\ldots,n$, $j=1,2,\ldots,m$, $x_{i}(t)$ ($i=1,2,\ldots,n$) and $x_{n+j}(t)$ ($j=1,2,\ldots,m$) stand for the population density of ith prey and jth predator at time t, respectively. $r_{i}$ ($i=1,2,\ldots,n$) and $\hat{r}_{j}(t)$ ($j=1,2,\ldots,m$) present the ith prey natural growth rate and the jth predator natural death rate at time t, respectively. $a_{ik}(t)$, $c_{ik}(t)$, $\hat{b}_{lj}(t)$ and $\hat{d}_{lj}(t)$ ($i,k=1,2,\ldots,n$; $j,l=1,2,\ldots,m$) stand for the intraspecific competition rate between preys at time t. $b_{il}(t)$ and $d_{il}(t)$ ($i=1,2,\ldots,n$; $l=1,2,\ldots,m$) are the predation rate of lth predator on ith prey at time t. $\hat{a}_{kj}(t)$ and $\hat{c}_{kj}$ ($k=1,2,\ldots,n$; $j=1,2,\ldots,m$) are the conversion rate of kth prey to jth predator at time t. $\tau _{ik}(t)$, $\sigma _{il}(t)$, $\hat{\tau }_{kj}(t)$ and $\sigma _{lj}(t)$ ($i,k=1,2,\ldots,n$; $j, l=1,2,\ldots,m$) are the discrete time delay at time t. The constants $\alpha _{ik}$, $\beta _{il}$, $\gamma _{ik}$, $\delta _{il}$, $\hat{\alpha }_{kj}$, $\hat{\beta }_{lj}$, $\hat{\gamma }_{kj}$ and $\hat{\delta }_{lj}$ ($i,k=1,2,\ldots,n$; $j, l=1,2,\ldots,m$) provide a nonlinear measure of intraspecific interference.

To the best of our knowledge, few authors have considered the existence and global exponential stability of positive almost periodic solutions for model (1.4). However, the previous papers deal with the permanence, periodic solution and their global attractivity of the Gilpin–Ayala population models. Indeed, the effects of an almost periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, we will establish some sufficient conditions of existence and global exponential stability of positive almost periodic solutions for model (1.4).

The remain of this paper is organized as follows. In Sect. 2, we give some notations and lemmas. In Sect. 3, some new sufficient criteria for the existence of positive almost periodic solutions have been established by means of Mawhin’s continuation theorem of coincidence degree theory. In Sect. 4, we also obtain the global exponential stability of the positive almost periodic solution of this system by constructing appropriate Lyapunov functionals and inequality techniques. As applications, an example and its simulation are given to illustrate the validity of our main results in Sect. 5. Finally, we conclude with the main results and their biological meaning in Sect. 6.

2 Preliminaries

In this section, we first recall some basic definitions and lemmas which are used in what follows.

Definition 2.1

([17])

Let $u(t):\mathbb{R} \rightarrow \mathbb{R}$ be continuous in t. $u(t)$ is said to be almost periodic on $\mathbb{R}$, if, for any $\epsilon >0$, the set $K(u,\epsilon)=\{\delta: \vert u(t+\delta)-u(t)\vert <\epsilon, \forall t\in \mathbb{R}\}$ is relatively dense, that is, for any $\epsilon >0$, it is possible to find a real number $l(\epsilon)>0$, for any interval with length $l(\epsilon)$, there exists a number $\delta =\delta (\epsilon)$ in this interval such that $\vert u(t+\delta)-u(t)\vert <\epsilon$, $\forall t\in \mathbb{R}$.

Definition 2.2

A solution $(x_{1}(t),\ldots,x_{n}(t),x_{n+1}(t),\ldots,x_{n+m}(t))^{T}$ of (1.1) is called an almost periodic solution if and only if $x_{i}(t)$ and $x_{n+j}(t)$ ($i=1,2,\ldots,n$; $j=1,2,\ldots,m$) are almost periodic.

For convenience, we denote $\operatorname{AP}(\mathbb{R})$ is the set of all real valued, almost periodic functions on $\mathbb{R}$. For $f\in \operatorname{AP}(\mathbb{R})$, define

$$\wedge (f)= \biggl\{ \tilde{\lambda }\in \mathbb{R}:\lim_{T\rightarrow \infty } \frac{1}{T} \int _{0}^{T}f(s)e^{-i\tilde{\lambda }s}\,ds\neq 0 \biggr\} $$

and

$$\operatorname{mod}(f)= \Biggl\{ \sum_{i=1}^{N} n_{i}\tilde{\lambda }_{i}:n_{i}\in Z, N\in N^{+},\tilde{\lambda }_{i}\in \wedge (f) \Biggr\} $$

to be the set of Fourier exponents and the module of f, respectively. $m(f)=\frac{1}{T}\int _{0}^{T}f(s)\,ds$ be the mean value of f on interval $[0,T]$, where $T>0$ is a constant. Clearly, $m(f)$ depends on T. $m[f]=\lim_{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}f(s)\,ds$. Suppose that $f(t,\phi)$ is almost periodic in t, uniformly with respect to $\phi \in C([-\sigma,0],\mathbb{R})$, let $K(f,\epsilon,S)$ denote the set of ϵ-almost periods with respect to $S\subset C([-\sigma,0],\mathbb{R})$ and $l(\epsilon,S)$ denote the length of inclusion interval.

Lemma 2.1

([17])

Suppose that f and g are almost periodic. Then the following statements are equivalent:

• (i) $\operatorname{mod}(f)\supset \operatorname{mod}(g)$,

• (ii) for any sequence $\{t_{n}^{*}\}$, if $\lim_{n\rightarrow \infty } f(t+t_{n}^{*})=f(t)$ for each $t\in \mathbb{R}$, then there exists a subsequence $\{t_{n}\}\subseteq \{t_{n}^{*}\}$ such that $\lim_{n\rightarrow \infty }g(t+t_{n})=f(t)$ for each $t\in \mathbb{R}$.

Lemma 2.2

([17])

If $u\in \operatorname{AP}(\mathbb{R})$, then $\int _{t-\tau } ^{t} u(s)\,ds$ is almost periodic.

Lemma 2.3

([17])

If $u(t)\in \operatorname{AP}(\mathbb{R})$, then $u(t)$ is bounded on $\mathbb{R}$.

Lemma 2.4

([18])

If $f(t)\in \operatorname{AP}(\mathbb{R})$, then there exists $t_{0}\in \mathbb{R}$ such that $f(t_{0})=m(f)$.

Lemma 2.5

([18])

Assume that $u(t)\in \operatorname{AP}(\mathbb{R})\cap C^{1}(\mathbb{R},\mathbb{R})$, then there exist two point sequences $\{\xi _{k}\}_{k=1}^{\infty }$ and $\{\eta _{k}\}_{k=1}^{\infty }$ such that $u'(\xi _{k})=u'(\eta _{k})=0$, $\lim_{k\rightarrow \infty }\xi _{k}=\infty$, $\lim_{k\rightarrow \infty }\eta _{k}=-\infty$.

Lemma 2.6

([18])

Assume that $u(t)\in \operatorname{AP}(\mathbb{R})\cap C^{1}(\mathbb{R},\mathbb{R})$, then $u(t)$ falls into one of the following four cases:

• (i) There are $\xi,\eta \in \mathbb{R}$ such that $u(\xi)=\sup_{t\in \mathbb{R}}u(t)$ and $u(\eta)=\inf_{t\in \mathbb{R}}u(t)$. In this case, $u'(\xi)=u'(\eta)=0$.

• (ii) There are no $\xi, \eta \in \mathbb{R}$ such that $u(\xi)=\sup_{t\in \mathbb{R}}u(t)$ and $u(\eta)=\inf_{t\in \mathbb{R}}u(t)$. In this case, for all $\epsilon >0$, there exist two points $\xi,\eta \in \mathbb{R}$ such that $u'(\xi)=u'(\eta)=0$, $u(\xi)>\sup_{t\in \mathbb{R}}u(t)-\epsilon$ and $u(\eta)<\inf_{t\in \mathbb{R}}u(t)+\epsilon$.

• (iii) There is a $\xi \in \mathbb{R}$ such that $u(\xi)=\sup_{t\in \mathbb{R}}u(t)$ and there is no $\eta \in \mathbb{R}$ such that $u(\eta)=\inf_{t\in \mathbb{R}}u(t)$. In this case, $u'(\xi)=0$ and for all $\epsilon >0$, there exists an $\eta \in \mathbb{R}$ such that $u'(\eta)=0$ and $u(\eta)<\inf_{t\in \mathbb{R}}u(t)+\epsilon$.

• (vi) There is an $\eta \in \mathbb{R}$ such that $u(\eta)=\inf_{t\in \mathbb{R}}u(t)$ and there is no $\xi \in \mathbb{R}$ such that $u(\eta)=\sup_{t\in \mathbb{R}}u(t)$. In this case, $u'(\eta)=0$ and for all $\epsilon >0$, there exists an $\xi \in \mathbb{R}$ such that $u'(\xi)=0$ and $u(\xi)>\sup_{t\in \mathbb{R}}u(t)-\epsilon$.

For the sake of convenience, we denote $v^{l}=\inf_{t\in \mathbb{R}}v(t)$, $v^{M}=\sup_{t\in \mathbb{R}}v(t)$, here $v(t)$ is a continuous almost periodic function on $\mathbb{R}$. For simplicity, we need to introduce some notations as follows:

$$\begin{aligned}& l_{i}^{+}= \biggl( \frac{r_{i}^{M}}{a_{ii}^{l}} \biggr) ^{\frac{1}{\alpha _{ii}}},\qquad l_{n+j}^{+}= \biggl( \frac{ \sum_{k=1}^{n}\hat{a}_{kj}^{M}(l_{k}^{+})^{\hat{\alpha }_{kj}} +\sum_{k=1}^{n}\hat{c}_{kj}^{M}(l_{k}^{+})^{\hat{\gamma }_{kj}}}{\hat{b}_{jj}^{l}} \biggr) ^{\frac{1}{\hat{\beta }_{jj}}}, \\& l_{i}^{-}= \biggl( \frac{r_{i}^{l}- \sum_{k=1,k\neq i}^{n}a_{ik}^{M}(l_{k}^{+})^{\alpha _{ik}} -\sum_{l=1}^{m}b_{il}^{M}(l_{n+l}^{+})^{\beta _{il}} -\sum_{k=1}^{n}c_{ik}^{M}(l_{k}^{+})^{\gamma _{ik}}-\sum_{l=1}^{m}d_{il}^{M}(l_{n+l}^{+})^{\delta _{il}}}{a_{ii}^{M}} \biggr) ^{\frac{1}{\alpha _{ii}}}, \\& l_{n+j}^{-}= \biggl( \frac{ \sum_{k=1}^{n}\hat{a}_{kj}^{l}(l_{k}^{-})^{\hat{\alpha }_{kj}} + \sum_{k=1}^{n}\hat{c}_{kj}^{l}(l_{k}^{-})^{\hat{\gamma }_{kj}}-\hat{r}_{j}^{M} - \sum_{l=1,l\neq j}^{m}\hat{b}_{lj}^{M}(l_{n+l}^{+})^{\hat{\beta }_{lj}} -\sum_{l=1}^{m}\hat{d}_{lj}^{M}(l_{n+l}^{+})^{\hat{\delta }_{lj}}}{\hat{b}_{jj}^{M}} \biggr) ^{\frac{1}{\hat{\beta }_{jj}}}, \end{aligned}$$

where $i=1,2,\ldots,n$, $j=1,2,\ldots,m$.

Throughout this paper, we need the following assumptions.

$(H_{1})$ :

$r_{i}(t)$, $\hat{r}_{j}(t)$, $a_{ik}(t)$, $b_{il}(t)$, $c_{ik}(t)$, $d_{il}(t)$, $\hat{a}_{kj}(t)$, $\hat{b}_{lj}(t)$, $\hat{c}_{kj}(t)$, $\hat{d}_{lj}(t)\ (i,k=1,2,\ldots,n;j, l=1,2,\ldots,m)\in C(\mathbb{R},\mathbb{R}^{+})$ and $\tau _{ik}(t)$, $\sigma _{il}(t)$, $\hat{\tau }_{kj}(t)$, $\sigma _{lj}(t)\ (i,k=1,2,\ldots,n;j, l=1,2,\ldots, m)\in C(\mathbb{R},\mathbb{R}_{0}^{+})$ are all continuous positive almost periodic functions with respect to the time variable t, where $\mathbb{R}^{-}=(-\infty,0)$, $\mathbb{R}^{+}=(0,\infty)$, $\mathbb{R}^{+}_{0}=[0,\infty)$, $\mathbb{R}=(-\infty,+\infty)$.

$(H_{2})$ :

The nonlinear measures of intraspecific interference $\alpha _{ik}$, $\beta _{il}$, $\gamma _{ik}$, $\delta _{il}$, $\hat{\alpha }_{kj}$, $\hat{\beta }_{lj}$, $\hat{\gamma }_{kj}$ and $\hat{\delta }_{lj}$ ($i,k=1,2,\ldots,n$; $j, l=1,2,\ldots,m$) are all the positive constants, that is, $\alpha _{ik}>0$, $\beta _{il}>0$, $\gamma _{ik}>0$, $\delta _{il}>0$, $\hat{\alpha }_{kj}>0$, $\hat{\beta }_{lj}>0$, $\hat{\gamma }_{kj}>0$, $\hat{\delta }_{lj}>0$.

