Full article: Global exponential stability of positive periodic solution of the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays

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ABSTRACT

In this paper, we study the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays. The existence of positive periodic solution is proved by employing the fixed point theorem on cones. By constructing appropriate Lyapunov functional, we also obtain the global exponential stability of the positive periodic solution of this system. As an application, an interesting example is provided to illustrate the validity of our main results.

KEYWORDS:

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

1. Introduction

In this paper, we mainly study the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays as follows: (1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) where the explanation of the variables is listed in the following table

Table

Display Table

By employing the fixed point theorem on cones and constructing appropriate Lyapunov functionals, we have researched the existence and global exponential stability of positive periodic solutions for (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ).

It is important and essential to investigate the model (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) attributing to its wide applications. On the one hand, system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) contain many mathematical population models. Some specific and important models of system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) are enumerated as follows. For example, when $α_{i j} = β_{i j} = γ_{i j} = 1 (i, j = 1, 2, \dots, n),$ system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) changes into the Lotka–Volterra competition population model of the form (2) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u (t_{k}), k = 1, 2, \dots . \end{aligned}$ (2) When $α_{i j} = β_{i j} = γ_{i j} = 1 (i, j = 1, 2, \dots, n, i \neq j),$ system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) changes into the Gilpin–Ayala competition model (Equation3(3) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - F_{i} (t, u (t))], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (3) ) of the form (3) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - F_{i} (t, u (t))], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (3) where $i = 1, 2, \dots, n,$ $u (t) = (u_{1} (t), u_{2} (t), \dots, u_{n} (t))^{T},$ $\begin{aligned} F_{i} (t, u (t)) & = a_{i i} (t) u_{i}^{α_{i i}} (t) + b_{i i} (t) u_{i}^{β_{i i}} (t - τ_{i i} (t)) + c_{i i} (t) \int_{- \infty}^{0} K_{i i} (s) u_{i}^{γ_{i i}} (t + s) d s \\ + \sum_{j = 1, j \neq i}^{n} a_{i j} (t) u_{j} (t) + \sum_{j = 1, j \neq i}^{n} b_{i j} (t) u_{j} (t - τ_{i j} (t)) \\ + \sum_{j = 1, j \neq i}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j} (t + s) d s . \end{aligned}$ In [Citation7], Gilpin and Ayala proposed a more realistic and complicated competition model as follows: (4) $u_{i}^{'} (t) = r_{i} u_{i} (t) [1 - {(\frac{u_{i} (t)}{K_{i}})}^{θ_{i}} - \sum_{j = 1, j \neq i}^{n} a_{i j} (t) \frac{u_{j} (t)}{K_{j}}], i = 1, 2, \dots, n,$ (4) where $u_{i} (t)$ 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, $θ_{i}$ provides a nonlinear measure of intraspecific interference, and $a_{i j} (t) (i \neq j)$ is the interspecific competition rate between the ith-species and the jth-species at time t. Compared with system (Equation4(4) $u_{i}^{'} (t) = r_{i} u_{i} (t) [1 - {(\frac{u_{i} (t)}{K_{i}})}^{θ_{i}} - \sum_{j = 1, j \neq i}^{n} a_{i j} (t) \frac{u_{j} (t)}{K_{j}}], i = 1, 2, \dots, n,$ (4) ), the model (Equation3(3) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - F_{i} (t, u (t))], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (3) ) is a more complicated and generalized model containing the model (Equation4(4) $u_{i}^{'} (t) = r_{i} u_{i} (t) [1 - {(\frac{u_{i} (t)}{K_{i}})}^{θ_{i}} - \sum_{j = 1, j \neq i}^{n} a_{i j} (t) \frac{u_{j} (t)}{K_{j}}], i = 1, 2, \dots, n,$ (4) ).

On the other hand, the model (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) is more advantageous compared with the Lotka–Volterra models. In fact, 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. However, 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 the Lotka–Volterra models. Therefore, as soon as it was put forward, the Gilpin–Ayala models have been widely focused and deeply studied by many scholars. There have been a number of papers dealing with the dynamics of the Gilpin–Ayala models (see [Citation1–6, Citation10–22, Citation24]).

In addition, the changing environment of ecosystem often presents a certain periodicity, for example, seasonal effects of weather, food supplies, mating habits, and so on. So it is more important and necessary to consider the effects of a periodically varying environment than a stable environment. The number of species in ecosystems sometimes changes suddenly because of the effects of natural disasters and human kill and so on. At the same time, the time delay effect is inevitable in ecosystem. To the best of our knowledge, few authors have considered the global exponential stability of positive periodic solutions for model (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ). Therefore, it is necessary to study the existence and global exponential stability of positive periodic solutions for (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ).

The rest of this paper is organized as follows. In Section 2, we shall give some notations and lemmas. In Section 3, we shall prove the existence of positive periodic solutions. In Section 4, we also obtain the global exponential stability of the positive periodic solution of this system by constructing appropriate Lyapunov functionals and inequality techniques. Finally, an example is given to illustrate the effectiveness of our results in Section 5.

2. Preliminaries

Throughout this paper, we always assume that the assumptions hold as follows:

$r_{i} (t),$ $a_{i j} (t),$ $b_{i j} (t),$ $c_{i j} (t) \in C (R, R^{+})$ and $τ_{i j} (t) \in C (R \times R^{n}, R^{+})$ are all ω-periodic with respect to the time variable t. $K_{i j} (s) \in C (R^{-}, R^{+})$ satisfy $\int_{- \infty}^{0} K_{i j} (s) d s = 1$ . $α_{i j} > 0,$ $β_{i j} > 0$ and $γ_{i j} > 0$ are the constants. Where $i, j = 1, 2, \dots, n,$ $k \in N,$ $R^{-} = (- \infty, 0),$ $R^{+} = (0, \infty),$ $R = (- \infty, + \infty),$ $ω > 0$ is a constant.
There exists a positive integer p such that $t_{k + p} = t_{k} + ω,$ $I_{i, k + p} = I_{i k}$ , $k \in Z,$ $i = 1, 2, \dots, n .$ Without loss of generality, we also assume that $[0, ω) \cap {t_{k} : k \in Z} = {t_{1}, t_{2}, \dots, t_{p}} .$
$I_{i k} > - 1, i = 1, 2, \dots, n, k \in N$ are the constants satisfying $\prod_{k = 1}^{p} (1 + I_{i k}) = 1$ .

Consider the following system (5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5)

Lemma 2.1

For systems (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) and (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ), the following results hold:

If $x_{i} (t) (i = 1, 2, \dots, n)$ is a solution of (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ), then $u_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k}) x_{i} (t) (i = 1, 2, \dots, n)$ is a solution of (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ).
If $u_{i} (t) (i = 1, 2, \dots, n)$ is a solution of (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ), then $x_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k})^{- 1} u_{i} (t) (i = 1, 2, \dots, n)$ is a solution of (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ).

Proof.

Suppose that $x_{i} (t) (i = 1, 2, \dots, n)$ is a solution of (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ). Let $u_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k}) x_{i} (t) (i = 1, 2, \dots, n)$ , then for any $t \neq t_{k}, k \in N,$ by substituting $x_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k})^{- 1} u_{i} (t)$ into system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ), we can easily verify that the first equation of (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) holds. For $t = t_{k}, k \in N, i = 1, 2, \dots, n,$ we obtain $\begin{aligned} u_{i} (t_{k}^{+}) & = lim_{t \to t_{k}^{+}} \prod_{0 < t_{k} < t} (1 + I_{i k}) x_{i} (t) = \prod_{0 < t_{h} \leq t_{k}} (1 + I_{i h}) x_{i} (t_{k}) \\ = (1 + I_{i k}) \prod_{0 < t_{h} < t_{k}} (1 + I_{i h}) x_{i} (t_{k}) = (1 + I_{i k}) u_{i} (t_{k}) . \end{aligned}$ So, the second equation of (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) also holds. Thus $u_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k}) x_{i} (t) (i = 1, 2, \dots, n)$ is a solution of (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ).
We first show that $x_{i}, i = 1, 2, \dots, n$ are continuous. Indeed, $x_{i}, i = 1, 2, \dots, n$ are continuous on each interval $(t_{k}, t_{k + 1}] .$ It is necessary to check the continuity of $x_{i} (t)$ at the impulse points $t = t_{k}, k \in N .$ Since $x_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k})^{- 1} u_{i} (t)$ , $i = 1, 2, \dots, n,$ we have $\begin{aligned} x_{i} (t_{k}^{+}) & = \prod_{0 < t_{h} \leq t_{k}} (1 + I_{i h})^{- 1} u_{i} (t_{k}^{+}) = \prod_{0 < t_{h} \leq t_{k}} (1 + I_{i h})^{- 1} (1 + I_{i k}) u_{i} (t_{k}) \\ = \prod_{0 < t_{h} < t_{k}} (1 + I_{i h})^{- 1} u_{i} (t_{k}) = x_{i} (t_{k}), \\ x_{i} (t_{k}^{-}) & = \prod_{0 < t_{h} \leq t_{k}} (1 + I_{i h})^{- 1} u_{i} (t_{k}^{-}) = \prod_{0 < t_{h} \leq t_{k}} (1 + I_{i h})^{- 1} (1 + I_{i k}) u_{i} (t_{k}) \\ = \prod_{0 < t_{h} < t_{k}} (1 + I_{i h})^{- 1} u_{i} (t_{k}) = x_{i} (t_{k}) . \end{aligned}$ Thus $x_{i} (t), i = 1, 2, \dots, n$ is continuous on $[0, + \infty)$ . It is easy to check that $x_{i} (t), i = 1, 2, \dots, n$ satisfies Equation (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ). Therefore, it is a solution of (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ). This completes the proof of Lemma 2.1.

It follows from Lemma 2.1 that if system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ) exists a globally exponentially stable positive periodic solution ${\bar{x}}_{i} (t), i = 1, 2, \dots, n,$ then ${\bar{u}}_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k}) {\bar{x}}_{i} (t) (i = 1, 2, \dots, n)$ is the globally exponentially stable positive periodic solution of system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ). So it is necessary to discuss the existence of globally exponentially stable positive periodic solution for system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ).

For convenience, we introduce some notations $θ = min_{1 \leq i, j \leq n} {α_{i j}, β_{i j}, γ_{i j}}, δ_{i} = e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}, f^{M} = max_{t \in [0, ω]} {f (t)}, f^{l} = max_{t \in [0, ω]} {f (t)},$ where $i = 1, 2, \dots, n$ and $f (t)$ is a continuous ω-periodic function on $R .$

Making the change of variable $y_{i} (t) = x_{i}^{θ} (t),$ then we can transform system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ) into (6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) The existence of globally exponentially stable positive periodic solution for system (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ) is equivalent to the existence of globally exponentially stable positive periodic solution of the corresponding integral system. So the following lemmas are important in our discussion.

Lemma 2.2

Let $r \in C (R, R),$ $a \in R$ and $y_{a} \in R,$ the unique solution of the initial value problem $v^{'} (t) = r (t) v (t) + h (t), v (a) = v_{a}$ is given by $v (t) = v_{a} e^{\int_{a}^{t} r (s) d s} + \int_{a}^{t} e^{- \int_{t}^{s} r (τ) d τ} h (s) d s .$

Lemma 2.3

If $(H_{1})$ – $(H_{3})$ hold, then $y (t) = (y_{1} (t), y_{2} (t), \dots, y_{n} (t))^{T}$ is an ω-periodic solution of (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ) is equivalent to $y (t)$ is an ω-periodic solution of the following integral system (7) $y_{i} (t) = θ \int_{t}^{t + ω} G_{i} (t, s) y_{i} (s) H_{i} (s, y (s)) d s, i = 1, 2, \dots, n,$ (7) where (8) $G_{i} (t, s) = \frac{e^{- θ \int_{t}^{s} r_{i} (τ) d τ}}{1 - e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}}, s \in [t, t + ω], i = 1, 2, \dots, n,$ (8) (9) $\begin{aligned} H_{i} (s, y (s)) & = \sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ, i = 1, 2, \dots, n . \end{aligned}$ (9)

Proof.

If $y (t) = (y_{1} (t), y_{2} (t), \dots, y_{n} (t))^{T}$ is an ω-periodic solution of (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ), $\forall t \in R,$ by applying Lemma 2.2 and (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ), for $s \in [t, + \infty),$ we have $y_{i} (s) = y_{i} (t) e^{θ \int_{t}^{s} r_{i} (τ) d τ} - θ \int_{t}^{s} e^{- θ \int_{s}^{τ} r_{i} (ξ) d ξ} y_{i} (τ) H_{i} (τ, y (τ)) d τ .$ Let $s = t + ω$ in the above equality and notice that $y_{i} (t) = y_{i} (t + ω),$ $r_{i} (t) = r_{i} (t + ω),$ we have $\begin{aligned} y_{i} (t) = y_{i} (t + ω) = y_{i} (t) e^{θ \int_{t}^{t + ω} r_{i} (τ) d τ} - θ \int_{t}^{t + ω} e^{- θ \int_{t + ω}^{τ} r_{i} (ξ) d ξ} y_{i} (τ) H_{i} (τ, y (τ)) d τ \\ = y_{i} (t) e^{θ \int_{0}^{ω} r_{i} (τ) d τ} - θ e^{θ \int_{0}^{ω} r_{i} (ξ) d ξ} \int_{t}^{t + ω} e^{- θ \int_{t}^{τ} r_{i} (ξ) d ξ} y_{i} (τ) H_{i} (τ, y (τ)) d τ, \end{aligned}$ which imply that $\begin{aligned} y_{i} (t) & = θ \int_{t}^{t + ω} \frac{e^{- θ \int_{t}^{τ} r_{i} (ξ) d ξ}}{1 - e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}} y_{i} (τ) H_{i} (τ, y (τ)) d τ \\ = θ \int_{t}^{t + ω} G_{i} (t, τ) y_{i} (τ) H_{i} (τ, y (τ)) d τ, \end{aligned}$ where $G_{i} (t, s)$ and $H_{i} (s, y (s))$ are defined by Equations (Equation8(8) $G_{i} (t, s) = \frac{e^{- θ \int_{t}^{s} r_{i} (τ) d τ}}{1 - e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}}, s \in [t, t + ω], i = 1, 2, \dots, n,$ (8) ) and (Equation9(9) $\begin{aligned} H_{i} (s, y (s)) & = \sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ, i = 1, 2, \dots, n . \end{aligned}$ (9) ). Thus, we conclude that $y (t)$ satisfies Equation (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ), and vice versa. The proof of Lemma 2.3 is complete.

Lemma 2.4

If the condition $(H_{1})$ – $(H_{3})$ hold, then $G_{i} (t, s) (i = 1, 2, \dots, n)$ and $H_{i} (s, y (s)) (i = 1, 2, \dots, n)$ defined by Equations (Equation8(8) $G_{i} (t, s) = \frac{e^{- θ \int_{t}^{s} r_{i} (τ) d τ}}{1 - e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}}, s \in [t, t + ω], i = 1, 2, \dots, n,$ (8) ) and (Equation9(9) $\begin{aligned} H_{i} (s, y (s)) & = \sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ, i = 1, 2, \dots, n . \end{aligned}$ (9) ) satisfies the following:

$δ_{i} / (1 - δ_{i}) \leq G_{i} (t, s) \leq 1 / (1 - δ_{i}),$ $\forall s \in [t, t + ω],$ where $δ_{i} = e^{- θ \int_{0}^{ω} r_{i} (τ) d τ},$ $i =, 1, 2, \dots, n .$
$G_{i} (t + ω, s + ω) = G_{i} (t, s), \forall s, t \in R, i = 1, 2, \dots, n$ .
$H_{i} (s + ω, y (s + ω)) = H_{i} (s, y (s)), \forall y_{i} (s + ω) = y_{i} (s), i = 1, 2, \dots, n$ .

Proof.

According to $(H_{1}),$ we know $θ r_{i} (t) > 0$ and $\partial G_{i} (t, s) / \partial s < 0,$ that is, $G_{i} (t, s)$ is monotone decreasing with respect to s. Thus, for $s \in [t, t + ω],$ we have $\frac{δ_{i}}{1 - δ_{i}} = \frac{e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}}{1 - δ_{i}} = \frac{e^{- θ \int_{t}^{t + ω} r_{i} (τ) d τ}}{1 - δ_{i}} \leq G_{i} (t, s) \leq \frac{e^{- θ \int_{t}^{t} r_{i} (τ) d τ}}{1 - δ_{i}} = \frac{1}{1 - δ_{i}} .$ Thus, the assertion $(1)$ holds. Now we show that the assertion $(2)$ also holds. In fact, by the integration by substitution, we have $\begin{aligned} G_{i} (t + ω, s + ω) & = \frac{e^{- θ \int_{t + ω}^{s + ω} r_{i} (τ) d τ}}{1 - δ_{i}} = \frac{e^{- θ (\int_{t + ω}^{t} + \int_{t}^{s} + \int_{s}^{s + ω}) r_{i} (τ) d τ}}{1 - δ_{i}} \\ = \frac{e^{- θ \int_{t}^{s} r_{i} (τ) d τ}}{1 - δ_{i}} = G_{i} (t, s) . \end{aligned}$ According to Equation (Equation9(9) $\begin{aligned} H_{i} (s, y (s)) & = \sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ, i = 1, 2, \dots, n . \end{aligned}$ (9) ) and $(H_{1})$ – $(H_{3})$ , we obtain $\begin{aligned} H_{i} (s + ω, y (s + ω)) \\ = \sum_{j = 1}^{n} a_{i j} (s + ω) \prod_{0 < t_{k} < s + ω} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s + ω) \\ + \sum_{j = 1}^{n} b_{i j} (s + ω) \prod_{0 < t_{k} < s + ω} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s + ω - τ_{i j} (s + ω)) \\ + \sum_{j = 1}^{n} c_{i j} (s + ω) \prod_{0 < t_{k} < s + ω} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + ω + τ) d τ \\ = \sum_{j = 1}^{n} a_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{α_{i j}} \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{β_{i j}} \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{γ_{i j}} \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ \\ = \sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ \\ = H_{i} (s, y (s)), i = 1, 2, \dots, n . \end{aligned}$ Thus Equation (Equation3(3) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - F_{i} (t, u (t))], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (3) ) also holds. This completes the proof of Lemma 2.4.