$(H_{3})$ :

$r_{i}^{l}> \sum_{k=1,k\neq i}^{n}a_{ik}^{M}(l_{k}^{+})^{\alpha _{ik}} +\sum_{l=1}^{m}b_{il}^{M}(l_{n+l}^{+})^{\beta _{il}} +\sum_{k=1}^{n}c_{ik}^{M}(l_{k}^{+})^{\gamma _{ik}}+\sum_{l=1}^{m}d_{il}^{M}(l_{n+l}^{+})^{\delta _{il}}$ and $\sum_{k=1}^{n}\hat{a}_{kj}^{l}(l_{k}^{-})^{\hat{\alpha }_{kj}} +\sum_{k=1}^{n}\hat{c}_{kj}^{l}(l_{k}^{-})^{\hat{\gamma }_{kj}}>\hat{r}_{j}^{M} + \sum_{l=1,l\neq j}^{m}\hat{b}_{lj}^{M}(l_{n+l}^{+})^{\hat{\beta }_{lj}} +\sum_{l=1}^{m}\hat{d}_{lj}^{M}(l_{n+l}^{+})^{\hat{\delta }_{lj}}$, $i=1,2,\ldots,n$; $j=1,2,\ldots,m$.

3 Existence of positive almost periodic solution

In this section, by using Mawhin’s continuation theorem, we shall show the existence of positive almost periodic solutions of (1.1).

Let X and Z be Banach spaces. $L : \operatorname{Dom}(L)\subset X\rightarrow Z$ be a linear mapping and $N: X \times [0,1]\rightarrow Z$ is a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if $\operatorname{dim} \operatorname{Ker}(L)=\operatorname{codim} \operatorname{Im}(L) < \infty$ and $\operatorname{Im}(L)$ is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors $P:X\rightarrow X$ and $Q:Z\rightarrow Z$ such that $\operatorname{Im}(P)=\operatorname{Ker}(L)$ and $\operatorname{Ker}(Q)=\operatorname{Im}(L)=\operatorname{Im}(I-Q)$, and $X=\operatorname{Ker}(L)\oplus \operatorname{Ker}(P)$, $Z=\operatorname{Im}(L)\oplus \operatorname{Im}(Q)$. It follows that $L \vert _{\operatorname{Dom}(L)\bigcap \operatorname{Ker}(P)}:(I-P)X\rightarrow \operatorname{Im}(L)$ is invertible and its inverse is denoted by $K_{P}$. If Ω is a bounded open subset of X, the mapping N is called L-compact on $\overline{\Omega }\times [0,1]$, if $QN(\overline{\Omega }\times [0,1])$ is bounded and $K_{p}(I-Q)N$: $\overline{\Omega }\times [0,1]\rightarrow X$ is compact, where I is the identity. Because $\operatorname{Im}(Q)$ is isomorphic to $\operatorname{Ker}(L)$, there exists an isomorphism J: $\operatorname{Im}(Q)\rightarrow \operatorname{Ker}(L)$.

Let L be a Fredholm linear mapping with index zero and let N be a L-compact mapping on Ω̅. Define mapping $F: \operatorname{Dom}(L)\cap \overline{\Omega }\rightarrow Z$ by $F=L-N$. If $Lx\neq Nx$ for all $x\in \operatorname{Dom}(L)\cap \partial \Omega$, then by using P, Q, $K_{P}$, J defined above, the coincidence degree of F in Ω with respect to L is defined by

$$\begin{aligned} \operatorname{deg}_{L}(F,\Omega)=\operatorname{deg}\bigl(I-P- \bigl(J^{-1}Q+K_{P}(I-Q)\bigr)N,\Omega,0\bigr), \end{aligned}$$

where $\operatorname{deg}(g,\Gamma,p)$ is the Leray–Schauder degree of g at p relative to Γ.

The Mawhin’s continuous theorem [19], p. 40, is given as follows:

Lemma 3.1

([19])

Let L be a Fredholm mapping of index zero and N be L-compact on $\overline{\Omega }\times [0,1]$. Assume

• (a) for each $\lambda \in (0,1)$, every solution x of $Lx=\lambda N(x,\lambda)$ is such that $x \notin \partial \Omega \cap \operatorname{Dom}(L)$;

• (b) $QN(x,0)\neq 0$ for each $x\in \partial \Omega \cap \operatorname{Ker}(L)$;

• (c) $\operatorname{deg}(JQN(x,0),\Omega \cap \operatorname{Ker}(L),0)\neq 0$.

Then $Lx=N(x,1)$ has at least one solution in $\overline{\Omega }\cap \operatorname{Dom}(L)$.

By making the substitution $x_{i}(t)=e^{u_{i}(t)}$ ($i=1,2,\ldots,n$), $x_{n+j}(t)=e^{u_{n+j}(t)}$ ($j=1,2,\ldots,m$), then system (1.4) is rewritten in the form of

$$\begin{aligned} \textstyle\begin{cases} u'_{i}(t)=r_{i}(t)- \sum_{k=1}^{n}a_{ik}(t)e^{\alpha _{ik}u_{k}(t)} -\sum_{l=1}^{m}b_{il}(t)e^{\beta _{il}u_{n+l}(t)}\\ \hphantom{u'_{i}(t)=}{}-\sum_{k=1}^{n}c_{ik}(t)e^{\gamma _{ik}u_{k}(t-\tau _{ik}(t))} \\ \hphantom{u'_{i}(t)=}{} -\sum_{l=1}^{m}d_{il}(t)e^{\delta _{il}u_{n+l}(t-\sigma _{il}(t))},\quad i=1,2,\ldots,n, \\ u'_{n+j}(t)=-\hat{r}_{j}(t)+ \sum_{k=1}^{n}\hat{a}_{kj}(t)e^{\hat{\alpha }_{kj}u_{k}(t)}-\sum_{l=1}^{m}\hat{b}_{lj}(t)e^{\hat{\beta }_{lj}u_{n+l}(t)} \\ \hphantom{u'_{n+j}(t)=}{}+\sum_{k=1}^{n}\hat{c}_{kj}(t)e^{\hat{\gamma }_{kj}u_{k}(t-\hat{\tau }_{kj}(t))} \\ \hphantom{u'_{n+j}(t)=}{} -\sum_{l=1}^{m}\hat{d}_{lj}(t)e^{\hat{\delta }_{lj}u_{n+l}(t-\hat {\sigma }_{lj}(t))}, \quad j=1,2,\ldots,m. \end{cases}\displaystyle \end{aligned}$$

(3.1)

Let $X=Z=V_{1}\oplus V_{2}$, where

$$\begin{aligned}& V_{1}= \bigl\{ w(t)=\bigl(w_{1}(t),w_{2}(t), \ldots,w_{n+m}(t)\bigr)^{T}: w_{k}(t)\in \operatorname{AP}( \mathbb{R}),\operatorname{mod}\bigl(w_{k}(t)\bigr)\subset \operatorname{mod}\bigl(F_{k}(t)\bigr), \\& \hphantom{V_{1}=}{}\forall \tilde{\lambda }_{k}\in \wedge \bigl(w_{k}(t) \bigr) \text{ satisfies } \vert \tilde{\lambda }_{k}\vert >\beta,k=1,2,\ldots,n+m \bigr\} , \\& V_{2}=\bigl\{ w(t)\equiv (c_{1},c_{2},\ldots,c_{n+m})\in \mathbb{R}^{n+m}\bigr\} , \\& F_{i}(t)=r_{i}(t)- \sum_{k=1}^{n}a_{ik}(t)e^{\alpha _{ik}\phi _{k}(0)}- \sum_{l=1}^{m}b_{il}(t)e^{\beta _{il}\phi _{n+l}(0)} -\sum_{k=1}^{n}c_{ik}(t)e^{\gamma _{ik}\phi _{k}(-\tau _{ik}(t))} \\& \hphantom{F_{i}(t)=}{}-\sum_{l=1}^{m}d_{il}(t)e^{\delta _{il}\phi _{n+l}(-\sigma _{il}(t))},\quad i=1,2,\ldots,n, \\& F_{n+j}(t)=-\hat{r}_{j}(t)+ \sum _{k=1}^{n}\hat{a}_{kj}(t)e^{\hat{\alpha }_{kj}\phi _{k}(0)}- \sum_{l=1}^{m}\hat{b}_{lj}(t)e^{\hat{\beta }_{lj}\phi _{n+l}(0)} +\sum_{k=1}^{n}\hat{c}_{kj}(t)e^{\hat{\gamma }_{kj}\phi _{k}(-\hat{\tau }_{kj}(t))} \\& \hphantom{F_{n+j}(t)=}{}-\sum_{l=1}^{m}\hat{d}_{lj}(t)e^{\hat{\delta }_{lj}\phi _{n+l}(-\hat{\sigma }_{lj}(t))}, \quad j=1,2,\ldots,m, \end{aligned}$$

in which $\phi _{k}\in C([-\sigma,0],\mathbb{R})$, $k=1,2,\ldots,n+m$, $\sigma = \max_{1\leq i,k\leq n,1\leq j,l\leq m}\sup_{t\in \mathbb{R}}\{\tau _{ik}(t),\sigma _{il}(t), \hat{\tau }_{ik}(t),\hat{\sigma }_{il}(t)\}$, and β is a given constant. Define the norm

$$\begin{aligned} \Vert w\Vert =\max_{1\leq k\leq n+m}\sup _{t\in \mathbb{R}}\bigl\vert w_{k}(t)\bigr\vert ,\quad \forall w=(w_{1},w_{2},\ldots,w_{n+m})^{T}\in X=Z. \end{aligned}$$

(3.2)

Lemma 3.2

$X=Z$ is a Banach space equipped with the norm $\Vert \cdot \Vert$ defined by (3.2).

Proof

Assume that $\{w^{\{k\}}=(w_{1}^{\{k\}},w_{2}^{\{k\}},\ldots,w_{n+m}^{\{k\}})^{T}\}\subset V_{1}$ converge to $\overline{w}=(\overline{w}_{1},\overline{w}_{2},\ldots,\overline{w}_{n+m})^{T}$, that is, $w_{h}^{\{k\}}\rightarrow \overline{w}_{h}$, as $k\rightarrow \infty$, $h=1,2,\ldots,n+m$. Then it is easy to show that $\overline{w}_{h} \in \operatorname{AP}(\mathbb{R})$ and $\operatorname{mod}(\overline{w}_{h})\in \operatorname{mod}(F_{h})$. For any $\vert \widetilde{\lambda }_{h}\vert \leq \beta$, we have

$$\begin{aligned} \lim_{T\rightarrow \infty }\frac{1}{T} \int _{0}^{T} w_{h}^{\{k\}}(t)e^{-i\widetilde{\lambda }_{h} t}\,dt=0, \quad h=1,2,\ldots,n+m, \end{aligned}$$

therefore,

$$\begin{aligned} \lim_{T\rightarrow \infty }\frac{1}{T} \int _{0}^{T} \overline{w}_{h}^{\{k\}}(t)e^{-i\widetilde{\lambda }_{h} t}\,dt=0,\quad h=1,2,\ldots,n+m, \end{aligned}$$

which implies $\overline{w}\in V_{1}$. Then it is not difficult to see that $V_{1}$ is a Banach space equipped with the norm $\Vert \cdot \Vert$. Thus, we can easily verify that X and Z are Banach spaces equipped with the norm $\Vert \cdot \Vert$. The proof is complete. □

Lemma 3.3

Let $L: X\rightarrow Z$, $Lw=\frac{dw(t)}{dt}$, then L is a Fredholm mapping of index zero.

Proof

Clearly, L is a linear operator and $\operatorname{Ker}(L)=V_{2}$. We claim that $\operatorname{Im}(L)=V_{1}$. In fact, we suppose that $z(t)=(z_{1}(t),z_{2}(t),\ldots,z_{n+m}(t))^{T}\in \operatorname{Im}(L)\subset Z$, then there exist $z^{\{1\}}(t)=(z_{1}^{\{1\}}(t),z_{2}^{\{1\}}(t),\ldots,z_{n+m}^{\{1\}}(t))^{T}\in V_{1}$ and constant vector $z^{\{2\}}=\{z_{1}^{\{2\}}(t),z_{2}^{\{2\}}(t),\ldots,z_{n+m}^{\{2\}}(t)\} \in V_{2}$ such that $z(t)=z^{\{1\}}(t)+z^{\{2\}}$, that is, $z_{h}(t)=z_{h}^{\{1\}}(t)+z_{h}^{\{2\}}$, $h=1,2,\ldots,n+m$. From Lemma 2.2 and the definitions of $z_{i}(t)$ and $z_{i}^{\{1\}}(t)$, we know that $\int _{t-\sigma }^{t} z_{h}(s)\,ds$ and $\int _{t-\sigma }^{t} z_{h}^{\{1\}}(s)\,ds$ are almost periodic functions. So we have $z_{h}^{\{2\}}\equiv 0$, $h=1,2,\ldots,n+m$, then $z^{\{2\}}\equiv (0,0,\ldots,0)^{T}$, which implies $z(t)\in V_{1}$, that is, $\operatorname{Im}(L)\subset V_{1}$.

On the other hand, if $w(t)=(w_{1}(t),w_{2}(t),\ldots,w_{n+m}(t))^{T}\in V_{1}\setminus \{(0,0,\ldots,0)^{T}\}$, then we have $\int _{0}^{t} w_{h}(s)\,ds \in \operatorname{AP}(\mathbb{R})$, $h=1,2,\ldots,n+m$. If $\widetilde{\lambda }_{h}\neq 0$, then we obtain

$$\begin{aligned} \lim_{T\rightarrow \infty }\frac{1}{T} \int _{0}^{T} \biggl( \int _{0}^{t}w_{h}(s)\,ds \biggr)e^{-i\widetilde{\lambda }_{j} t}\,dt=\frac{1}{i\widetilde{\lambda }_{h}}\lim_{T\rightarrow \infty } \frac{1}{T} \int _{0}^{T} w_{h}(t)e^{-i\widetilde{\lambda }_{h}t}\,dt, \end{aligned}$$

$h=1,2,\ldots,n+m$. It follows that

$$\begin{aligned} \wedge \biggl[ \int _{0} ^{t} w_{h}(s)\,ds-m \biggl( \int _{0}^{t}w_{h}(s)\,ds \biggr) \biggr] =\wedge \bigl(w_{h}(t)\bigr),\quad h=1,2,\ldots,n+m, \end{aligned}$$

hence

$$\begin{aligned} \int _{0}^{t} w(s)\,ds -m \biggl( \int _{0}^{t} w(s)\,ds \biggr) \in V_{1} \subset X. \end{aligned}$$

Note that $\int _{0}^{t} w(s)\,ds-m(\int _{0}^{t} w(s)\,ds)$ is the primitive of $w(t)$ in X, we have $w(t)\in \operatorname{Im}(L)$, that is, $V_{1} \subset \operatorname{Im}(L)$. Therefore, $\operatorname{Im}(L)=V_{1}$.

Furthermore, one easily shows that $\operatorname{Im}(L)$ is closed in Z and

$$\begin{aligned} \operatorname{dim} \operatorname{ker}(L)=n+m=\operatorname{codim} \operatorname{Im}(L). \end{aligned}$$

Therefore, L is a Fredholm mapping of index zero. The proof is complete. □

Lemma 3.4

Let $N:X\times (0,1)\rightarrow Z$, $N(u(t),\lambda)=(N_{1}^{u}, N_{2}^{u},\ldots,N_{n+m}^{u})^{T}$, where

$$\begin{aligned} N_{i}^{u}={}&r_{i}(t)-a_{ii}(t)e^{\alpha _{ii}u_{i}(t)}- \lambda \sum_{k=1,k\neq i}^{n}a_{ik}(t)e^{\alpha _{ik}u_{k}(t)} -\lambda \sum_{l=1}^{m}b_{il}(t)e^{\beta _{il}u_{n+l}(t)} \\ &{}-\lambda \sum_{k=1}^{n}c_{ik}(t)e^{\gamma _{ik}u_{k}(t-\tau _{ik}(t))} -\lambda \sum_{l=1}^{m}d_{il}(t)e^{\delta _{il}u_{n+l}(t-\sigma _{il}(t))},\quad i=1,2,\ldots,n, \end{aligned}$$

and

$$\begin{aligned} N_{n+j}^{u}={}&-\hat{r}_{j}(t)- \hat{b}_{jj}(t)e^{\hat{\beta }_{jj}u_{n+j}(t)}+ \sum_{k=1}^{n} \hat{a}_{kj}(t)e^{\hat{\alpha }_{kj}u_{k}(t)} -\lambda \sum _{l=1,l\neq j}^{m}\hat{b}_{lj}(t)e^{\hat{\beta }_{lj}u_{n+l}(t)} \\ &{}+\sum_{k=1}^{n}\hat{c}_{kj}(t)e^{\hat{\gamma }_{kj}u_{k}(t-\hat{\tau }_{kj}(t))} -\lambda \sum_{l=1}^{m} \hat{d}_{lj}(t)e^{\hat{\delta }_{lj}u_{n+l}(t-\hat{\sigma }_{lj}(t))}, \quad j=1,2,\ldots,m. \end{aligned}$$

Define the projectors

$$\begin{aligned} &P:X\rightarrow Z,\qquad Pu=\bigl(m(u_{1}),m(u_{2}),\ldots,m(u_{n+m})\bigr)^{T}, \\ &Q:Z\rightarrow Z,\qquad Qz=\bigl(m[z_{1}],m[z_{2}],\ldots,m[z_{n+m}]\bigr)^{T}. \end{aligned}$$

Then N is L-compact on Ω̅, where Ω is an open bounded subset of X.

Proof

Obviously, P and Q are continuous projectors such that

$$\begin{aligned} \operatorname{Im}(P)=\operatorname{Ker}(L), \qquad \operatorname{Im}(L)=\operatorname{Ker}(Q). \end{aligned}$$

It is clear that $(I-Q)V_{2}=\{(0,0,\ldots,0)\}$, $(I-Q)V_{1}=V_{1}$. Hence

$$\begin{aligned} \operatorname{Im}(I-Q)=V_{1}=\operatorname{Im}(L). \end{aligned}$$

In view of

$$\begin{aligned} \operatorname{Im}(P)=\operatorname{Ker}(L),\qquad \operatorname{Im}(L)=\operatorname{Ker}(Q)=\operatorname{Im}(I-Q), \end{aligned}$$

we find that the inverse $K_{P}$: $\operatorname{Im}(L)\rightarrow \operatorname{Ker}(P)\cap \operatorname{Dom}(L)$ of $L_{P}$ exists and is given by

$$\begin{aligned} K_{P}(z)= \int _{0}^{t} z(s)\,ds-m \biggl[ \int _{0}^{t} z(s)\,ds \biggr] . \end{aligned}$$

Thus,

$$\begin{aligned}& QNu=\bigl(m\bigl[H_{1}^{u}\bigr],m\bigl[H_{2}^{u} \bigr],\ldots,m\bigl[H_{n+m}^{u}\bigr]\bigr)^{T}, \\& K_{P}(I-Q)Nu=(f(u_{1})-Q(f(u_{1})),f(u_{2})-Q(f(u_{2})),\ldots,f(u_{n+m})-Q(f(u_{n+m})))^{T}, \end{aligned}$$

where

$$\begin{aligned} f(u_{h})= \int _{0}^{t} \bigl(H_{h}^{u}-m \bigl[H_{h}^{u}\bigr]\bigr)\,ds,\quad h=1,2,\ldots,n+m. \end{aligned}$$

Clearly, QN and $(I-Q)N$ are continuous. Now we show that $K_{P}$ is also continuous. By assumptions, for any $0<\epsilon <1$, let $l_{h}(\epsilon _{h},S)$ be the length of the inclusion interval of $K_{h}(F_{h},\epsilon _{h},S)$, $h=1,2,\ldots,n+m$. Suppose that $\{z^{k}(t)\}\subset \operatorname{Im}(L)=V_{1}$ and $z^{k}(t)=(z_{1}^{k}(t),z_{2}^{k}(t),\ldots,z_{n+m}^{k}(t))^{T}$ uniformly converges to $\overline{z}(t)=(\overline{z}_{1}(t),\overline{z}_{2}(t),\ldots,\overline{z}_{n+m}(t))^{T}$, that is $z_{h}^{k}\rightarrow \overline{z}_{h}$, as $k\rightarrow \infty$, $h=1,2,\ldots,n+m$. Because of $\int _{0}^{t} z^{k}(s)\,ds\in Z$, $k=1,2,\ldots$ , there exists $\sigma _{h}$ ($0<\sigma _{h} <\epsilon _{h}$) such that $K_{h}(F_{h},\sigma _{h},S)\subset K_{h}(\int _{0}^{t} z_{h}^{k}(s)\,ds,\sigma _{h})$, $h=1,2,\ldots,n+m$. Let $l_{h}(\sigma _{h},S)$ be the length of the inclusion interval of $K_{h}(F_{h},\sigma _{h},S)$, $h=1,2,\ldots,n+m$ and $l_{h}=\max \{l_{h}(\epsilon _{h},S),l_{h}(\sigma _{h},S)\}$, $h=1,2,\ldots,n+m$. It is easy to see that $l_{h}$ is the length of the inclusion interval of $K_{h}(F_{h},\sigma _{h},S)$ and $K_{h}(F_{h},\epsilon _{h},S)$, $h=1,2,\ldots,n+m$. Hence, for any $t \notin [0,l_{h}]$, there exists $\xi _{t}\in K_{h}(F_{h},\sigma _{h},S)\subset K_{h}(\int _{0}^{t}z_{h}^{k}(s)\,ds,\sigma _{h},S)$ such that $t+\xi _{t} \in [0,l_{h}]$, $h=1,2,\ldots,n+m$. So, by the definition of almost periodic function we have

$$\begin{aligned} &\biggl\Vert \int _{0}^{t} z^{k}(s)\,ds\biggr\Vert \\ &\quad = \max_{1\leq h\leq n+m}\sup_{t\in \mathbb{R}} \biggl\vert \int _{0}^{t} z_{h}^{k}(s)\,ds \biggr\vert \\ &\quad \leq \max_{1\leq h\leq n+m} \sup_{t\in [0,l_{h}]} \biggl\vert \int _{0}^{t} z_{h}^{k}(s)\,ds \biggr\vert \\ &\qquad {} +\max_{1\leq h\leq n+m} \sup_{t \notin [0,l_{h}]} \biggl\vert \int _{0}^{t} z_{h}^{k}(s)\,ds- \int _{0}^{t+\xi _{l}} z_{h}^{k}(s)\,ds+ \int _{0}^{t+\xi _{l}} z_{h}^{k}(s)\,ds \biggr\vert \\ &\quad \leq 2\max_{1\leq h\leq n+m} \sup_{t\in [0,l_{h}]} \biggl\vert \int _{0}^{t} z_{h}^{k}(s)\,ds \biggr\vert \\ &\qquad {}+\max_{1\leq h\leq n+m} \sup_{t \notin [0,l_{h}]} \biggl\vert \int _{0}^{t} z_{h}^{k}(s)\,ds- \int _{0}^{t+\xi _{l}} z_{h}^{k}(s)\,ds \biggr\vert \\ & \quad \leq 2\max_{1\leq h\leq n+m} \biggl\vert \int _{0}^{l_{h}} z_{h}^{k}(s)\,ds \biggr\vert +\max_{1\leq h\leq n+m} \epsilon _{h}. \end{aligned}$$

(3.3)

From (3.3), we conclude that $\int _{0}^{t} z(s)\,ds$ is continuous, where $z(t)=(z_{1}(t),z_{2}(t), \ldots,z_{n+m}(t))^{T}\in \operatorname{Im}(L)$. Consequently, $K_{P}$ and $K_{P} (I-Q)Ny$ are continuous. Meanwhile, we also have $\int _{0}^{t} z(s)\,ds$ and $K_{P}(I-P)Nu$ are uniformly bounded in Ω̅. Further, it is not difficult to verify that $QN(\overline{\Omega })$ is bounded and $K_{P}(I-Q)Nu$ is equicontinuous in Ω̅. By the Arzela–Ascoli theorem, we have immediately conclude that $K_{P}(I-Q)N(\overline{\Omega })$ is compact. Thus N is L-compact on Ω̅. The proof is complete. □

Theorem 3.1

Assume that $(H_{1})$–$(H_{3})$ hold. Then system (1.4) has at least one positive almost periodic solution.

Proof

In order to use Lemma 3.1 establishing the existence of positive almost periodic solutions of (1.1), we set Banach space X and Z the same as those in Lemma 3.2 and set mappings L, N, P, Q the same as those in Lemma 3.3 and Lemma 3.4, respectively. Then we find that L is a Fredholm mapping of index zero and N is a continuous operator which is L-compact on Ω̄.

Now we are in the position of searching for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation

$$\begin{aligned} Lu=\lambda N(u,\lambda),\quad \lambda \in (0,1), \end{aligned}$$

we obtain

$$\begin{aligned} u'_{i}(t)={}&\lambda \Biggl[r_{i}(t)-a_{ii}(t)e^{\alpha _{ii}u_{i}(t)}-\lambda \sum _{k=1,k\neq i}^{n}a_{ik}(t)e^{\alpha _{ik}u_{k}(t)} -\lambda \sum_{l=1}^{m}b_{il}(t)e^{\beta _{il}u_{n+l}(t)} \\ &{}-\lambda \sum_{k=1}^{n}c_{ik}(t)e^{\gamma _{ik}u_{k}(t-\tau _{ik}(t))} -\lambda \sum_{l=1}^{m}d_{il}(t)e^{\delta _{il}u_{n+l}(t-\sigma _{il}(t))} \Biggr],\quad i=1,2,\ldots,n, \end{aligned}$$

(3.4)

and

$$\begin{aligned} u'_{n+j}(t)={}&\lambda \Biggl[- \hat{r}_{j}(t)-\hat{b}_{jj}(t)e^{\hat{\beta }_{jj}u_{n+j}(t)}+ \sum _{k=1}^{n}\hat{a}_{kj}(t)e^{\hat{\alpha }_{kj}u_{k}(t)} -\lambda \sum_{l=1,l\neq j}^{m} \hat{b}_{lj}(t)e^{\hat{\beta }_{lj}u_{n+l}(t)} \\ &{}+\sum_{k=1}^{n}\hat{c}_{kj}(t)e^{\hat{\gamma }_{kj}u_{k}(t-\hat{\tau }_{kj}(t))} -\lambda \sum_{l=1}^{m} \hat{d}_{lj}(t)e^{\hat{\delta }_{lj}u_{n+l}(t-\hat{\sigma }_{lj}(t))} \Biggr],\quad j=1,2,\ldots,m. \end{aligned}$$

(3.5)

Assume that $u\in X$ is an almost periodic solution of (3.4)–(3.5), for some $\lambda \in (0,1)$. Then, by Lemma 2.6, for any $\epsilon >0$, there exist $\xi _{h},\eta _{h}\in \mathbb{R}$ such that $u_{h}(\xi _{h})>u_{h}^{M}-\epsilon$, $u_{h}(\eta _{h})< u_{h}^{l}+\epsilon$ and $u'_{h}(\xi _{h})=u'_{h}(\eta _{h})=0$, $h=1,2,\ldots,n+m$. From this and (3.4)–(3.5), for $i=1,2,\ldots,n$, $j=1,2,\ldots,m$, we have

$$\begin{aligned} &r_{i}(\xi _{i})-a_{ii}( \xi _{i})e^{\alpha _{ii}u_{i}(\xi _{i})}-\lambda \sum_{k=1,k\neq i}^{n}a_{ik}( \xi _{i})e^{\alpha _{ik}u_{k}(\xi _{i})} -\lambda \sum_{l=1}^{m}b_{il}( \xi _{i})e^{\beta _{il}u_{n+l}(\xi _{i})} \\ &\quad {}-\lambda \sum_{k=1}^{n}c_{ik}( \xi _{i})e^{\gamma _{ik}u_{k}(\xi _{i}-\tau _{ik}(\xi _{i}))} -\lambda \sum_{l=1}^{m}d_{il}( \xi _{i})e^{\delta _{il}u_{n+l}(\xi _{i}-\sigma _{il}(\xi _{i}))}=0, \end{aligned}$$