Lemma 2.5

When $ϑ \geq 1$ and $0 < u \leq 1,$ then $0 < u^{ϑ} \leq u \leq 1;$ When $ϑ \geq 1$ and $u \geq 1,$ then $1 \leq u \leq u^{ϑ}$ .

Proof.

Let $f (u) = u^{ϑ - 1},$ when $ϑ \geq 1$ and $u > 0,$ then $f^{'} (u) = (ϑ - 1) u^{ϑ - 2} \geq 0,$ that is, $f (u)$ is monotone increasing on $(0, \infty) .$ Thus, when $0 < u \leq 1,$ we have $0 < u^{ϑ} \leq 1$ and $u^{ϑ - 1} = f (u) \leq f (1) = 1,$ namely, $0 < u^{ϑ} \leq u \leq 1;$ when $u \geq 1,$ we have $u^{ϑ} \geq 1$ and $f (u) = u^{ϑ - 1} \geq f (1) = 1,$ namely, $1 \leq u \leq u^{ϑ}$ . The proof is complete.

Let X be a real Banach space, and K be a closed non-empty subset of X. Then K is a cone provided

$k α + l β \in K$ for all $α, β \in K$ and all $k, l \geq 0;$
$α, - α \in K$ imply $α = θ,$ here θ is the zero element of X.

The following lemma is useful for the proof of existence of positive periodic solution for system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ).

Lemma 2.6

[Citation9]

Let K be a cone in the Banach space X, and $Ω_{1}, Ω_{2}$ be two bounded open balls of X centred at the origin with $0 \in Ω_{1}$ and ${\bar{Ω}}_{1} \subset Ω_{2} .$ Suppose that $Φ : K \cap ({\bar{Ω}}_{2} ∖ Ω_{1}) \to K$ is a completely continuous operator such that either

$∥ Φ u ∥ \leq ∥ u ∥, u \in K \cap \partial Ω_{1}$ and $∥ Φ u ∥ \geq ∥ u ∥, u \in K \cap \partial Ω_{2},$ or
$∥ Φ u ∥ \geq ∥ u ∥, u \in K \cap \partial Ω_{1}$ and $∥ Φ u ∥ \leq ∥ u ∥, u \in K \cap \partial Ω_{2}$

hold. Then Φ has at least one fixed point in

K \cap ({\bar{Ω}}_{2} ∖ Ω_{1}) .

3. Existence of positive periodic solutions

In this section, we shall apply Lemma 2.6 to prove the existence of positive ω-periodic solutions for system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ). To do so, define $X = {y = (y_{1}, y_{2}, \dots, y_{n}) \in C (R, R^{n}) : y_{i} (t + ω) = y_{i} (t), t \in R, i = 1, 2, \dots, n}$ equipped with the norm defined by $∥ y ∥ = \sum_{i = 1}^{n} | y_{i} |_{0},$ where $| y_{i} |_{0} = sup_{t \in [0, ω]} {| y_{i} (t) |}$ , $i = 1, 2, \dots, n .$ Then X is a Banach space. In view of Lemma 2.4, we define the cone K in X as $K = \{y = (y_{1}, \dots, y_{n}) \in X : y_{i} (t) \geq δ_{i} ∥ y ∥, \forall t \in [0, ω], i = 1, 2, \dots, n\} .$ Let the map Φ be defined by (10) $(Φ y) (t) = ((Φ_{1} y) (t), (Φ_{2} y) (t), \dots, (Φ_{n} y) (t))^{T},$ (10) where $y \in K, t \in R,$ $(Φ_{i} y) (t) = θ \int_{t}^{t + ω} G_{i} (t, s) y_{i} (s) H_{i} (s, y (s)) d s, i = 1, 2, \dots, n,$ $G_{i} (t, s) (i = 1, 2, \dots, n)$ and $H_{i} (s, y (s))$ defined by Equations (Equation8(8) $G_{i} (t, s) = \frac{e^{- θ \int_{t}^{s} r_{i} (τ) d τ}}{1 - e^{- θ \int_{0}^{ω} r_{i} (τ) d τ}}, s \in [t, t + ω], i = 1, 2, \dots, n,$ (8) ) and (Equation9(9) $\begin{aligned} H_{i} (s, y (s)) & = \sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ, i = 1, 2, \dots, n . \end{aligned}$ (9) ).

Lemma 3.1

Assume that $(H_{1})$ – $(H_{3})$ hold, then $Φ : K \to K$ defined by Equation (Equation10(10) $(Φ y) (t) = ((Φ_{1} y) (t), (Φ_{2} y) (t), \dots, (Φ_{n} y) (t))^{T},$ (10) ) is well defined, namely, $Φ (K) \subset K .$

Proof.

For any $y \in K,$ it is clear that $Φ y \in C (R, R^{n}) .$ In view of Lemma 2.4 and Equation (Equation10(10) $(Φ y) (t) = ((Φ_{1} y) (t), (Φ_{2} y) (t), \dots, (Φ_{n} y) (t))^{T},$ (10) ), we obtain $\begin{aligned} (Φ_{i} y) (t + ω) = θ \int_{t + ω}^{t + 2 ω} G_{i} (t + ω, s) y_{i} (s) H_{i} (s, y (s)) d s \\ = θ \int_{t}^{t + ω} G_{i} (t + ω, τ + ω) y_{i} (τ + ω) H_{i} (τ + ω, y (τ + ω)) d τ \\ = θ \int_{t}^{t + ω} G_{i} (t, τ) y_{i} (τ) H_{i} (τ, y (τ)) d τ = (Φ_{i} y) (t), \end{aligned}$ that is, $(Φ_{i} y) (t + ω) = (Φ_{i} y) (t), \forall t \in R, i = 1, 2, \dots, n .$ So $Φ y \in X .$ For any $y \in K,$ we have $| Φ_{i} y |_{0} \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) H_{i} (s, y (s)) d s, i = 1, 2, \dots, n$ and $(Φ_{i} y) (t) \geq \frac{θ δ_{i}}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) H_{i} (s, y (s)) d s \geq δ_{i} | Φ_{i} y |_{0}, i = 1, 2, \dots, n .$ So $Φ u \in K .$ This completes the proof of Lemma 3.1.

Lemma 3.2

Assume that $(H_{1})$ – $(H_{3})$ hold, then $Φ : K \to K$ defined by Equation (Equation10(10) $(Φ y) (t) = ((Φ_{1} y) (t), (Φ_{2} y) (t), \dots, (Φ_{n} y) (t))^{T},$ (10) ) is completely continuous.

Proof.