(3.6)

$$\begin{aligned} &{-}\hat{r}_{j}(\xi _{n+j})- \hat{b}_{jj}(\xi _{n+j})e^{\hat{\beta }_{jj}u_{n+j}(\xi _{n+j})} + \sum _{k=1}^{n}\hat{a}_{kj}(\xi _{n+j})e^{\hat{\alpha }_{kj}u_{k}(\xi _{n+j})} -\lambda \sum_{l=1,l\neq j}^{m} \hat{b}_{lj}(\xi _{n+j})e^{\hat{\beta }_{lj}u_{n+l}(\xi _{n+j})} \\ &\quad {}+\sum_{k=1}^{n}\hat{c}_{kj}( \xi _{n+j})e^{\hat{\gamma }_{kj}u_{k}(\xi _{n+j}-\hat{\tau }_{kj}(\xi _{n+j}))} -\lambda \sum_{l=1}^{m} \hat{d}_{lj}(\xi _{n+j})e^{\hat{\delta }_{lj}u_{n+l}(\xi _{n+j}-\hat{\sigma }_{lj}(\xi _{n+j}))}=0, \end{aligned}$$

(3.7)

$$\begin{aligned} &r_{i}(\eta _{i})-a_{ii}( \eta _{i})e^{\alpha _{ii}u_{i}(\eta _{i})}-\lambda \sum_{k=1,k\neq i}^{n}a_{ik}( \eta _{i})e^{\alpha _{ik}u_{k}(\eta _{i})} -\lambda \sum_{l=1}^{m}b_{il}( \eta _{i})e^{\beta _{il}u_{n+l}(\eta _{i})} \\ &\quad {}-\lambda \sum_{k=1}^{n}c_{ik}( \eta _{i})e^{\gamma _{ik}u_{k}(\eta _{i}-\tau _{ik}(\eta _{i}))} -\lambda \sum_{l=1}^{m}d_{il}( \eta _{i})e^{\delta _{il}u_{n+l}(\eta _{i}-\sigma _{il}(\eta _{i}))}=0, \end{aligned}$$

(3.8)

and

$$\begin{aligned} &{-}\hat{r}_{j}(\eta _{n+j})- \hat{b}_{jj}(\eta _{n+j})e^{\hat{\beta }_{jj}u_{n+j}(\eta _{n+j})} + \sum _{k=1}^{n}\hat{a}_{kj}(\eta _{n+j})e^{\hat{\alpha }_{kj}u_{k}(\eta _{n+j})} -\lambda \sum_{l=1,l\neq j}^{m} \hat{b}_{lj}(\eta _{n+j})e^{\hat{\beta }_{lj}u_{n+l}(\eta _{n+j})} \\ &\quad {}+\sum_{k=1}^{n}\hat{c}_{kj}( \eta _{n+j})e^{\hat{\gamma }_{kj}u_{k}(\eta _{n+j}-\hat{\tau }_{kj}(\eta _{n+j}))} -\lambda \sum_{l=1}^{m} \hat{d}_{lj}(\eta _{n+j})e^{\hat{\delta }_{lj}u_{n+l}(\eta _{n+j}-\hat{\sigma }_{lj} (\eta _{n+j}))}=0. \end{aligned}$$

(3.9)

On the one hand, according to (3.6), we have

$$\begin{aligned} a_{ii}^{l}e^{\alpha _{ii}u_{i}(\xi _{i})} \leq& a_{ii}(\xi _{i})e^{\alpha _{ii}u_{i}(\xi _{i})} \\ < &a_{ii}(\xi _{i})e^{\alpha _{ii}u_{i}(\xi _{i})}+\lambda \sum_{k=1,k\neq i}^{n}a_{ik}( \xi _{i})e^{\alpha _{ik}u_{k}(\xi _{i})} +\lambda \sum_{l=1}^{m}b_{il}( \xi _{i})e^{\beta _{il}u_{n+l}(\xi _{i})} \\ &{}+\lambda \sum_{k=1}^{n}c_{ik}( \xi _{i})e^{\gamma _{ik}u_{k}(\xi _{i}-\tau _{ik}(\xi _{i}))} +\lambda \sum_{l=1}^{m}d_{il}( \xi _{i})e^{\delta _{il}u_{n+l}(\xi _{i}-\sigma _{il}(\xi _{i}))} \\ =&r_{i}(\xi _{i})\leq r_{i}^{M}, \end{aligned}$$

(3.10)

which implies that

$$\begin{aligned} u_{i}(\xi _{i})< \ln l_{i}^{+},\quad i=1,2,\ldots,n. \end{aligned}$$

(3.11)

By means of (3.7), we obtain

$$\begin{aligned} \hat{b}_{jj}^{l}e^{\hat{\beta }_{jj}u_{n+j}(\xi _{n+j})} \leq& \hat{b}_{jj}(\xi _{n+j})e^{\hat{\beta }_{jj}u_{n+j}(\xi _{n+j})} \\ < & \hat{r}_{j}(\xi _{n+j})+\hat{b}_{jj}(\xi _{n+j})e^{\hat{\beta }_{jj}u_{n+j}(\xi _{n+j})} \\ &{}+\lambda \sum_{l=1,l\neq j}^{m} \hat{b}_{lj}(\xi _{n+j})e^{\hat{\beta }_{lj}u_{n+l}(\xi _{n+j})} +\lambda \sum _{l=1}^{m}\hat{d}_{lj}(\xi _{n+j})e^{\hat{\delta }_{lj}u_{n+l}(\xi _{n+j}-\hat{\sigma }_{lj}(\xi _{n+j}))} \\ =& \sum_{k=1}^{n}\hat{a}_{kj}( \xi _{n+j})e^{\hat{\alpha }_{kj}u_{k}(\xi _{n+j})} +\sum_{k=1}^{n} \hat{c}_{kj}(\xi _{n+j})e^{\hat{\gamma }_{kj}u_{k}(\xi _{n+j}-\hat{\tau }_{kj}(\xi _{n+j}))} \\ \leq & \sum_{k=1}^{n}\hat{a}_{kj}^{M} \bigl(l_{k}^{+}\bigr)^{\hat{\alpha }_{kj}} +\sum _{k=1}^{n}\hat{c}_{kj}^{M} \bigl(l_{k}^{+}\bigr)^{\hat{\gamma }_{kj}}, \end{aligned}$$

(3.12)

which indicates that

$$\begin{aligned} u_{n+j}(\xi _{n+j})< \ln l_{n+j}^{+},\quad j=1,2,\ldots,m. \end{aligned}$$

(3.13)

Similar to (3.10)–(3.13), it follows from (3.8) and (3.9) that

$$\begin{aligned} u_{i}(\eta _{i})< \ln l_{i}^{+},\quad i=1,2,\ldots,n, \end{aligned}$$

(3.14)

and

$$\begin{aligned} u_{n+j}(\eta _{n+j})< \ln l_{n+j}^{+},\quad j=1,2,\ldots,m. \end{aligned}$$

(3.15)

On the other hand, in view of (3.8), we obtain

$$\begin{aligned} r_{i}^{l} \leq & r_{i}(\eta _{i})=a_{ii}(\eta _{i})e^{\alpha _{ii}u_{i}(\eta _{i})}+\lambda \sum_{k=1,k\neq i}^{n}a_{ik}( \eta _{i})e^{\alpha _{ik}u_{k}(\eta _{i})} +\lambda \sum_{l=1}^{m}b_{il}( \eta _{i})e^{\beta _{il}u_{n+l}(\eta _{i})} \\ &{}+\lambda \sum_{k=1}^{n}c_{ik}( \eta _{i})e^{\gamma _{ik}u_{k}(\eta _{i}-\tau _{ik}(\eta _{i}))} +\lambda \sum_{l=1}^{m}d_{il}( \eta _{i})e^{\delta _{il}u_{n+l}(\eta _{i}-\sigma _{il}(\eta _{i}))} \\ \leq &a_{ii}^{M}e^{\alpha _{ii}u_{i}(\eta _{i})}+ \sum _{k=1,k\neq i}^{n}a_{ik}^{M} \bigl(l_{k}^{+}\bigr)^{\alpha _{ik}} +\sum _{l=1}^{m}b_{il}^{M} \bigl(l_{n+l}^{+}\bigr)^{\beta _{il}} \\ &{} + \sum _{k=1}^{n}c_{ik}^{M} \bigl(l_{k}^{+}\bigr)^{\gamma _{ik}} +\sum _{l=1}^{m}d_{il}^{M} \bigl(l_{n+l}^{+}\bigr)^{\delta _{il}}, \end{aligned}$$

(3.16)

which implies that

$$\begin{aligned} u_{i}(\eta _{i})\geq \ln l_{i}^{-},\quad i=1,2,\ldots,n. \end{aligned}$$

(3.17)

According to (3.9), we get

$$\begin{aligned} & \sum_{k=1}^{n} \hat{a}_{kj}^{l}\bigl(l_{k}^{-} \bigr)^{\hat{\alpha }_{kj}} +\sum_{k=1}^{n} \hat{c}_{kj}^{l}\bigl(l_{k}^{-} \bigr)^{\hat{\gamma }_{kj}} \\ &\quad \leq \sum_{k=1}^{n} \hat{a}_{kj}(\eta _{n+j})e^{\hat{\alpha }_{kj}u_{k}(\eta _{n+j})} +\sum _{k=1}^{n}\hat{c}_{kj}(\eta _{n+j})e^{\hat{\gamma }_{kj}u_{k}(\eta _{n+j}-\hat{\tau }_{kj}(\eta _{n+j}))} \\ &\quad =\hat{r}_{j}(\eta _{n+j})+\hat{b}_{jj}(\eta _{n+j})e^{\hat{\beta }_{jj}u_{n+j}(\eta _{n+j})} +\lambda \sum_{l=1,l\neq j}^{m} \hat{b}_{lj}(\eta _{n+j})e^{\hat{\beta }_{lj}u_{n+l}(\eta _{n+j})} \\ &\qquad {} +\lambda \sum_{l=1}^{m} \hat{d}_{lj}(\eta _{n+j})e^{\hat{\delta }_{lj}u_{n+l}(\eta _{n+j}-\hat{\sigma }_{lj}(\eta _{n+j}))} \\ &\quad \leq\hat{r}_{j}^{M}+\hat{b}_{jj}^{M}e^{\hat{\beta }_{jj}u_{n+j}(\eta _{n+j})} + \sum_{l=1,l\neq j}^{m}\hat{b}_{lj}^{M} \bigl(l_{n+l}^{+}\bigr)^{\hat{\beta }_{lj}} +\sum _{l=1}^{m}\hat{d}_{lj}^{M} \bigl(l_{n+l}^{+}\bigr)^{\hat{\delta }_{lj}}, \end{aligned}$$

(3.18)

which implies that

$$\begin{aligned} u_{n+j}(\eta _{n+j})\geq \ln l_{n+j}^{-},\quad j=1,2,\ldots,m. \end{aligned}$$

(3.19)

Similar to (3.16)–(3.19), it follows from (3.6) and (3.7) that

$$\begin{aligned} u_{i}(\eta _{i})\geq \ln l_{i}^{-},\quad i=1,2,\ldots,n \end{aligned}$$

(3.20)

and

$$\begin{aligned} u_{n+j}(\eta _{n+j})\geq \ln l_{n+j}^{-},\quad j=1,2,\ldots,m. \end{aligned}$$

(3.21)

By (3.11), (3.13)–(3.15), (3.17), (3.19)–(3.21) and noticing that $0< l_{h}^{-}< l_{h}^{+}$ ($h=1,2,\ldots,n+m$), we have, for all $t\in \mathbb{R}$,

$$\begin{aligned} \ln l_{h}^{-}\leq u_{h}(t)< \ln l_{h}^{+}, \quad h=1,2,\ldots,n+m. \end{aligned}$$

(3.22)

Denote the open bounded set $\Omega \subset X$ by $\Omega =I_{1}\times I_{2}\times \cdots \times I_{n+m}$, where $I_{h}=(\ln l_{h}^{-}-\theta _{h}),\ln l_{h}^{+}+\theta _{h})$, $\theta _{h}\in (0,\infty)$ ($h=1,2,\ldots,n+m$). Clearly, $l_{h}^{+}$, $l_{h}^{-}$ and $\theta _{h}$ ($h=1,2,\ldots,n+m$) are independent of λ. Thus Ω satisfies the requirement (a) in Lemma 3.1.