It is easy to see that Φ is continuous and bounded. Now we show that Φ maps bounded sets into relatively compact sets. Let $Ω \subset K$ be an arbitrary open bounded set in K, then there exists a number R>0 such that $∥ y ∥ < R$ for any $y = (y_{1}, y_{2}, \dots, y_{n})^{T} \in Ω$ . We prove that $\bar{Φ (Ω)}$ is compact. In fact, for any $y \in Ω$ and $t \in [0, ω]$ , by $(H_{1})$ – $(H_{3})$ , we have $\begin{aligned} | (Φ_{i} y) (t) | = θ \int_{t}^{t + ω} G_{i} (t, s) y_{i} (s) H_{i} (s, y (s)) d s \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) H_{i} (s, y (s)) d s \\ = \frac{θ}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) (\sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ) d s \\ \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} | y_{i} (s) | (\sum_{j = 1}^{n} a_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{α_{i j}} | y_{j} (s) |^{α_{i j} / θ} \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{β_{i j}} | y_{j} (s - τ_{i j} (s)) |^{β_{i j} / θ} \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) | y_{j} (s + τ) |^{γ_{i j} / θ} d τ) d s \\ \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} | y_{i} |_{0} (\sum_{j = 1}^{n} a_{i j}^{M} | y_{j} |_{0}^{α_{i j} / θ} \\ + \sum_{j = 1}^{n} b_{i j}^{M} | y_{j} |_{0}^{β_{i j} / θ} + \sum_{j = 1}^{n} c_{i j}^{M} \int_{- \infty}^{0} K_{i j} (τ) | y_{j} |_{0}^{γ_{i j} / θ} d τ) d s \\ \leq \frac{θ ω ∥ y ∥}{1 - δ_{i}} (\sum_{j = 1}^{n} a_{i j}^{M} ∥ y ∥^{α_{i j} / θ} + \sum_{j = 1}^{n} b_{i j}^{M} ∥ y ∥^{β_{i j} / θ} + \sum_{j = 1}^{n} c_{i j}^{M} ∥ y ∥^{γ_{i j} / θ}) \\ < \frac{θ ω R}{1 - δ_{i}} (\sum_{j = 1}^{n} a_{i j}^{M} R^{α_{i j} / θ} + \sum_{j = 1}^{n} b_{i j}^{M} R^{β_{i j} / θ} + \sum_{j = 1}^{n} c_{i j}^{M} R^{γ_{i j} / θ}) ≜ A_{i}, i = 1, 2, \dots, n \end{aligned}$ and $\begin{aligned} | (Φ_{i} y)^{'} (t) | = | θ r_{i} (t) (Φ_{i} x) (t) - θ y_{i} (t) H_{i} (s, y (t)) | \\ = θ |r_{i} (t) (Φ_{i} y) (t) - y_{i} (s) [\sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ + \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ + \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (t + τ) d τ]| \\ \leq θ r_{i}^{M} A_{i} + θ ∥ y ∥ [\sum_{j = 1}^{n} a_{i j}^{M} {(\prod_{k = 1}^{p} (1 + I_{i k})^{α_{i j}})}^{m} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i j}} ∥ y ∥^{α_{i j} / θ} \\ + \sum_{j = 1}^{n} b_{i j}^{M} {(\prod_{k = 1}^{p} (1 + I_{i k})^{β_{i j}})}^{m} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{β_{i j}} ∥ y ∥^{β_{i j} / θ} \\ + \sum_{j = 1}^{n} c_{i j}^{M} {(\prod_{k = 1}^{p} (1 + I_{i k})^{γ_{i j}})}^{m} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) ∥ y ∥^{γ_{i j} / θ} d τ] \\ \leq θ r_{i}^{M} A_{i} + θ ∥ y ∥ [\sum_{j = 1}^{n} a_{i j}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i j}} ∥ y ∥^{α_{i j} / θ} \\ + \sum_{j = 1}^{n} b_{i j}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{β_{i j}} ∥ y ∥^{β_{i j} / θ} + \sum_{j = 1}^{n} c_{i j}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{γ_{i j}} ∥ y ∥^{γ_{i j} / θ}] \\ < θ r_{i}^{M} A_{i} + θ R [\sum_{j = 1}^{n} a_{i j}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i j}} R^{α_{i j} / θ} + \sum_{j = 1}^{n} b_{i j}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{β_{i j}} ∥ R^{β_{i j} / θ} \\ + \sum_{j = 1}^{n} c_{i j}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{γ_{i j}} R^{γ_{i j} / θ}] ≜ B_{i}, i = 1, 2, \dots, n, \end{aligned}$ where $m = [t / ω]$ . Hence, $∥ (Φ y) ∥ \leq max {A_{1}, A_{2}, \dots, A_{n}},$ $∥ (Φ y)^{'} ∥ \leq max {B_{1}, B_{2}, \dots, B_{n}} .$ It follows from Lemma 2.4 in [Citation23] that $Φ (\bar{Ω})$ is relatively compact in X. Therefore, $Φ : K \to K$ defined by Equation (Equation10(10) $(Φ y) (t) = ((Φ_{1} y) (t), (Φ_{2} y) (t), \dots, (Φ_{n} y) (t))^{T},$ (10) ) is completely continuous.

Theorem 3.1

Assume that $(H_{1})$ – $(H_{3})$ hold. Assume further that

$(θ ω δ_{i}^{2} / (1 - δ_{i})) \sum_{j = 1}^{n} (a_{i j}^{l} δ_{j}^{α_{i j} / θ} + b_{i j}^{l} δ_{j}^{β_{i j} / θ} + c_{i j}^{l} δ_{j}^{γ_{i j} / θ}) < 1,$ $i = 1, 2, \dots, n .$

Then system (Equation1) has at least one positive ω-periodic solution.

Proof.