Now we show that (b) of Lemma 3.1 holds, namely, we prove when $u\in \partial \Omega \cap \operatorname{Ker}(L)=\partial \Omega \cap \mathbb{R}^{n+m}$, $QN(u,0)\neq (0,0,\ldots,0)^{T}$. If it is not true, then when $u\in \partial \Omega \cap \operatorname{Ker}(L)=\partial \Omega \cap \mathbb{R}^{n+m}$, constant vector $u=(u_{1},u_{2},\ldots,u_{n+m})^{T}$ with $u\in \partial \Omega$ satisfies

$$\begin{aligned} &m \Biggl( r_{i}(t)- \sum_{k=1}^{n}a_{ik}(t)e^{\alpha _{ik}u_{k}}- \sum_{l=1}^{m}b_{il}(t)e^{\beta _{il}u_{n+l}} -\sum_{k=1}^{n}c_{ik}(t)e^{\gamma _{ik}u_{k}} -\sum_{l=1}^{m}d_{il}(t)e^{\delta _{il}u_{n+l}} \Biggr) =0, \\ &m \Biggl( -\hat{r}_{j}(t)+ \sum_{k=1}^{n} \hat{a}_{kj}(t)e^{\hat{\alpha }_{kj}u_{k}}-\sum_{l=1}^{m} \hat{b}_{lj}(t)e^{\hat{\beta }_{lj}u_{n+l}} +\sum_{k=1}^{n} \hat{c}_{kj}(t)e^{\hat{\gamma }_{kj}u_{k}} -\sum_{l=1}^{m} \hat{d}_{lj}(t)e^{\hat{\delta }_{lj}u_{n+l}} \Biggr) =0. \end{aligned}$$

In view of the mean value theorem of calculous, there exist $n+m$ points $t_{i}$, $\hat{t}_{j}$ ($i=1,2,\ldots,n$; $j=1,2,\ldots,m$) such that

$$\begin{aligned} &r_{i}(t_{i})- \sum _{k=1}^{n}a_{ik}(t_{i})e^{\alpha _{ik}u_{k}}- \sum_{l=1}^{m}b_{il}(t_{i})e^{\beta _{il}u_{n+l}} -\sum_{k=1}^{n}c_{ik}(t_{i})e^{\gamma _{ik}u_{k}} -\sum_{l=1}^{m}d_{il}(t_{i})e^{\delta _{il}u_{n+l}} \\ &\quad =0, \end{aligned}$$

(3.23)

$$\begin{aligned} &{-}\hat{r}_{j}(\hat{t}_{j})+ \sum _{k=1}^{n}\hat{a}_{kj}( \hat{t}_{j})e^{\hat{\alpha }_{kj}u_{k}}-\sum_{l=1}^{m} \hat{b}_{lj}(\hat{t}_{j})e^{\hat{\beta }_{lj}u_{n+l}} +\sum _{k=1}^{n}\hat{c}_{kj}( \hat{t}_{j})e^{\hat{\gamma }_{kj}u_{k}} -\sum_{l=1}^{m} \hat{d}_{lj}(\hat{t}_{j})e^{\hat{\delta }_{lj}u_{n+l}} \\ &\quad =0. \end{aligned}$$

(3.24)

By the analogous argument of (3.10)–(3.21), it follows from (3.23)–(3.24) that $\ln l_{h}^{-}-\theta _{h}< u_{h}<\ln l_{h}^{+}+\theta _{h}$ ($h=1,2,\ldots,n+m$), that is, $u\in \Omega \cap \mathbb{R}^{n+m}$. This contradicts the fact that $u\in \partial \Omega \cap \mathbb{R}^{n+m}$. So (b) of Lemma 3.1 holds.

Next we show that (c) in Lemma 3.1 holds. Consider the following algebraic equation:

$$\begin{aligned} &r_{i}(t)-a_{ii}(t)e^{\alpha _{ii}u_{i}}- \sum_{k=1,k\neq i}^{n}a_{ik}(t) \bigl(l_{k}^{+}\bigr)^{\alpha _{ik}} -\sum _{l=1}^{m}b_{il}(t) \bigl(l_{n+j}^{+} \bigr)^{\beta _{il}} \\ &\quad {}- \sum_{k=1}^{n}c_{ik}(t) \bigl(l_{k}^{+}\bigr)^{\gamma _{ik}} -\sum _{l=1}^{m}d_{il}(t) \bigl(l_{n+j}^{+} \bigr)^{\delta _{il}}=0, \quad i=1,2,\ldots,n, \end{aligned}$$

(3.25)

and

$$\begin{aligned} &{-}\hat{r}_{j}(t)-\hat{b}_{jj}(t)e^{\hat{\beta }_{jj}u_{n+j}}+ \sum_{k=1}^{n}\hat{a}_{kj}(t) \bigl(l_{k}^{-}\bigr)^{\hat{\alpha }_{kj}} -\lambda \sum _{l=1,l\neq j}^{m}\hat{b}_{lj}(t) \bigl(l_{n+l}^{+}\bigr)^{\hat{\beta }_{lj}} \\ &\quad {}+\sum_{k=1}^{n}\hat{c}_{kj}(t) \bigl(l_{k}^{-}\bigr)^{\hat{\gamma }_{kj}} -\lambda \sum _{l=1}^{m}\hat{d}_{lj}(t) \bigl(l_{n+l}^{+}\bigr)^{\hat{\delta }_{lj}}=0, \quad j=1,2,\ldots,m. \end{aligned}$$

(3.26)

Obviously, (3.25)–(3.26) has an unique solution $(u_{1}^{*},u_{2}^{*},\ldots,u^{*}_{n+m})$. It is easy to verify that $\ln l_{h}^{-}-\theta _{h}< u^{*}_{h}<\ln l_{h}^{+}+\theta _{h}$ ($h=1,2,\ldots,n+m$). Therefore, $(u^{*}_{1},u^{*}_{2},\ldots,u^{*}_{n+m})\in \Omega$. Since $\operatorname{Ker}(L)=\operatorname{Im}(Q)$, we can take $J=I$. A direct computation gives

$$\begin{aligned} &\operatorname{deg}\bigl\{ JQN(u,0),\Omega \cap \operatorname{Ker}(L),(0,0,\ldots,0)^{T}\bigr\} \\ &\quad =\operatorname{sign}\Biggl[ \prod _{i=1} ^{n}\prod_{j=1} ^{m} \bigl( -a_{ii}(t)u^{*}_{i} \bigr) \bigl( -\hat{b}_{jj}(t)u^{*}_{n+j} \bigr) \Biggr] =\pm 1. \end{aligned}$$

So far, we have proved that Ω satisfies all the assumptions in Lemma 3.1. Hence, system (3.1) has at least an almost periodic solution $(\overline{u}_{1}(t),\overline{u}_{2}(t),\ldots,\overline{u}_{n+m}(t))$. Therefore, system (1.4) has at least one positive almost periodic solution $(e^{\overline{u}_{1}(t)},e^{\overline{u}_{2}(t)},\ldots,e^{\overline{u}_{n+m}(t)})$. The proof is complete. □

4 Global exponential stability

The aim of this section is to derive the sufficient condition of a unique globally exponentially stable positive almost periodic solution of (1.4).

Under the assumption of Theorem 3.1, we know that system (1.4) has at least one positive almost periodic solution $(\overline{x}_{1}(t),\overline{x}_{2}(t),\ldots,\overline{x}_{n+m}(t))$ satisfying $l_{h}^{-}\leq \overline{x}_{h}(t)< l_{h}^{+}$ ($h=1,2,\ldots,n+m$). Now let ρ be a positive constant satisfying $0<\rho <\min_{1\leq h\leq n+m}\{l_{h}^{-}\}$. We assume further that

$(H_{4})$ :

$\tau _{ik}(t)$, $\sigma _{il}(t)$, $\hat{\tau }_{kj}(t)$, $\sigma _{lj}(t)(i,k=1,2,\ldots,n;j, l=1,2,\ldots,m)\in C^{1}(\mathbb{R},\mathbb{R}_{0}^{+})$ satisfy $0\leq \tau '_{ik}(t)$, $\sigma '_{il}(t)$, $\hat{\tau }'_{kj}(t)$ $\sigma '_{lj}(t)<1$ ($i,k=1,2,\ldots,n$; $j, l=1,2,\ldots,m$).

$(H_{5})$ :

$\alpha _{ii}\geq \max_{1\leq k\leq n,1\leq l\leq m} \{ \alpha _{ik},\beta _{il},\gamma _{ik},\delta _{il} \}$, $\hat{\beta }_{jj}\geq \max_{1\leq k\leq n,1\leq l\leq m} \{ \hat{\alpha }_{kj},\hat{\beta }_{lj},\hat{\gamma }_{kj},\hat{\delta }_{lj} \}$, $i=1, 2,\ldots,n$, $j=1,2,\ldots,m$.

$(H_{6})$ :

$-\alpha _{ii}\rho a_{ii}^{l}+ \sum_{k=1,k\neq i}^{n}\alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} + \sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} +\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M} +\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M}<0$ and $-\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l} + \sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} +\sum_{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} +\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M} +\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M}<0$, $i=1,2,\ldots,n$; $j=1,2,\ldots,m$.

Making the change of variable $y_{i}(t)=\frac{1}{\rho }x_{i}^{\alpha _{ii}}(t)$ ($i=1,2,\ldots,n$), $y_{n+j}(t)=\frac{1}{\rho }x_{n+j}^{\hat{\beta }_{jj}}(t)$ ($j=1,2,\ldots,m$), then system (1.4) is transformed into

$$\begin{aligned} \textstyle\begin{cases} y'_{i}(t)=\alpha _{ii}y_{i}(t) [r_{i}(t)- \sum_{k=1}^{n}\rho ^{\frac{\alpha _{ik}}{\alpha _{ii}}}a_{ik}(t)y_{k}^{\frac{\alpha _{ik}}{\alpha _{ii}}}(t) -\sum_{l=1}^{m}\rho ^{\frac{\beta _{il}}{\alpha _{ii}}}b_{il}(t)y_{n+l}^{\frac{\beta _{il}}{\alpha _{ii}}}(t) \\ \hphantom{y'_{i}(t)=}{} -\sum_{k=1}^{n}\rho ^{\frac{\gamma _{ik}}{\alpha _{ii}}}c_{ik}(t)y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(t-\tau _{ik}(t)) -\sum_{l=1}^{m}\rho ^{\frac{\delta _{il}}{\alpha _{ii}}}d_{il}(t)y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(t-\sigma _{il}(t)) ], \\ y'_{n+j}(t)=\hat{\beta }_{jj}y_{n+j}(t) [-\hat{r}_{j}(t)+ \sum_{k=1}^{n}\rho ^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}\hat{a}_{kj}(t)y_{k}^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}(t) -\sum_{l=1}^{m}\rho ^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}\hat{b}_{lj}(t)y_{n+l}^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}(t) \\ \hphantom{y'_{n+j}(t)=}{} +\sum_{k=1}^{n}\rho ^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\hat{c}_{kj}(t)y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(t-\hat{\tau }_{kj}(t)) -\sum_{l=1}^{m}\rho ^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\hat{d}_{lj}(t)y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(t-\hat{\sigma }_{lj}(t)) ]. \end{cases}\displaystyle \end{aligned}$$

(4.1)

Obviously, $\overline{y}(t)=(\overline{y}_{1}(t),\ldots,\overline{y}_{n}(t),\overline{y}_{n+1}(t),\ldots,\overline{y}_{n+m}(t))^{T}$ is the positive almost periodic solution of system (4.1), where $\overline{y}_{i}(t)=\frac{1}{\rho }\overline{x}_{i}^{\alpha _{ii}}(t)$ ($i=1,2,\ldots,n$), $\overline{y}_{n+j}(t)=\frac{1}{\rho }\overline{x}_{n+j}^{\hat{\beta }_{jj}}(t)$ ($j=1,2,\ldots,m$). From Theorem 3.1, we know that $l_{h}^{-}\leq \overline{x}_{h}(t)< l_{h}^{+}$ ($h=1,2,\ldots,n+m$). Therefore,

$$\begin{aligned} &1< \frac{1}{\rho }\bigl(l_{i}^{-} \bigr)^{\alpha _{ii}}\leq \overline{y}_{i}(t)< \frac{1}{\rho } \bigl(l_{i}^{+}\bigr)^{\alpha _{ii}},\qquad 1< \frac{1}{\rho } \bigl(l_{n+j}^{-}\bigr)^{\hat{\beta }_{jj}}\leq \overline{y}_{n+j}(t)< \frac{1}{\rho }\bigl(l_{n+j}^{+} \bigr)^{\hat{\beta }_{jj}}. \end{aligned}$$

(4.2)

Theorem 4.1

Assume that $(H_{1})$–$(H_{6})$ hold. Then for system (1.4) there exists a unique positive almost periodic solution which is globally exponentially stable.

Proof

According to conditions $(H_{1})$–$(H_{3})$, it follows from Theorem 3.1 that system (4.1) has a positive almost periodic solution $\overline{y}(t)=(\overline{y}_{1}(t),\overline{y}_{2}(t),\ldots,\overline{y}_{n+m}(t))^{T}$. Let $y(t)=(y_{1}(t),y_{2}(t),\ldots, y_{n+m}(t))^{T}$ be any positive solution of system (4.1). Now we construct a Lyapunov functional $V(t)=V_{1}(t)+V_{2}(t)$, where

$$\begin{aligned} V_{1}(t)=\sum_{i=1}^{n} \bigl\vert \ln y_{i}(t)-\ln\overline{y}_{i}(t)\bigr\vert + \sum_{j=1}^{m}\bigl\vert \ln y_{n+j}(t)-\ln\overline{y}_{n+j}(t)\bigr\vert \end{aligned}$$

(4.3)

and

$$\begin{aligned} V_{2}(t)={} &\sum_{i=1}^{n} \sum_{k=1}^{n}\alpha _{ii}\rho ^{\frac{\gamma _{ik}}{\alpha _{ii}}}c_{ik}^{M} \int _{t-\tau _{ik}(t)}^{t}\bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(s) -\overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(s)\bigr\vert \,ds \\ &{}+\sum_{i=1}^{n}\sum _{l=1}^{m}\alpha _{ii}\rho ^{\frac{\delta _{il}}{\alpha _{ii}}}d_{il}^{M} \int _{t-\sigma _{il}(t)}^{t}\bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(s) -\overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(s)\bigr\vert \,ds \\ &{}+\sum_{j=1}^{m}\sum _{k=1}^{n}\hat{\beta }_{jj}\rho ^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\hat{c}_{kj}^{M} \int _{t-\hat{\tau }_{kj}(t)}^{t}\bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(s) -\overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(s)\bigr\vert \,ds \\ &{}+\sum_{j=1}^{m}\sum _{l=1}^{m}\hat{\beta }_{jj}\rho ^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\hat{d}_{lj}^{M} \int _{t-\hat{\sigma }_{lj}(t)}^{t}\bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(s) -\overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(s)\bigr\vert \,ds. \end{aligned}$$

(4.4)