Define the operator $Φ : K \to K$ as Equation (Equation10(10) $(Φ y) (t) = ((Φ_{1} y) (t), (Φ_{2} y) (t), \dots, (Φ_{n} y) (t))^{T},$ (10) ). According to Lemmas 3.1–3.2, we know that $Φ : K \to K$ is completely continuous. According to $(H_{4})$ , we take $\begin{aligned} r_{1} & = min_{1 \leq i \leq n} \{\frac{1 - δ_{i}}{θ ω \sum_{j = 1}^{n} (a_{i j}^{M} + b_{i j}^{M} + c_{i j}^{M}) + 1}\}, \\ r_{2} & = max_{1 \leq i \leq n} \{{[\frac{θ ω δ_{i}^{2}}{1 - δ_{i}} \sum_{j = 1}^{n} (a_{i j}^{l} δ_{j}^{α_{i j} / θ} + b_{i j}^{l} δ_{j}^{β_{i j} / θ} + c_{i j}^{l} δ_{j}^{γ_{i j} / θ})]}^{- 1}\} . \end{aligned}$ Obviously, $0 < r_{1} < 1 < r_{2} .$ Define $Ω_{i} = {u \in K : ∥ u ∥ < r_{i}}, i = 1, 2$ . By Lemma 2.3, it is easy to see that if there exists $\bar{y} (t) \in K$ such that $(Φ \bar{y}) (t) = \bar{y} (t),$ then $\bar{y} (t)$ is one positive ω-periodic solution of system (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ). Now, we shall prove that condition $(i)$ of Lemma 2.6 holds. In fact, For all $y \in K \cap \partial Ω_{1},$ namely, $∥ y ∥ = r_{1} < 1,$ from $(H_{1})$ – $(H_{4})$ and Lemma 2.4–2.5 and noticing that $α_{i j} / θ \geq 1,$ $β_{i j} / θ \geq 1$ and $γ_{i j} / θ \geq 1, i, j = 1, 2, \dots, n,$ we have $\begin{aligned} | (Φ_{i} y) (t) | & = θ \int_{t}^{t + ω} G_{i} (t, s) y_{i} (s) H_{i} (s, y (s)) d s \\ \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) H_{i} (s, y (s)) d s \\ = \frac{θ}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) [\sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ] d s \\ \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} | y_{i} (s) | [\sum_{j = 1}^{n} a_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{α_{i j}} | y_{j} (s) |^{α_{i j} / θ} \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{β_{i j}} | y_{j} (s - τ_{i j} (s)) |^{β_{i j} / θ} \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) | y_{j} (s + τ) |^{γ_{i j} / θ} d τ] d s \\ \leq \frac{θ}{1 - δ_{i}} \int_{0}^{ω} | y_{i} |_{0} [\sum_{j = 1}^{n} a_{i j}^{M} | y_{j} |_{0}^{α_{i j} / θ} + \sum_{j = 1}^{n} b_{i j}^{M} | y_{j} |_{0}^{β_{i j} / θ} \\ + \sum_{j = 1}^{n} c_{i j}^{M} \int_{- \infty}^{0} K_{i j} (τ) | y_{j} |_{0}^{γ_{i j} / θ} d τ] d s \\ \leq \frac{θ ω ∥ y ∥}{1 - δ_{i}} [\sum_{j = 1}^{n} a_{i j}^{M} ∥ y ∥^{α_{i j} / θ} + \sum_{j = 1}^{n} b_{i j}^{M} ∥ y ∥^{β_{i j} / θ} + \sum_{j = 1}^{n} c_{i j}^{M} ∥ y ∥^{γ_{i j} / θ}] \\ \leq \frac{θ ω ∥ y ∥}{1 - δ_{i}} [\sum_{j = 1}^{n} a_{i j}^{M} ∥ y ∥ + \sum_{j = 1}^{n} b_{i j}^{M} ∥ y ∥ + \sum_{j = 1}^{n} c_{i j}^{M} ∥ y ∥] \\ = \frac{θ ω r_{1}^{2}}{1 - δ_{i}} \sum_{j = 1}^{n} (a_{i j}^{M} + b_{i j}^{M} + c_{i j}^{M}) \\ \leq \frac{θ ω r_{1}}{1 - δ_{i}} \sum_{j = 1}^{n} (a_{i j}^{M} + b_{i j}^{M} + c_{i j}^{M}) \times \frac{1 - δ_{i}}{θ ω \sum_{j = 1}^{n} (a_{i j}^{M} + b_{i j}^{M} + c_{i j}^{M}) + 1} < r_{1} = ∥ u ∥, \end{aligned}$ which indicates that (11) $∥ Φ u ∥ < ∥ u ∥, u \in K \cap \partial Ω_{1} .$ (11) On the other hand, for all $u \in K \cap \partial Ω_{2},$ namely, $∥ y ∥ = r_{2} > 1,$ according to $(H_{1})$ – $(H_{4})$ and Lemma 2.4–2.5 and noticing that $α_{i j} / θ \geq 1,$ $β_{i j} / θ \geq 1$ and $γ_{i j} / θ \geq 1, i, j = 1, 2, \dots, n,$ we obtain $\begin{aligned} | (Φ_{i} y) (t) | & = θ \int_{t}^{t + ω} G_{i} (t, s) y_{i} (s) H_{i} (s, y (s)) d s \\ \geq \frac{θ δ_{i}}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) H_{i} (s, y (s)) d s \\ = \frac{θ δ_{i}}{1 - δ_{i}} \int_{0}^{ω} y_{i} (s) [\sum_{j = 1}^{n} a_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (s) \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (s - τ_{i j} (s)) \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{0 < t_{k} < s} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) y_{j}^{γ_{i j} / θ} (s + τ) d τ] d s \\ \geq \frac{θ δ_{i}}{1 - δ_{i}} \int_{0}^{ω} δ_{i} ∥ y ∥ [\sum_{j = 1}^{n} a_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{α_{i j}} (δ_{j} ∥ y ∥)^{α_{i j} / θ} \\ + \sum_{j = 1}^{n} b_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{β_{i j}} (δ_{j} ∥ y ∥)^{β_{i j} / θ} \\ + \sum_{j = 1}^{n} c_{i j} (s) \prod_{k = 1}^{p} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (τ) (δ_{j} ∥ y ∥)^{γ_{i j} / θ} d τ] d s \\ \geq \frac{θ ω δ_{i}^{2} ∥ y ∥}{1 - δ_{i}} [\sum_{j = 1}^{n} a_{i j}^{l} δ_{j}^{α_{i j} / θ} ∥ y ∥^{α_{i j} / θ} + \sum_{j = 1}^{n} b_{i j}^{l} δ_{j}^{β_{i j} / θ} ∥ y ∥^{β_{i j} / θ} + \sum_{j = 1}^{n} c_{i j}^{l} δ_{j}^{γ_{i j} / θ} ∥ y ∥^{γ_{i j} / θ}] \\ \geq \frac{θ ω δ_{i}^{2} ∥ y ∥}{1 - δ_{i}} [\sum_{j = 1}^{n} a_{i j}^{l} δ_{j}^{α_{i j} / θ} ∥ y ∥ + \sum_{j = 1}^{n} b_{i j}^{l} δ_{j}^{β_{i j} / θ} ∥ y ∥ + \sum_{j = 1}^{n} c_{i j}^{l} δ_{j}^{γ_{i j} / θ} ∥ y ∥] \\ = \frac{θ ω δ_{i}^{2} r_{2}^{2}}{1 - δ_{i}} \sum_{j = 1}^{n} (a_{i j}^{l} δ_{j}^{α_{i j} / θ} + b_{i j}^{l} δ_{j}^{β_{i j} / θ} + c_{i j}^{l} δ_{j}^{γ_{i j} / θ}) \\ \geq \frac{θ ω δ_{i}^{2} r_{2}}{1 - δ_{i}} \sum_{j = 1}^{n} (a_{i j}^{l} δ_{j}^{α_{i j} / θ} + b_{i j}^{l} δ_{j}^{β_{i j} / θ} + c_{i j}^{l} δ_{j}^{γ_{i j} / θ}) \\ \times \frac{1 - δ_{i}}{θ ω δ_{i}^{2} \sum_{j = 1}^{n} (a_{i j}^{l} δ_{j}^{α_{i j} / θ} + b_{i j}^{l} δ_{j}^{β_{i j} / θ} + c_{i j}^{l} δ_{j}^{γ_{i j} / θ})} = r_{2} = ∥ u ∥, \end{aligned}$ which implies that (12) $∥ Φ u ∥ \geq ∥ u ∥, u \in K \cap \partial Ω_{2} .$ (12) From Equations (Equation11(11) $∥ Φ u ∥ < ∥ u ∥, u \in K \cap \partial Ω_{1} .$ (11) )–(Equation12(12) $∥ Φ u ∥ \geq ∥ u ∥, u \in K \cap \partial Ω_{2} .$ (12) ),we know that condition $(i)$ of Lemma 2.5 holds. By Lemma 2.6, we see that Φ has at least one non-zero fixed point $\bar{y} (t) = ({\bar{y}}_{1} (t), {\bar{y}}_{2} (t), \dots, {\bar{y}}_{n} (t))^{T} \in K \cap ({\bar{Ω}}_{2} ∖ Ω_{1}) .$ Therefore, system (Equation6(6) $\begin{aligned} y_{i}^{'} (t) & = θ y_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} y_{j}^{α_{i j} / θ} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} y_{j}^{β_{i j} / θ} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) y_{j}^{γ_{i j} / θ} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (6) ) has at least one positive ω-periodic solution $\bar{y} (t) = ({\bar{y}}_{1} (t), {\bar{y}}_{2} (t), \dots, {\bar{y}}_{n} (t))^{T} \in K \cap ({\bar{Ω}}_{2} ∖ Ω_{1})$ . Thus, $({\bar{x}}_{1} (t), {\bar{x}}_{2} (t), \dots, {\bar{x}}_{n} (t))^{T} = ({\bar{y}}_{1}^{1 / θ} (t), {\bar{y}}_{2}^{1 / θ} (t), \dots, {\bar{y}}_{n}^{1 / θ} (t))^{T}$ is a positive ω-periodic solution of system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ). From Lemma 2.1, we know that ${\bar{u}}_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k}) {\bar{x}}_{i} (t) = \prod_{0 < t_{k} < t} (1 + I_{i k}) {\bar{y}}_{i}^{1 / θ} (t) (i = 1, 2, \dots, n)$ is a positive ω-periodic solution of system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ). This completes the proof of Theorem 3.1.

4. Global exponential stability

The aim of this section is to derive the sufficient condition of a unique globally exponentially stable positive periodic solution of (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ).

According to Lemma 2.1, we know that the global exponential stability of positive periodic solution for system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) and system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ) is equivalent. So we are mainly prepared to investigate the global exponential stability of positive periodic solution for system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ). Under the assumption of Theorem 3.1, we find that system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ) has at least one positive ω-periodic solution $({\bar{x}}_{1} (t), {\bar{x}}_{2} (t), \dots, {\bar{x}}_{n} (t))^{T}$ and there exist positive constants ${\underline{C}}_{i}, {\bar{C}}_{i}$ such that ${\underline{C}}_{i} \leq {\bar{x}}_{i} (t) \leq {\bar{C}}_{i},$ where ${\underline{C}}_{i} = ((1 - δ_{i}) / θ ω \sum_{j = 1}^{n} (a_{i j}^{M} + b_{i j}^{M} + c_{i j}^{M}) + 1)^{1 / θ},$ ${\bar{C}}_{i} = [(θ ω δ_{i}^{2} / (1 - δ_{i})) \sum_{j = 1}^{n} (a_{i j}^{l} δ_{j}^{α_{i j} / θ} + b_{i j}^{l} δ_{j}^{β_{i j} / θ} + c_{i j}^{l} δ_{j}^{γ_{i j} / θ})]^{- 1 / θ},$ $i = 1, 2, \dots, n .$ Now let ρ be a positive constant satisfying $0 < ρ < min_{1 \leq i \leq n} {{\underline{C}}_{i}}$ . We assume further that

$τ_{i k} (t) (i, j = 1, 2, \dots, n) \in C^{1} (R, R^{+})$ satisfy $0 \leq τ_{i k}^{'} (t) < 1 (i = 1, 2, \dots, n)$ .
$α_{i i} \geq max_{1 \leq j \leq n} {α_{i j}, β_{i j}, γ_{i j}},$ $i = 1, 2, \dots, n .$
$- α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} < 0,$ where $m = [t / ω]$ , $i = 1, 2, \dots, n .$

Making the change of variable $z_{i} (t) = (1 / ρ) x_{i}^{α_{i i}} (t),$ $i = 1, 2, \dots, n,$ then system (Equation5(5) $\begin{aligned} x_{i}^{'} (t) & = x_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} x_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} x_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) x_{j}^{γ_{i j}} (t + s) d s], i = 1, 2, \dots, n . \end{aligned}$ (5) ) is transformed into (13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) Obviously, $\bar{z} (t) = ({\bar{z}}_{1} (t), {\bar{z}}_{2} (t), \dots, {\bar{z}}_{n} (t))^{T}$ is the positive ω-periodic solution of system (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ), where ${\bar{z}}_{i} (t) = (1 / ρ) {\bar{x}}_{i}^{α_{i i}} (t), i = 1, 2, \dots, n .$ From Theorem 3.1, we know that ${\underline{C}}_{i} < {\bar{x}}_{i} (t) < {\bar{C}}_{i} (i = 1, 2, \dots, n) .$ Therefore, (14) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} \leq {\bar{z}}_{i} (t) \leq \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}}, i = 1, 2, \dots, n .$ (14)

Theorem 4.1

If $(H_{1})$ – $(H_{6})$ hold, then system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) exists a unique positive periodic solution which is globally exponentially stable.

Proof.

According to conditions $(H_{1})$ – $(H_{3}),$ it follows from Theorem 3.1 that system (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ) has a positive periodic solution $\bar{z} (t) = ({\bar{z}}_{1} (t), {\bar{z}}_{2} (t), \dots, {\bar{z}}_{n} (t))^{T} .$ Let $z (t) = (y_{1} (t), z_{2} (t),$ $\dots, z_{n} (t))^{T}$ be any positive solution of system (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ). Now we construct a Lyapunov functional $V (t) = V_{1} (t) + V_{2} (t),$ where (15) $V_{1} (t) = \sum_{i = 1}^{n} | \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) |$ (15) and (16) $\begin{aligned} V_{2} (t) & = \sum_{i = 1}^{n} \sum_{j = 1}^{n} α_{i i} ρ^{β_{i j} / α_{i i}} b_{i k}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} \int_{t - τ_{i j} (t)}^{t} |z_{j}^{β_{i j} / α_{i i}} (s) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (s)| d s \\ + \sum_{i = 1}^{n} \sum_{j = 1}^{n} α_{i i} ρ^{γ_{i j} / α_{i i}} c_{i j}^{M} \\ \times \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) [\int_{t}^{t - s} |z_{j}^{γ_{i j} / α_{i i}} (τ + s) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (τ + s)| d τ] d s . \end{aligned}$ (16) From the definition of $V (t),$ we easily see that $V (0) < + \infty$ and $V (t) \geq V_{1} (t) .$ By direct computation, we find that (17) $\begin{aligned} D^{+} (| \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) |) \leq - α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | + \sum_{j = 1, j \neq i}^{n} α_{i i} ρ^{α_{i j} / α_{i i}} a_{i j}^{M} \\ \times \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} |z_{j}^{α_{i j} / α_{i i}} (t) - {\bar{z}}_{j}^{α_{i j} / α_{i i}} (t)| + \sum_{j = 1}^{n} α_{i i} ρ^{β_{i j} / α_{i i}} b_{i j}^{M} \\ \times |z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t))| \\ \times \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} + \sum_{j = 1}^{n} α_{i i} c_{i j}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} \\ \times |z_{j}^{γ_{i j} / α_{i i}} (t + s) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (t + s)| d s . \end{aligned}$ (17) By the condition $(H_{4}),$ we obtain (18) $\begin{aligned} {(\int_{t - τ_{i k} (t)}^{t} |z_{j}^{β_{i j} / α_{i i}} (s) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (s)| d s)}^{'} = |z_{j}^{β_{i j} / α_{i i}} (t) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (t)| \\ - |z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t))| \\ \times (1 - τ_{i j}^{'} (t)) \leq |z_{j}^{β_{i j} / α_{i i}} (t) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (t)| - |z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - {\bar{z}}_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t))| \end{aligned}$ (18) and (19) $\begin{aligned} {(\int_{t}^{t - s} |z_{j}^{γ_{i j} / α_{i i}} (τ + s) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (τ + s)| d τ)}^{'} & = |z_{j}^{γ_{i j} / α_{i i}} (t) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (t)| \\ - |z_{j}^{γ_{i j} / α_{i i}} (t + s) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (t + s)| . \end{aligned}$ (19) By $(H_{5}),$ we have $0 < α_{i j} / α_{i i}, β_{i j} / α_{i i}, γ_{i j} / α_{i i} \leq 1.$ Observe that $g (x) = | a^{x} - b^{x} |$ is an increasing function for $a \geq 1$ and x>0. Therefore, according to Equations (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) )–(Equation19(19) $\begin{aligned} {(\int_{t}^{t - s} |z_{j}^{γ_{i j} / α_{i i}} (τ + s) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (τ + s)| d τ)}^{'} & = |z_{j}^{γ_{i j} / α_{i i}} (t) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (t)| \\ - |z_{j}^{γ_{i j} / α_{i i}} (t + s) - {\bar{z}}_{j}^{γ_{i j} / α_{i i}} (t + s)| . \end{aligned}$ (19) ), we derive (20) $\begin{aligned} D^{+} V (t) \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | \\ + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ \times |z_{i}^{α_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{α_{j i} / α_{j j}} (t)| + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} |z_{i}^{β_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{β_{j i} / α_{j j}} (t)| \\ + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \int_{- \infty}^{0} K_{j i} (s) |z_{i}^{γ_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{γ_{j i} / α_{j j}} (t)| d s] \\ \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ \times |z_{i} (t) - {\bar{z}}_{i} (t)| + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} |z_{i} (t) - {\bar{z}}_{i} (t)| \\ + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \int_{- \infty}^{0} K_{j i} (s) |z_{i} (t) - {\bar{z}}_{i} (t)| d s] \\ = \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}}] |z_{i} (t) - {\bar{z}}_{i} (t)| \\ = \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}}] |z_{i} (t) - {\bar{z}}_{i} (t)|, \end{aligned}$ (20) where $m = [t / ω] .$ In addition, from the condition $(H_{6}),$ there exists a positive constant κ such that (21) $\begin{aligned} - α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} + κ < 0. \end{aligned}$ (21) Equations (Equation20(20) $\begin{aligned} D^{+} V (t) \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | \\ + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ \times |z_{i}^{α_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{α_{j i} / α_{j j}} (t)| + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} |z_{i}^{β_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{β_{j i} / α_{j j}} (t)| \\ + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \int_{- \infty}^{0} K_{j i} (s) |z_{i}^{γ_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{γ_{j i} / α_{j j}} (t)| d s] \\ \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ \times |z_{i} (t) - {\bar{z}}_{i} (t)| + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} |z_{i} (t) - {\bar{z}}_{i} (t)| \\ + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \int_{- \infty}^{0} K_{j i} (s) |z_{i} (t) - {\bar{z}}_{i} (t)| d s] \\ = \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}}] |z_{i} (t) - {\bar{z}}_{i} (t)| \\ = \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}}] |z_{i} (t) - {\bar{z}}_{i} (t)|, \end{aligned}$ (20) ) and (Equation21(21) $\begin{aligned} - α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} + κ < 0. \end{aligned}$ (21) ) give (22) $D^{+} V (t) \leq - κ \sum_{i = 1}^{n} | z_{i} (t) - {\bar{z}}_{i} (t) | .$ (22) Integrating both sides of Equation (Equation22(22) $D^{+} V (t) \leq - κ \sum_{i = 1}^{n} | z_{i} (t) - {\bar{z}}_{i} (t) | .$ (22) ) with respect to t, we have (23) $V (t) + κ \int_{0}^{t} \sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) | d s \leq V (0) < + \infty, t \geq 0.$ (23) Equation (Equation23(23) $V (t) + κ \int_{0}^{t} \sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) | d s \leq V (0) < + \infty, t \geq 0.$ (23) ) leads to (24) $\int_{0}^{t} \sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) | d s \leq \frac{V (0)}{κ} < + \infty, t \geq 0,$ (24) which implies that (25) $\sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) | \in L^{1} [0, + \infty) .$ (25) Equation (Equation14(14) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} \leq {\bar{z}}_{i} (t) \leq \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}}, i = 1, 2, \dots, n .$ (14) ) indicates that ${\bar{z}}_{i} (t) (h = 1, 2, \dots, n)$ is uniformly bounded from below and above, and so $\ln {\bar{z}}_{i} (t)$ is bounded. From $| \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) | \leq V_{1} (t) \leq V (t) \leq V (0),$ we get (26) ${\bar{z}}_{i} (t) e^{- V (0)} \leq z_{i} (t) \leq {\bar{z}}_{i} (t) e^{V (0)} .$ (26) Equations (Equation26(26) ${\bar{z}}_{i} (t) e^{- V (0)} \leq z_{i} (t) \leq {\bar{z}}_{i} (t) e^{V (0)} .$ (26) ) and (Equation14(14) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} \leq {\bar{z}}_{i} (t) \leq \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}}, i = 1, 2, \dots, n .$ (14) ) show that $z_{i} (t), {\bar{z}}_{i} (t), h = 1, 2, \dots, n$ are uniformly bounded. This fact together with (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ) lead to $z_{i}^{'} (t), {\bar{z}}_{i}^{'} (t), i = 1, 2, \dots, n$ are uniformly bounded on $[0, + \infty) .$ Therefore $\sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) |$ is uniformly continuous on $[0, + \infty) .$ From Equation (Equation24(24) $\int_{0}^{t} \sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) | d s \leq \frac{V (0)}{κ} < + \infty, t \geq 0,$ (24) ) we know that $\sum_{i = 1}^{n} | z_{i} (s) - {\bar{z}}_{i} (s) |$ is integrable on $[0, + \infty) .$ By Barbalat's Lemma [Citation8, Lemmas 1.2.2 and 1.2.3], we can conclude that (27) $lim_{t \to + \infty} | z_{i} (t) - {\bar{z}}_{i} (t) | = 0, i = 1, 2, \dots, n .$ (27) Thus, we have proved that the positive ω-periodic solution $({\bar{z}}_{1} (t), {\bar{z}}_{2} (t), \dots, {\bar{z}}_{n} (t))^{T}$ of system (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ) is globally attractive.