From the definition of $V(t)$, we easily see that $V(0)<+\infty$ and $V(t)\geq V_{1}(t)$. Noticing that $\operatorname{sgn}(\ln y_{h}(t)-\ln \overline{y}_{h}(t))=\operatorname{sgn}(y_{h}(t)-\overline{y}_{h}(t))$ ($h=1,2,\ldots,n+m$), we have

$$\begin{aligned} & D^{+}\bigl(\bigl\vert \ln y_{i}(t)-\ln \overline{y}_{i}(t)\bigr\vert \bigr) \\ &\quad = \operatorname{sgn}\bigl(\ln y_{i}(t)-\ln \overline{y}_{i}(t) \bigr) \biggl( \frac{y'_{i}(t)}{y_{i}(t)}- \frac{\overline{y}'_{i}(t)}{\overline{y}_{i}(t)} \biggr) \\ &\quad =\operatorname{sgn}\bigl( y_{i}(t)-\overline{y}_{i}(t) \bigr) \biggl( \frac{y'_{i}(t)}{y_{i}(t)}-\frac{\overline{y}'_{i}(t)}{\overline{y}_{i}(t)} \biggr) \\ &\quad =\operatorname{sgn}\bigl( y_{i}(t)-\overline{y}_{i}(t)\bigr) \Biggl[-\alpha _{ii}\rho a_{ii}(t) \bigl(y_{i}(t)- \overline{y}_{i}(t)\bigr) \\ &\qquad {}- \sum_{k=1,k\neq i}^{n}\alpha _{ii}\rho ^{\frac{\alpha _{ik}}{\alpha _{ii}}}a_{ik}(t) \bigl( y_{k}^{\frac{\alpha _{ik}}{\alpha _{ii}}}(t)-\overline{y}_{k}^{\frac{\alpha _{ik}}{\alpha _{ii}}}(t) \bigr) -\sum_{l=1}^{m}\alpha _{ii}\rho ^{\frac{\beta _{il}}{\alpha _{ii}}}b_{il}(t) \bigl( y_{n+l}^{\frac{\beta _{il}}{\alpha _{ii}}}(t)-\overline{y}_{n+l}^{\frac{\beta _{il}}{\alpha _{ii}}}(t) \bigr) \\ &\qquad {}-\sum_{k=1}^{n}\alpha _{ii}\rho ^{\frac{\gamma _{ik}}{\alpha _{ii}}}c_{ik}(t) \bigl( y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) - \overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) \bigr) \\ &\qquad {}-\sum_{l=1}^{m}\alpha _{ii}\rho ^{\frac{\delta _{il}}{\alpha _{ii}}}d_{il}(t) \bigl( y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t)\bigr) - \overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t) \bigr) \bigr) \Biggr] \\ &\quad \leq-\alpha _{ii}\rho a_{ii}^{l} \bigl\vert y_{i}(t)-\overline{y}_{i}(t)\bigr\vert + \sum_{k=1,k\neq i}^{n}\alpha _{ii}\rho ^{\frac{\alpha _{ik}}{\alpha _{ii}}}a_{ik}^{M} \bigl\vert y_{k}^{\frac{\alpha _{ik}}{\alpha _{ii}}}(t)-\overline{y}_{k}^{\frac{\alpha _{ik}}{\alpha _{ii}}}(t) \bigr\vert \\ &\qquad {}+\sum_{l=1}^{m}\alpha _{ii}\rho ^{\frac{\beta _{il}}{\alpha _{ii}}}b_{il}^{M} \bigl\vert y_{n+l}^{\frac{\beta _{il}}{\alpha _{ii}}}(t)-\overline{y}_{n+l}^{\frac{\beta _{il}}{\alpha _{ii}}}(t) \bigr\vert +\sum_{k=1}^{n}\alpha _{ii}\rho ^{\frac{\gamma _{ik}}{\alpha _{ii}}}c_{ik}^{M} \bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) - \overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) \bigr\vert \\ &\qquad {}+\sum_{l=1}^{m}\alpha _{ii} \rho ^{\frac{\delta _{il}}{\alpha _{ii}}}d_{il}^{M} \bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t)\bigr) - \overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t) \bigr)\bigr\vert , \quad i=1,2,\ldots,n, \end{aligned}$$

(4.5)

and

$$\begin{aligned} & D^{+}\bigl(\bigl\vert \ln y_{n+j}(t)-\ln \overline{y}_{n+j}(t)\bigr\vert \bigr) \\ &\quad = \operatorname{sgn}\bigl(\ln y_{n+j}(t)-\ln \overline{y}_{n+j}(t) \bigr) \biggl( \frac{y'_{n+j}(t)}{y_{n+j}(t)}-\frac{\overline{y}'_{n+j}(t)}{\overline{y}_{n+j}(t)} \biggr) \\ &\quad =\operatorname{sgn}\bigl( y_{n+j}(t)-\overline{y}_{n+j}(t) \bigr) \biggl( \frac{y'_{n+j}(t)}{y_{n+j}(t)}-\frac{\overline{y}'_{n+j}(t)}{\overline{y}_{n+j}(t)} \biggr) \\ &\quad =\operatorname{sgn}\bigl( y_{n+j}(t)- \overline{y}_{n+j}(t)\bigr) \\ &\qquad {}\times \Biggl[-\hat{\beta }_{jj}\rho \hat{b}_{jj}(t) \bigl(y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr) + \sum _{k=1}^{n}\hat{\beta }_{jj}\rho ^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}\hat{a}_{kj}(t) \bigl( y_{k}^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}(t)- \overline{y}_{k}^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}(t) \bigr) \\ &\qquad {}-\sum_{l=1,l\neq j}^{m}\hat{\beta }_{jj}\rho ^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}\hat{b}_{lj}(t) \bigl( y_{n+l}^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}(t) -\overline{y}_{n+l}^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}(t) \bigr) \\ &\qquad {}+\sum_{k=1}^{n}\hat{\beta }_{jj}\rho ^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\hat{c}_{kj}(t) \bigl( y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\tau }_{kj}(t)\bigr) - \overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\tau }_{kj}(t) \bigr) \bigr) \\ &\qquad {}-\sum_{l=1}^{m}\hat{\beta }_{jj}\rho ^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\hat{d}_{lj}(t) \bigl( y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\sigma }_{lj}(t)\bigr) - \overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\sigma }_{lj}(t)\bigr) \bigr) \Biggr] \\ &\quad \leq-\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l} \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert + \sum_{k=1}^{n}\hat{\beta }_{jj} \rho ^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}\hat{a}_{kj}^{M} \bigl\vert y_{k}^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}(t)-\overline{y}_{k}^{\frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}}}(t) \bigr\vert \\ &\qquad {}+\sum_{l=1,l\neq j}^{m}\hat{\beta }_{jj}\rho ^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}\hat{b}_{lj}^{M} \bigl\vert y_{n+l}^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}(t) -\overline{y}_{n+l}^{\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}}}(t) \bigr\vert \\ &\qquad {}+\sum_{k=1}^{n}\hat{\beta }_{jj}\rho ^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\hat{c}_{kj}^{M} \bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\tau }_{kj}(t)\bigr) -\overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t- \hat{\tau }_{kj}(t)\bigr)\bigr\vert \\ &\qquad {}+\sum_{l=1}^{m}\hat{\beta }_{jj}\rho ^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\hat{d}_{lj}^{M} \bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\sigma }_{lj}(t)\bigr) -\overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t- \hat{\sigma }_{lj}(t)\bigr)\bigr\vert , \quad j=1,2,\ldots,m. \end{aligned}$$

(4.6)

By the condition $(H_{4})$, we obtain

$$\begin{aligned} & \biggl( \int _{t-\tau _{ik}(t)}^{t}\bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(s) -\overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(s)\bigr\vert \,ds \biggr) ' \\ &\quad =\bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(t)- \overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(t)\bigr\vert -\bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) - \overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) \bigr\vert \bigl(1-\tau '_{ik}(t)\bigr) \\ &\quad \leq \bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(t) -\overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}(t) \bigr\vert -\bigl\vert y_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t-\tau _{ik}(t)\bigr) -\overline{y}_{k}^{\frac{\gamma _{ik}}{\alpha _{ii}}}\bigl(t- \tau _{ik}(t)\bigr)\bigr\vert , \end{aligned}$$

(4.7)

$$\begin{aligned} & \biggl( \int _{t-\sigma _{il}(t)}^{t}\bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(s) -\overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(s)\bigr\vert \,ds \biggr) ' \\ &\quad =\bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(t)- \overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(t)\bigr\vert -\bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t)\bigr) - \overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t) \bigr)\bigr\vert \bigl(1-\sigma '_{il}(t)\bigr) \\ &\quad \leq \bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(t) -\overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}(t) \bigr\vert -\bigl\vert y_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t-\sigma _{il}(t)\bigr) -\overline{y}_{n+l}^{\frac{\delta _{il}}{\alpha _{ii}}}\bigl(t- \sigma _{il}(t)\bigr)\bigr\vert , \end{aligned}$$

(4.8)

$$\begin{aligned} & \biggl( \int _{t-\hat{\tau }_{kj}(t)}^{t}\bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(s) -\overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(s)\bigr\vert \,ds \biggr) ' \\ &\quad =\bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(t) - \overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(t)\bigr\vert -\bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\tau }_{kj}(t)\bigr) - \overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\tau }_{kj}(t) \bigr)\bigr\vert \bigl(1-\hat{\tau }'_{kj}(t)\bigr) \\ &\quad {}\leq \bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(t) -\overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}(t) \bigr\vert -\bigl\vert y_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\tau }_{kj}(t)\bigr) -\overline{y}_{k}^{\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}}\bigl(t- \hat{\tau }_{kj}(t)\bigr)\bigr\vert \end{aligned}$$

(4.9)

and

$$\begin{aligned} & \biggl( \int _{t-\hat{\sigma }_{lj}(t)}^{t}\bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(s) -\overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(s)\bigr\vert \,ds \biggr) ' \\ &\quad =\bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(t)- \overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(t)\bigr\vert -\bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\sigma }_{lj}(t)\bigr) - \overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\sigma }_{lj}(t)\bigr)\bigr\vert \bigl(1-\hat{\sigma }'_{lj}(t)\bigr) \\ &\quad \leq \bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(t) -\overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}(t) \bigr\vert -\bigl\vert y_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t-\hat{\sigma }_{lj}(t)\bigr) -\overline{y}_{n+l}^{\frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}}\bigl(t- \hat{\sigma }_{lj}(t)\bigr)\bigr\vert . \end{aligned}$$

(4.10)

By $(H_{5})$, we have $0<\frac{\alpha _{ik}}{\alpha _{ii}},\frac{\beta _{il}}{\alpha _{ii}},\frac{\gamma _{ik}}{\alpha _{ii}},\frac{\delta _{il}}{\alpha _{ii}}, \frac{\hat{\alpha }_{kj}}{\hat{\beta }_{jj}},\frac{\hat{\beta }_{lj}}{\hat{\beta }_{jj}},\frac{\hat{\gamma }_{kj}}{\hat{\beta }_{jj}}, \frac{\hat{\delta }_{lj}}{\hat{\beta }_{jj}}\leq 1$. Observe that $g(x)=\vert a^{x}-b^{x}\vert$ is an increasing function for $a\geq 1$ and $x >0$. Therefore, according to (4.1) and (4.5)–(4.10), we derive

$$\begin{aligned} D^{+}V(t) \leq &\sum _{i=1}^{n} \Biggl[-\alpha _{ii}\rho a_{ii}^{l}\bigl\vert y_{i}(t)- \overline{y}_{i}(t)\bigr\vert + \sum_{k=1,k\neq i}^{n} \alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} \bigl\vert y_{i}^{\frac{\alpha _{ki}}{\alpha _{kk}}}(t)-\overline{y}_{i}^{\frac{\alpha _{ki}}{\alpha _{kk}}}(t) \bigr\vert \\ &{}+ \sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} \bigl\vert y_{i}^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}(t)-\overline{y}_{i}^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}(t) \bigr\vert +\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M}\bigl\vert y_{i}^{\frac{\gamma _{ki}}{\alpha _{kk}}}(t)-\overline{y}_{i}^{\frac{\gamma _{ki}}{\alpha _{kk}}}(t) \bigr\vert \\ &{}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M}\bigl\vert y_{i}^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}(t)-\overline{y}_{i}^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}(t) \bigr\vert \Biggr] +\sum_{j=1}^{m} \Biggl[-\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l} \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert \\ &{}+\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} \bigl\vert y_{n+j}^{\frac{\beta _{jk}}{\alpha _{kk}}}(t)-\overline{y}_{n+j}^{\frac{\beta _{jk}}{\alpha _{kk}}}(t) \bigr\vert +\sum_{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} \bigl\vert y_{n+j}^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}(t)-\overline{y}_{n+j}^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}(t) \bigr\vert \\ &{}+\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M}\bigl\vert y_{n+j}^{\frac{\delta _{jk}}{\alpha _{kk}}}(t)-\overline{y}_{n+j}^{\frac{\delta _{jk}}{\alpha _{kk}}}(t) \bigr\vert +\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M}\bigl\vert y_{n+j}^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}(t)-\overline{y}_{n+j}^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}(t) \bigr\vert \Biggr] \\ \leq &\sum_{i=1}^{n} \Biggl[- \alpha _{ii}\rho a_{ii}^{l}\bigl\vert y_{i}(t)-\overline{y}_{i}(t)\bigr\vert + \sum _{k=1,k\neq i}^{n}\alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} \bigl\vert y_{i}(t)- \overline{y}_{i}(t)\bigr\vert \\ &{}+ \sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} \bigl\vert y_{i}(t)-\overline{y}_{i}(t)\bigr\vert +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M} \bigl\vert y_{i}(t)- \overline{y}_{i}(t)\bigr\vert \\ &{}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M} \bigl\vert y_{i}(t)-\overline{y}_{i}(t)\bigr\vert \Biggr] + \sum_{j=1}^{m} \Biggl[-\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l}\bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert \\ &{}+\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert +\sum _{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert \\ &{}+\sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M} \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert +\sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M} \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert \Biggr] \\ =&\sum_{i=1}^{n} \Biggl[-\alpha _{ii}\rho a_{ii}^{l}+ \sum _{k=1,k\neq i}^{n}\alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} + \sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M} \\ &{}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M} \Biggr]\bigl\vert y_{i}(t)-\overline{y}_{i}(t)\bigr\vert +\sum_{j=1}^{m} \Biggl[-\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} \\ &{}+\sum_{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M} \\ &{}+\sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M} \Biggr]\bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert . \end{aligned}$$