Next, we shall prove that the positive ω-periodic solution $({\bar{z}}_{1} (t), {\bar{z}}_{2} (t), \dots, {\bar{z}}_{n} (t))^{T}$ of system (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ) is globally exponentially stable. In fact, in the light of Equations (Equation14(14) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} \leq {\bar{z}}_{i} (t) \leq \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}}, i = 1, 2, \dots, n .$ (14) ) and (Equation27(27) $lim_{t \to + \infty} | z_{i} (t) - {\bar{z}}_{i} (t) | = 0, i = 1, 2, \dots, n .$ (27) ), we know that for any $ε > 0$ (ϵ enough small), there exists T>0 such that for t>T (28) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε \leq {\bar{z}}_{i} (t) - ε < z_{i} (t) \leq {\bar{z}}_{i} (t) + ε < \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε .$ (28) By Equation (Equation28(28) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε \leq {\bar{z}}_{i} (t) - ε < z_{i} (t) \leq {\bar{z}}_{i} (t) + ε < \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε .$ (28) ) and the mean value theorem of calculous, we have (29) $| \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) | = |\frac{1}{ξ_{i}}| | z_{i} (t) - {\bar{z}}_{i} (t) | \leq \frac{| z_{i} (t) - {\bar{z}}_{i} (t) |}{\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε}, i = 1, 2, \dots, n,$ (29) where $ξ_{i} (i = 1, 2, \dots, n)$ lies between $z_{i} (t)$ and ${\bar{z}}_{i} (t)$ . According to $(H_{6})$ , there exists a constant $μ > 0$ such that (30) $\begin{aligned} - α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} + \frac{μ}{\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε} < 0. \end{aligned}$ (30) Construct Lyapunov functional $W (t) = e^{μ t} V_{1} (t) + V_{2} (t) .$ Applying Equations (Equation20(20) $\begin{aligned} D^{+} V (t) \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | \\ + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ \times |z_{i}^{α_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{α_{j i} / α_{j j}} (t)| + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} |z_{i}^{β_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{β_{j i} / α_{j j}} (t)| \\ + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \int_{- \infty}^{0} K_{j i} (s) |z_{i}^{γ_{j i} / α_{j j}} (t) - {\bar{z}}_{i}^{γ_{j i} / α_{j j}} (t)| d s] \\ \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} | z_{i} (t) - {\bar{z}}_{i} (t) | + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ \times |z_{i} (t) - {\bar{z}}_{i} (t)| + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} |z_{i} (t) - {\bar{z}}_{i} (t)| \\ + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \int_{- \infty}^{0} K_{j i} (s) |z_{i} (t) - {\bar{z}}_{i} (t)| d s] \\ = \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{0 < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{0 < t_{k} < t} (1 + I_{j k})^{γ_{j i}}] |z_{i} (t) - {\bar{z}}_{i} (t)| \\ = \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}}] |z_{i} (t) - {\bar{z}}_{i} (t)|, \end{aligned}$ (20) ) and (Equation30(30) $\begin{aligned} - α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \\ \times \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} + \frac{μ}{\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε} < 0. \end{aligned}$ (30) ), we have (31) $\begin{aligned} D^{+} W (t) \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \\ + \frac{μ}{\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε}] \times | z_{i} (t) - {\bar{z}}_{i} (t) | < 0. \end{aligned}$ (31) Equation (Equation31(31) $\begin{aligned} D^{+} W (t) \leq \sum_{i = 1}^{n} [- α_{i i} ρ a_{i i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α_{i i}} + \sum_{j = 1, j \neq i}^{n} α_{j j} ρ^{α_{j i} / α_{j j}} a_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{α_{j i}} \\ + \sum_{j = 1}^{n} α_{j j} ρ^{β_{j i} / α_{j j}} b_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β_{j i}} + \sum_{j = 1}^{n} α_{j j} ρ^{γ_{j i} / α_{j j}} c_{j i}^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ_{j i}} \\ + \frac{μ}{\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε}] \times | z_{i} (t) - {\bar{z}}_{i} (t) | < 0. \end{aligned}$ (31) ) means that (32) $W (t) < W (T) = e^{μ T} V_{1} (T) + V_{2} (T) \leq e^{μ T} V (T) \leq e^{μ T} V (0), t > 0.$ (32) Similar to Equations (Equation28(28) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε \leq {\bar{z}}_{i} (t) - ε < z_{i} (t) \leq {\bar{z}}_{i} (t) + ε < \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε .$ (28) )–(Equation29(29) $| \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) | = |\frac{1}{ξ_{i}}| | z_{i} (t) - {\bar{z}}_{i} (t) | \leq \frac{| z_{i} (t) - {\bar{z}}_{i} (t) |}{\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε}, i = 1, 2, \dots, n,$ (29) ), we also have (33) $| \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) | > \frac{| z_{i} (t) - {\bar{z}}_{i} (t) |}{\frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε}, i = 1, 2, \dots, n .$ (33) Equations (Equation32(32) $W (t) < W (T) = e^{μ T} V_{1} (T) + V_{2} (T) \leq e^{μ T} V (T) \leq e^{μ T} V (0), t > 0.$ (32) ) and (Equation33(33) $| \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) | > \frac{| z_{i} (t) - {\bar{z}}_{i} (t) |}{\frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε}, i = 1, 2, \dots, n .$ (33) ) lead to $\begin{aligned} \frac{e^{μ t}}{\frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε} | z_{i} (t) - {\bar{z}}_{i} (t) | & \leq e^{μ t} \sum_{i = 1}^{n} \frac{| z_{i} (t) - {\bar{z}}_{i} (t) |}{\frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε} < e^{μ t} \sum_{i = 1}^{n} | \ln z_{i} (t) - \ln {\bar{z}}_{i} (t) | \\ = e^{μ t} V_{1} (t) \leq W (t) < e^{μ T} V (0), i = 1, 2, \dots, n . \end{aligned}$ Letting $ε \to 0^{+}$ in the above inequality, we have (34) $| z_{i} (t) - {\bar{z}}_{i} (t) | < M e^{- μ t}, i = 1, 2, \dots, n, t > 0,$ (34) where $M = e^{μ T} V (0) max_{1 \leq i \leq n} {(1 / ρ) {\bar{C}}_{i}^{α_{i i}}}$ . Equation (Equation34(34) $| z_{i} (t) - {\bar{z}}_{i} (t) | < M e^{- μ t}, i = 1, 2, \dots, n, t > 0,$ (34) ) indicates that the unique positive ω-periodic solution $({\bar{z}}_{1} (t), {\bar{z}}_{2} (t), \dots, {\bar{z}}_{n} (t))^{T}$ of system (Equation13(13) $\begin{aligned} z_{i}^{'} (t) & = α_{i i} z_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{α_{i j}} ρ^{α_{i j} / α_{i i}} z_{j}^{α_{i j}} (t) \\ - \sum_{j = 1}^{n} b_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{β_{i j}} ρ^{β_{i j} / α_{i i}} \\ \times z_{j}^{β_{i j} / α_{i i}} (t - τ_{i j} (t)) - \sum_{j = 1}^{n} c_{i j} (t) \prod_{0 < t_{k} < t} (1 + I_{i k})^{γ_{i j}} \int_{- \infty}^{0} K_{i j} (s) ρ^{γ_{i j} / α_{i i}} z_{j}^{γ_{i j} / α_{i i}} (t + s) d s] . \end{aligned}$ (13) ) is globally exponentially stable.

Now let us to show that the unique positive ω-periodic solution $({\bar{u}}_{1} (t), {\bar{u}}_{2} (t), \dots, {\bar{u}}_{n} (t))^{T}$ of system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) is globally exponentially stable.