(4.11)

In addition, from the condition $(H_{6})$, there exists a positive constant κ such that

$$\begin{aligned} -\alpha _{ii}\rho a_{ii}^{l}+ \sum _{k=1,k\neq i}^{n}\alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} + \sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M} +\sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M}+\kappa < 0 \end{aligned}$$

and

$$\begin{aligned} -\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l} + \sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} +\sum _{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M} +\sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M}+\kappa < 0. \end{aligned}$$

The above two inequalities together with (4.11) lead to

$$\begin{aligned} D^{+}V(t)\leq -\kappa \sum _{h=1}^{n+m}\bigl\vert y_{h}(t)- \overline{y}_{h}(t)\bigr\vert . \end{aligned}$$

(4.12)

Integrating both sides of (4.12) with respect to t, we have

$$\begin{aligned} V(t)+\kappa \int _{0}^{t}\sum_{h=1}^{n+m} \bigl\vert y_{h}(s)-\overline{y}_{h}(s)\bigr\vert \,ds \leq V(0)< +\infty,\quad t\geq 0. \end{aligned}$$

(4.13)

Equation (4.13) shows that

$$\begin{aligned} \int _{0}^{t}\sum_{h=1}^{n+m} \bigl\vert y_{h}(s)-\overline{y}_{h}(s)\bigr\vert \,ds \leq \frac{V(0)}{\kappa }< +\infty, \quad t\geq 0, \end{aligned}$$

(4.14)

which implies that

$$\begin{aligned} \sum_{h=1}^{n+m}\bigl\vert y_{h}(s)-\overline{y}_{h}(s)\bigr\vert \in L^{1}[0,+\infty). \end{aligned}$$

(4.15)

(4.2) means that $\overline{y}_{h}(t)$ ($h=1,2,\ldots,n+m$) is uniformly bounded from below and above, and so $\ln \overline{y}_{h}(t)$ is bounded. From $\vert \ln y_{h}(t)-\ln \overline{y}_{h}(t)\vert \leq V_{1}(t)\leq V(t)\leq V(0)$, we get

$$\begin{aligned} \overline{y}_{h}(t)e^{-V(0)}\leq y_{h}(t)\leq \overline{y}_{h}(t)e^{V(0)}. \end{aligned}$$

(4.16)

(4.16) and (4.2) show that $y_{h}(t)$, $\overline{y}_{h}(t)$, $h=1,2,\ldots,n+m$ are uniformly bounded. This fact together with (4.1) lead to $y'_{h}(t)$, $\overline{y}'_{h}(t)$, $h=1,2,\ldots,n+m$ are uniformly bounded on $[0,+\infty)$. Therefore $\sum_{h=1}^{n+m}\vert y_{h}(s)-\overline{y}_{h}(s)\vert$ is uniformly continuous on $[0,+\infty)$. From (4.14) we know that $\sum_{h=1}^{n+m}\vert y_{h}(s)-\overline{y}_{h}(s)\vert$ is integrable on $[0,+\infty)$. By Barbalat’s lemma (Lemma 1.2.2 and Lemma 1.2.3, [20]), we can conclude that

$$\begin{aligned} \lim_{t\rightarrow +\infty }\bigl\vert y_{h}(t)- \overline{y}_{h}(t)\bigr\vert =0,\quad h=1,2,\ldots,n+m. \end{aligned}$$

(4.17)

Thus, we have proved that the positive almost periodic solution $(\overline{y}_{1}(t),\overline{y}_{2}(t),\ldots,\overline{y}_{n+m}(t))^{T}$ of system (4.1) is globally attractive.

Next, we shall prove that the positive almost periodic solution $(\overline{y}_{1}(t),\overline{y}_{2}(t),\ldots,\overline{y}_{n+m}(t))^{T}$ of system (4.1) is globally exponentially stable. In fact, in the light of (4.2) and (4.17), we know that, for any $\varepsilon >0$ (ε enough small), there exists $T>0$ such that for $t>T$

$$\begin{aligned} 1< \frac{1}{\rho }\bigl(l_{i}^{-} \bigr)^{\alpha _{ii}}-\varepsilon \leq \overline{y}_{i}(t)- \varepsilon < y_{i}(t)< \overline{y}_{i}(t)+\varepsilon < \frac{1}{\rho }\bigl(l_{i}^{+}\bigr)^{\alpha _{ii}}+ \varepsilon \end{aligned}$$

(4.18)

and

$$\begin{aligned} 1< \frac{1}{\rho }\bigl(l_{n+j}^{-} \bigr)^{\hat{\beta }_{jj}}-\varepsilon \leq \overline{y}_{n+j}(t)- \varepsilon < y_{n+j}(t)< \overline{y}_{n+j}(t)+\varepsilon < \frac{1}{\rho }\bigl(l_{n+j}^{+}\bigr)^{\hat{\beta }_{jj}}+ \varepsilon. \end{aligned}$$

(4.19)

By (4.18)–(4.19) and the mean value theorem of calculus, we have

$$\begin{aligned} \bigl\vert \ln y_{i}(t)-\ln \overline{y}_{i}(t) \bigr\vert = \biggl\vert \frac{1}{A_{i}}\biggr\vert \bigl\vert y_{i}(t)-\overline{y}_{i}(t)\bigr\vert \leq \frac{\vert y_{i}(t)-\overline{y}_{i}(t)\vert }{\frac{1}{\rho }(l_{i}^{-})^{\alpha _{ii}}-\varepsilon } \end{aligned}$$

(4.20)

and

$$\begin{aligned} \bigl\vert \ln y_{n+j}(t)-\ln \overline{y}_{n+j}(t) \bigr\vert =\biggl\vert \frac{1}{A_{n+j}}\biggr\vert \bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\bigr\vert \leq \frac{\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\vert }{\frac{1}{\rho }(l_{n+j}^{-})^{\hat{\beta }_{jj}}-\varepsilon }, \end{aligned}$$

(4.21)

where $i=1,2,\ldots,n$, $j=1,2,\ldots,m$, $A_{h}$ ($h=1,2,\ldots,n+m$) lies between $y_{h}(t)$ and $\overline{y}_{h}(t)$. According to $(H_{6})$, there exists a constant $\mu >0$ such that

$$\begin{aligned} &{-}\alpha _{ii}\rho a_{ii}^{l}+ \sum_{k=1,k\neq i}^{n}\alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} + \sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M} \\ &\quad {}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M} + \frac{\mu }{\frac{1}{\rho }(l_{i}^{-})^{\alpha _{ii}}-\varepsilon }< 0,\quad i=1,2,\ldots,n, \end{aligned}$$

(4.22)

and

$$\begin{aligned} &{-}\hat{\beta }_{jj}\rho \hat{b}_{jj}^{l} + \sum_{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} +\sum _{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M} \\ &\quad {}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M} + \frac{\mu }{\frac{1}{\rho }(l_{n+j}^{-})^{\hat{\beta }_{jj}}-\varepsilon }< 0,\quad j=1,2,\ldots,m. \end{aligned}$$

(4.23)

Construct the Lyapunov functional $W(t)=e^{\mu t}V_{1}(t)+V_{2}(t)$. Applying (4.11), (4.22) and (4.23), we have

$$\begin{aligned} D^{+}W(t)\leq{} &\sum _{i=1}^{n} \Biggl[-\alpha _{ii}\rho a_{ii}^{l}+ \sum_{k=1,k\neq i}^{n} \alpha _{kk}\rho ^{\frac{\alpha _{ki}}{\alpha _{kk}}}a_{ki}^{M} + \sum _{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\alpha }_{li}}{\hat{\beta }_{ll}}}\hat{a}_{li}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\gamma _{ki}}{\alpha _{kk}}}c_{ki}^{M} \\ &{}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\gamma }_{li}}{\hat{\beta }_{ll}}}\hat{c}_{li}^{M} + \frac{\mu }{\frac{1}{\rho }(l_{i}^{-})^{\alpha _{ii}}-\varepsilon } \Biggr]\bigl\vert y_{i}(t)-\overline{y}_{i}(t) \bigr\vert \\ &{}+\sum_{j=1}^{m} \Biggl[-\hat{ \beta }_{jj}\rho \hat{b}_{jj}^{l} + \sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\beta _{jk}}{\alpha _{kk}}}b_{jk}^{M} +\sum _{l=1,l\neq j}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\beta }_{jl}}{\hat{\beta }_{ll}}}\hat{b}_{jl}^{M} +\sum _{k=1}^{n}\alpha _{kk}\rho ^{\frac{\delta _{jk}}{\alpha _{kk}}}d_{jk}^{M} \\ &{}+\sum_{l=1}^{m}\hat{\beta }_{ll}\rho ^{\frac{\hat{\delta }_{jl}}{\hat{\beta }_{ll}}}\hat{d}_{jl}^{M} + \frac{\mu }{\frac{1}{\rho }(l_{n+j}^{-})^{\hat{\beta }_{jj}}-\varepsilon } \Biggr]\bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t) \bigr\vert < 0. \end{aligned}$$

(4.24)

Equation (4.24) means that

$$\begin{aligned} W(t)< W(T)=e^{\mu T}V_{1}(T)+V_{2}(T) \leq e^{\mu T}V(T)\leq e^{\mu T}V(0),\quad t>0. \end{aligned}$$

(4.25)

Similar to (4.18)–(4.21), we also have

$$\begin{aligned} \begin{aligned} &\bigl\vert \ln y_{i}(t)-\ln \overline{y}_{i}(t) \bigr\vert > \frac{\vert y_{i}(t)-\overline{y}_{i}(t)\vert }{\frac{1}{\rho }(l_{i}^{+}) ^{\alpha _{ii}}+\varepsilon }, \\ & \bigl\vert \ln y_{n+j}(t)-\ln \overline{y}_{n+j}(t)\bigr\vert > \frac{\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\vert }{\frac{1}{\rho }(l_{n+j}^{+})^{\hat{\beta }_{jj}}+\varepsilon }. \end{aligned} \end{aligned}$$

(4.26)

(4.25) and (4.26) lead to

$$\begin{aligned} \frac{e^{\mu t}}{\frac{1}{\rho }(l_{i}^{+})^{\alpha _{ii}}+\varepsilon }\bigl\vert y_{i}(t)-\overline{y}_{i}(t) \bigr\vert \leq e^{\mu t}\sum_{i=1}^{n} \frac{\vert y_{i}(t)-\overline{y}_{i}(t)\vert }{\frac{1}{\rho }(l_{i}^{+})^{\alpha _{ii}}+\varepsilon }< e^{\mu t}V_{1}(t)\leq W(t)< e^{\mu T}V(0) \end{aligned}$$

and

$$\begin{aligned} \frac{e^{\mu t}}{\frac{1}{\rho }(l_{n+j}^{+})^{\hat{\beta }_{jj}}+\varepsilon }\bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t) \bigr\vert \leq e^{\mu t}\sum_{j=1}^{m} \frac{\vert y_{n+j}(t)-\overline{y}_{n+j}(t)\vert }{\frac{1}{\rho }(l_{n+j}^{+})^{\hat{\beta }_{jj}}+\varepsilon }< e^{\mu t}V_{1}(t)\leq W(t)< e^{\mu T}V(0). \end{aligned}$$

Letting $\varepsilon \rightarrow 0^{+}$ in the above two inequalities, we have

$$\begin{aligned} \bigl\vert y_{h}(t)-\overline{y}_{h}(t) \bigr\vert < Me^{-\mu t},\quad h=1,2,\ldots,n+m, t>0, \end{aligned}$$

(4.27)

where $M=e^{\mu T}V(0)\max_{1\leq i\leq n,1\leq j\leq m}\{\frac{1}{\rho }(l_{i}^{+})^{\alpha _{ii}}, \frac{1}{\rho }(l_{n+j}^{+})^{\hat{\beta }_{jj}}\}$. (4.27) indicates that the unique positive almost periodic solution $(\overline{y}_{1}(t),\overline{y}_{2}(t),\ldots,\overline{y}_{n+m}(t))^{T}$ of system (4.1) is globally exponentially stable.

Now let us to show that the unique positive almost periodic solution $(\overline{x}_{1}(t),\overline{x}_{2}(t),\ldots, \overline{x}_{n+m}(t))^{T}$ of system (1.4) is globally exponentially stable.

Indeed, by the mean value theorem of calculus and (4.18)–(4.19), we have

$$\begin{aligned} \bigl\vert x_{i}(t)-\overline{x}_{i}(t) \bigr\vert =\rho ^{\frac{1}{\alpha _{ii}}}\bigl\vert y_{i}^{\frac{1}{\alpha _{ii}}}(t)- \overline{y}_{i}^{\frac{1}{\alpha _{ii}}}(t)\bigr\vert \leq \frac{N_{i}(\varepsilon)}{\alpha _{ii}} \rho ^{\frac{1}{\alpha _{ii}}}\bigl\vert y_{i}(t)-\overline{y}_{i}(t) \bigr\vert \end{aligned}$$

(4.28)

and

$$\begin{aligned} \bigl\vert x_{n+j}(t)-\overline{x}_{n+j}(t) \bigr\vert =\rho ^{\frac{1}{\hat{\beta }_{jj}}}\bigl\vert y_{n+j}^{\frac{1}{\hat{\beta }_{jj}}}(t)- \overline{y}_{n+j}^{\frac{1}{\hat{\beta }_{jj}}}(t)\bigr\vert \leq \frac{N_{n+j}(\varepsilon)}{\hat{\beta }_{jj}} \rho ^{\frac{1}{\hat{\beta }_{jj}}}\bigl\vert y_{n+j}(t)-\overline{y}_{n+j}(t) \bigr\vert , \end{aligned}$$

(4.29)

where $i=1,2,\ldots,n$, $j=1,2,\ldots,m$,

$$\begin{aligned}& N_{i}(\varepsilon)=\max \biggl\{ \biggl( \frac{1}{\rho } \bigl(l_{i}^{-}\bigr)^{\alpha _{ii}}-\varepsilon \biggr) ^{\frac{1-\alpha _{ii}}{\alpha _{ii}}}, \biggl( \frac{1}{\rho }\bigl(l_{i}^{+} \bigr)^{\alpha _{ii}}+\varepsilon \biggr) ^{\frac{1-\alpha _{ii}}{\alpha _{ii}}} \biggr\} , \\& N_{n+j}(\varepsilon)=\max \biggl\{ \biggl( \frac{1}{\rho } \bigl(l_{n+j}^{-}\bigr)^{\hat{\beta }_{jj}}-\varepsilon \biggr) ^{\frac{1-\hat{\beta }_{jj}}{\hat{\beta }_{jj}}}, \biggl( \frac{1}{\rho }\bigl(l_{n+j}^{+} \bigr)^{\hat{\beta }_{jj}}+\varepsilon \biggr) ^{\frac{1-\hat{\beta }_{jj}}{\hat{\beta }_{jj}}} \biggr\} . \end{aligned}$$

Letting $\varepsilon \rightarrow 0^{+}$ in (4.28)–(4.29), and employing (4.27)–(4.29), we obtain

$$\begin{aligned} \bigl\vert x_{h}(t)-\overline{x}_{h}(t) \bigr\vert < M^{*}e^{-\mu t},\quad h=1,2,\ldots,n+m, t>0, \end{aligned}$$

(4.30)

where $M^{*}=M\max_{1\leq i\leq n,1\leq j\leq m} \{ \frac{N_{i}(0)}{\alpha _{ii}}\rho ^{\frac{1}{\alpha _{ii}}}, \frac{N_{n+j}(0)}{\hat{\beta }_{jj}}\rho ^{\frac{1}{\hat{\beta }_{jj}}} \}$. Thus, we have proved that the unique positive almost periodic solution $(\overline{x}_{1}(t),\overline{x}_{2}(t),\ldots,\overline{x}_{n+m}(t))^{T}$ of system (1.4) is globally exponentially stable. The proof of Theorem 4.1 is complete. □

5 Illustrative example

Consider the following two-layer Gilpin–Ayala predator–prey model with time delays:

$$ \textstyle\begin{cases} x'_{1}(t)=x_{1}(t) [ r(t)-a(t)x_{1}^{\alpha }(t)-b(t)x_{2}^{\beta }(t) -c(t)x_{1}^{\gamma }(t-\tau (t)) \\ \hphantom{x'_{1}(t)=}{} -d(t)x_{2}^{\delta }(t-\sigma (t)) ] , \\ x'_{2}(t)=x_{2}(t) [ -\hat{r}(t)+\hat{a}(t)x_{1}^{\hat{\alpha }}(t)-\hat{b}(t)x_{2}^{\hat{\beta }}(t) +\hat{c}(t)x_{1}^{\hat{\gamma }}(t-\hat{\tau }(t)) \\ \hphantom{x'_{2}(t)=}{} -\hat{d}(t)x_{2}^{\hat{\delta }}(t-\hat{\sigma }(t)) ] , \end{cases} $$

(5.1)

where $\alpha =\frac{3}{4}$, $\beta =\frac{1}{2}$, $\gamma =\frac{1}{3}$, $\delta =\frac{1}{4}$, $\hat{\alpha }=3$, $\hat{\beta }=4$, $\hat{\gamma }=\frac{3}{2}$, $\hat{\delta }=1$, $r(t)=10+\cos \sqrt{2} t$, $\hat{r}(t)=\frac{2+\sin \sqrt{3}t}{20}$, $a(t)=6+\sin \sqrt{2}t$, $b(t)=\frac{2-\sin \sqrt{3}t}{10}$, $c(t)=\frac{1+\vert \sin \sqrt{2}t\vert }{10}$, $d(t)=\frac{2+\cos \sqrt{5}t}{10}$, $\hat{a}(t)=\frac{3-\cos \sqrt{2}t}{5}$, $\hat{b}(t)=\frac{7+\sin \sqrt{7}t}{2}$, $\hat{c}(t)=\frac{1+2\vert \sin \sqrt{2} t\vert }{40}$, $\hat{d}(t)=\frac{2+\cos \sqrt{3}t}{10}$ and

$$\begin{aligned} &\tau (t)=\textstyle\begin{cases} \frac{1+\sin \sqrt{2}t}{4}, & \frac{2k\pi }{\sqrt{2}}-\frac{\pi }{2\sqrt{2}}\leq t\leq \frac{2k\pi }{\sqrt{2}}+\frac{\pi }{2\sqrt{2}}, k\in \mathbb{Z}, \\ \frac{1-\sin \sqrt{2}t}{4}, & \frac{2k\pi }{\sqrt{2}}+\frac{\pi }{2\sqrt{2}}\leq t\leq \frac{2k\pi }{\sqrt{2}}+\frac{3\pi }{2\sqrt{2}}, k\in \mathbb{Z}, \end{cases}\displaystyle \\ &\sigma (t)=\textstyle\begin{cases} \frac{1+\sin \sqrt{3}t}{3}, & \frac{2k\pi }{\sqrt{3}}-\frac{\pi }{2\sqrt{3}}\leq t\leq \frac{2k\pi }{\sqrt{3}}+\frac{\pi }{2\sqrt{3}}, k\in \mathbb{Z}, \\ \frac{1-\sin \sqrt{3}t}{3}, & 2\frac{k\pi }{\sqrt{3}}+\frac{\pi }{2\sqrt{3}}\leq t\leq \frac{2k\pi }{\sqrt{3}}+\frac{3\pi }{2\sqrt{3}}, k\in \mathbb{Z}, \end{cases}\displaystyle \\ &\hat{\tau }(t)=\textstyle\begin{cases} \frac{1-\cos \sqrt{2}t}{3}, & \frac{2k\pi }{\sqrt{2}}\leq t\leq \frac{2k\pi }{\sqrt{2}}+\frac{\pi }{\sqrt{2}}, k\in \mathbb{Z}, \\ \frac{1+\cos \sqrt{2}t}{3}, & \frac{2k\pi }{\sqrt{2}}+\frac{\pi }{\sqrt{2}}\leq t\leq \frac{2k\pi }{\sqrt{2}}+\frac{2\pi }{\sqrt{2}}, k\in \mathbb{Z}, \end{cases}\displaystyle \\ &\hat{\sigma }(t)=\textstyle\begin{cases} \frac{2-\cos \sqrt{3}t}{4}, & \frac{2k\pi }{\sqrt{3}}\leq t\leq \frac{2k\pi }{\sqrt{3}}+\frac{\pi }{\sqrt{3}}, k\in \mathbb{Z}, \\ \frac{2+\cos \sqrt{3}t}{4}, & 2\frac{k\pi }{\sqrt{3}}+\frac{\pi }{\sqrt{3}}\leq t\leq \frac{2k\pi }{\sqrt{3}}+\frac{2\pi }{\sqrt{3}}, k\in \mathbb{Z}. \end{cases}\displaystyle \end{aligned}$$

Obviously, $r(t)$, $\hat{r}(t)$, $a(t)$, $b(t)$, $c(t)$, $d(t)$, $\hat{a}(t)$, $\hat{b}(t)$, $\hat{c}(t)$, $\hat{d}(t)$, $\tau (t)$, $\sigma (t)$, $\hat{\tau }(t)$ and $\hat{\sigma }(t)$ are all positive almost periodic functions. By simple calculation, we have $r^{M}=11$, $r^{l}=9$, $\hat{r}^{M}=\frac{3}{20}$, $\hat{r}^{l}=\frac{1}{20}$, $a^{M}=7$, $a^{l}=5$, $b^{M}=\frac{3}{10}$, $b^{l}=\frac{1}{10}$, $c^{M}=\frac{1}{5}$, $c^{l}=\frac{1}{10}$, $d^{M}=\frac{3}{10}$, $d^{l}=\frac{1}{10}$, $\hat{a}^{M}=\frac{4}{5}$, $\hat{a}^{l}=\frac{2}{5}$, $\hat{b}^{M}=4$, $\hat{b}^{l}=3$, $\hat{c}^{M}=\frac{3}{40}$, $\hat{c}^{l}=\frac{1}{40}$, $\hat{d}^{M}=\frac{3}{10}$, $\hat{d}^{l}=\frac{1}{10}$ and

$$\begin{aligned} &l_{1}^{+}= \biggl( \frac{r^{M}}{a^{l}} \biggr) ^{\frac{1}{\alpha }}= \biggl( \frac{11}{2} \biggr) ^{\frac{4}{3}}\thickapprox 2.8613, \\ &l_{2}^{+}= \biggl( \frac{\hat{a}^{M}(l_{1}^{+})^{\hat{\alpha }} +\hat{c}^{M}(l_{1}^{+})^{\hat{\gamma }}}{\hat{b}^{l}} \biggr) ^{\frac{1}{\hat{\beta }}}\thickapprox 1.590, \\ &l_{1}^{-}= \biggl( \frac{r^{l} -b^{M}(l_{2}^{+})^{\beta } -c^{M}(l_{1}^{+})^{\gamma }-d^{M}(l_{2}^{+})^{\delta }}{a^{M}} \biggr) ^{\frac{1}{\alpha }}\thickapprox 1.1951, \\ &l_{2}^{-}= \biggl( \frac{\hat{a}^{l}(l_{1}^{-})^{\hat{\alpha }} +\hat{c}^{l}(l_{1}^{-})^{\hat{\gamma }}-\hat{r}^{M} -\hat{d}^{M}(l_{2}^{+})^{\hat{\delta }}}{\hat{b}^{M}} \biggr) ^{\frac{1}{\hat{\beta }}}\thickapprox 1.1303, \\ &9=r^{l}>b^{M}\bigl(l_{2}^{+} \bigr)^{\beta } +c^{M}\bigl(l_{1}^{+} \bigr)^{\gamma }+d^{M}\bigl(l_{2}^{+} \bigr)^{\delta }\thickapprox 0.9991, \\ &\hat{a}^{l}\bigl(l_{1}^{-} \bigr)^{\hat{\alpha }} +\hat{c}^{l}\bigl(l_{1}^{-} \bigr)^{\hat{\gamma }}\thickapprox 0.7155>\hat{r}^{M} + \hat{d}^{M}\bigl(l_{2}^{+}\bigr)^{\hat{\delta }} \thickapprox 0.6270. \end{aligned}$$

Thus, the assumptions $(H_{1})$–$(H_{3})$ are satisfied. So we derive from Theorem 3.1 that system (5.1) has at least one positive almost periodic solution $(\overline{x}_{1}(t),\overline{x}_{2}(t))$ satisfying $1.1951\leq \overline{x}_{1}(t)<2.8613$ and $1.1303\leq \overline{x}_{2}(t)<1.590$.

In addition, obviously, $0<1-\tau '(t)=1-\frac{\sqrt{2}\vert \cos \sqrt{2}t\vert }{4}<1$, $0<1-\sigma '(t)=1-\frac{\sqrt{3}\vert \cos \sqrt{3}t\vert }{3}<1$, $0<1-\hat{\tau }'(t)=1-\frac{\sqrt{2}\vert \sin \sqrt{2}t\vert }{3}<1$, $0<1-\hat{\sigma }'(t)=1-\frac{\sqrt{3}\vert \sin \sqrt{3}t\vert }{4}<1$, $\alpha =\frac{3}{4}>\max \{\beta,\gamma,\delta \}=\max \{\frac{1}{2},\frac{1}{3},\frac{1}{4}\}=\frac{1}{2}$, $\hat{\beta }=4>\max \{\hat{\alpha },\hat{\gamma },\hat{\delta }\}=\max \{3,\frac{3}{2},1\}=3$. Take $\rho =1<\min \{l_{1}^{-},l_{2}^{-}\}\thickapprox 1.1303$, we have

$$\begin{aligned} &{-}\alpha \rho a^{l} +\hat{\beta }\rho ^{\frac{\hat{\alpha }}{\hat{\beta }}} \hat{a}^{M} +\alpha \rho ^{\frac{\gamma }{\alpha }}c^{M} +\hat{\beta } \rho ^{\frac{\hat{\gamma }}{\hat{\beta }}}\hat{c}^{M}=-0.10< 0, \\ &{-}\hat{\beta }\rho \hat{b}^{l} +\alpha \rho ^{\frac{\beta }{\alpha }}b^{M} +\alpha \rho ^{\frac{\delta }{\alpha }}d^{M} +\hat{\beta }\rho ^{\frac{\hat{\delta }}{\hat{\beta }}} \hat{d}^{M}=-10.35< 0. \end{aligned}$$

Thus we verify that $(H_{4})$–$(H_{6})$ hold. Therefore, according to Theorem 4.1, we conclude that the unique positive almost periodic solution $(\overline{x}_{1}(t),\overline{x}_{2}(t))$ of system (5.1) is globally exponentially stable.

6 Conclusions

Compared with the Lotka–Volterra models, the Gilpin–Ayala models are more advantageous because the rate of change in the size of each species is a nonlinear function of the sizes of the interacting species in the Gilpin–Ayala models and the rate of change in the size of each species is a linear function of the sizes of the interacting species the Lotka–Volterra models. It is more precise to describe some ecosystems by the Gilpin–Ayala models than by the Lotka–Volterra models. Therefore, we mainly study a class of two-layer Gilpin–Ayala predator–prey model with time delays in this paper. By means of Mawhin’s continuation theorem of coincidence degree theory, we obtain some new sufficient conditions of the existence of positive almost periodic solutions. We also obtain the global exponential stability of the positive almost periodic solution for this system by constructing appropriate Lyapunov functionals and smart transformations. Our results provide a theoretical basis for the detection and artificial control of some ecological systems with periodic or almost periodic phenomena.