Indeed, by the mean value theorem of calculous and Equation (Equation28(28) $1 < \frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε \leq {\bar{z}}_{i} (t) - ε < z_{i} (t) \leq {\bar{z}}_{i} (t) + ε < \frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε .$ (28) ), we have (35) $\begin{aligned} | x_{i} (t) - {\bar{x}}_{i} (t) | & = ρ^{1 / α_{i i}} |z_{i}^{1 / α_{i i}} (t) - {\bar{z}}_{i}^{1 / α_{i i}} (t)| \\ \leq \frac{N_{i} (ε)}{α_{i i}} ρ^{1 / α_{i i}} |z_{i} (t) - {\bar{z}}_{i} (t)|, i = 1, 2, \dots, n, \end{aligned}$ (35) where $N_{i} (ε) = max \{{(\frac{1}{ρ} {\underline{C}}_{i}^{α_{i i}} - ε)}^{(1 - α_{i i}) / α_{i i}}, {(\frac{1}{ρ} {\bar{C}}_{i}^{α_{i i}} + ε)}^{(1 - α_{i i}) / α_{i i}}\}, i = 1, 2, \dots, n .$ Letting $ε \to 0^{+}$ in Equation (Equation35(35) $\begin{aligned} | x_{i} (t) - {\bar{x}}_{i} (t) | & = ρ^{1 / α_{i i}} |z_{i}^{1 / α_{i i}} (t) - {\bar{z}}_{i}^{1 / α_{i i}} (t)| \\ \leq \frac{N_{i} (ε)}{α_{i i}} ρ^{1 / α_{i i}} |z_{i} (t) - {\bar{z}}_{i} (t)|, i = 1, 2, \dots, n, \end{aligned}$ (35) ) and employing (Equation34(34) $| z_{i} (t) - {\bar{z}}_{i} (t) | < M e^{- μ t}, i = 1, 2, \dots, n, t > 0,$ (34) ), we obtain (36) $| x_{i} (t) - {\bar{x}}_{i} (t) | < M^{*} e^{- μ t}, i = 1, 2, \dots, n, t > 0,$ (36) where $M^{*} = M max_{1 \leq i \leq n} {(N_{i} (0) / α_{i i}) ρ^{1 / α_{i i}}}$ . In view of Lemma 2.1, we find from Equation (Equation36(36) $| x_{i} (t) - {\bar{x}}_{i} (t) | < M^{*} e^{- μ t}, i = 1, 2, \dots, n, t > 0,$ (36) ) that (37) $\begin{aligned} | u_{i} (t) - {\bar{u}}_{i} (t) | & = |\prod_{0 < t_{k} < t} (1 + I_{i k}) x_{i} (t) - \prod_{0 < t_{k} < t} (1 + I_{i k}) {\bar{x}}_{i} (t)| \\ = \prod_{m ω < t_{k} < t} (1 + I_{i k}) |x_{i} (t) - {\bar{x}}_{i} (t)| \\ < \prod_{m ω < t_{k} < t} (1 + I_{i k}) M^{*} e^{- μ t} = M^{* *} e^{- μ t}, i = 1, 2, \dots, n, t > 0, \end{aligned}$ (37) where $m = [t / ω],$ $M^{* *} = \prod_{m ω < t_{k} < t} (1 + I_{i k}) M^{*} .$ It is worth noticing that $\prod_{0 < t_{k} < ω} (1 + I_{i k}) = \prod_{k = 1}^{p} (1 + I_{i k}) = 1.$ So $\prod_{m ω < t_{k} < t} (1 + I_{i k}) < + \infty$ . Therefore, $M^{* *} > 0$ is a constant. Thus, we have proved that the unique positive ω-periodic solution $({\bar{u}}_{1} (t), {\bar{u}}_{2} (t), \dots, {\bar{u}}_{n} (t))^{T}$ of system (Equation1(1) $\begin{aligned} u_{i}^{'} (t) & = u_{i} (t) [r_{i} (t) - \sum_{j = 1}^{n} a_{i j} (t) u_{j}^{α_{i j}} (t) - \sum_{j = 1}^{n} b_{i j} (t) u_{j}^{β_{i j}} (t - τ_{i j} (t)) \\ - \sum_{j = 1}^{n} c_{i j} (t) \int_{- \infty}^{0} K_{i j} (s) u_{j}^{γ_{i j}} (t + s) d s], t \neq t_{k}, \\ u_{i} (t_{k}^{+}) & = u_{i} (t_{k}^{-}) + I_{i k} u_{i} (t_{k}), k = 1, 2, \dots, \end{aligned}$ (1) ) is globally exponentially stable. The proof of Theorem 4.1 is complete.

5. Illustrative example

Consider the following single species Gilpin–Ayala competition model with time delay (38) $\begin{aligned} u^{'} (t) & = u (t) [r (t) - a (t) u^{α} (t) - b (t) u^{β} (t - τ (t)) - c (t) \int_{- \infty}^{0} K (s) u^{γ} (t + s) d s], \\ u (t_{k}^{+}) & = u (t_{k}^{-}) + I_{k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (38) where $r (t) = | \sin t |,$ $a (t) = (82 + \sin t) / π,$ $b (t) = (3 + | \cos 2 t |) / π,$ $c (t) = (3 - \sin 3 t) / π,$ $K (s) = e^{s},$ $α = \ln 3,$ $β = \ln 2,$ $γ = \frac{1}{2},$ $t_{k} = 2 + (- 1)^{k}$ and $\begin{aligned} I_{k} & = \{\begin{cases} 2, & k is odd, \\ - \frac{2}{3}, & k is even, \end{cases}, k \in N^{+}, \\ τ (t) & = \{\begin{cases} \frac{1 + \sin t}{2}, & 2 κ π - \frac{π}{2} \leq t \leq 2 κ π + \frac{π}{2}, κ = Z, \\ \frac{1 - \sin t}{2}, & 2 κ π + \frac{π}{2} \leq t \leq 2 κ π + \frac{3 π}{2}, κ = Z . \end{cases} \end{aligned}$ Obviously, $r (t), a (t), b (t), c (t), τ (t)$ are all positive continuous $2 π$ -periodic functions. By the simple computation, we have $θ = min {α, β, γ} = \frac{1}{2},$ $δ = e^{- θ \int_{0}^{2 π} r (s) d s} = e^{- 2},$ $a^{M} = 83 / π, a^{l} = 81 / π,$ $b^{M} = 4 / π,$ $b^{l} = 3 / π,$ $c^{M} = 4 / π,$ $c^{l} = 2 / π,$ $\int_{- \infty}^{0} K (s) d s = 1$ and $\begin{aligned} \underline{C} = {(\frac{1 - δ}{θ ω (a^{M} + b^{M} + c^{M}) + 1})}^{1 / θ} \approx 0.0095, \\ \bar{C} = {[\frac{θ ω δ^{2}}{1 - δ} (a_{i j}^{l} δ^{α / θ} + b^{l} δ_{j}^{β / θ} + c^{l} δ_{j}^{γ / θ})]}^{- 1 / θ} \approx 1047.3288, \\ \frac{θ ω δ^{2}}{1 - δ} (a^{l} δ^{α / θ} + b^{l} δ^{β / θ} + c^{l} δ^{γ / θ}) \approx 0.0309 < 1, \end{aligned}$ Thus, the assumptions $(H_{1})$ – $(H_{3})$ are satisfied. So we conclude from Theorem 3.1 that system (Equation38(38) $\begin{aligned} u^{'} (t) & = u (t) [r (t) - a (t) u^{α} (t) - b (t) u^{β} (t - τ (t)) - c (t) \int_{- \infty}^{0} K (s) u^{γ} (t + s) d s], \\ u (t_{k}^{+}) & = u (t_{k}^{-}) + I_{k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (38) ) has at least one positive $2 π$ periodic solution $\bar{u} (t)$ satisfying $0.0285 \leq \bar{u} (t) \leq 3141.9864$ .

In addition, clearly $0 < 1 - τ^{'} (t) = 1 - | \cos t | / 2 < 1,$ $α = \ln 3 > max {β, γ} = \ln 2,$ $p = 2,$ $\prod_{k = 1}^{2} (1 + I_{1}) (1 + I_{2}) = (1 + 2) (1 - \frac{2}{3}) = 1.$ Take $ρ = 0.009 < \underline{C}$ , we have $\begin{aligned} - α ρ a^{M} \prod_{m ω < t_{k} < t} (1 + I_{i k})^{α} + α ρ^{β / α} b^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{β} \\ + α ρ^{γ / α} c^{M} \prod_{m ω < t_{k} < t} (1 + I_{j k})^{γ} \approx - 0.436 < 0. \end{aligned}$ Thus we verify that $(H_{4})$ – $(H_{6})$ hold. Therefore, According to Theorem 4.1, we assert that the unique positive $2 π$ -periodic solution $\bar{u} (t)$ of system (Equation38(38) $\begin{aligned} u^{'} (t) & = u (t) [r (t) - a (t) u^{α} (t) - b (t) u^{β} (t - τ (t)) - c (t) \int_{- \infty}^{0} K (s) u^{γ} (t + s) d s], \\ u (t_{k}^{+}) & = u (t_{k}^{-}) + I_{k} u (t_{k}), k = 1, 2, \dots, \end{aligned}$ (38) ) is globally exponentially stable.

Acknowledgements

The author thanks the referees for a number of suggestions which have improved many aspects of this article.

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No potential conflict of interest was reported by the author.

Additional information

Funding

This work is supported by the National Natural Sciences Foundation of People's Republic of China [Grant Nos 11161025, 11661047].

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Global exponential stability of positive periodic solution of the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays

ABSTRACT

1. Introduction

2. Preliminaries

[Citation9]

3. Existence of positive periodic solutions

4. Global exponential stability

5. Illustrative example

Acknowledgements

Disclosure statement

References

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Global exponential stability of positive periodic solution of the n-species impulsive Gilpin–Ayala competition model with discrete and distributed time delays

ABSTRACT

1. Introduction

2. Preliminaries

[Citation9]

3. Existence of positive periodic solutions

4. Global exponential stability

5. Illustrative example

Acknowledgements

Disclosure statement

Additional information

Funding

References

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