Skip to main content

Dynamical analysis of almost periodic solution for a multispecies predator-prey model with mutual interference and time delays

Abstract

In this paper, we build a multispecies predator-prey model with mutual interference and time delays. By means of the comparison theorem, Ascoli theorem and Lebesgue dominated convergence theorem, we establish the sufficient conditions of permanence and investigate the existence of a unique almost periodic solution. By constructing a suitable Lyapunov function, we obtain that the positive almost periodic solution is globally attractive. Finally, we give numerical simulations to indicate the complex dynamical behaviors of this system.

Introduction

In population dynamics, the linkages between predator and prey are usually expressed by different functional response functions, which reflect different dynamical behaviors. Holling [1] carried out a large number of experiments on predator and prey and got some different functional response functions. For example, the mathematical expression of Holling \(x_{i}\) (\(i = 1,2 \)) model is as follows [2]:

$$ \Phi(X)=\frac{\alpha X^{2}}{\beta^{2}+X^{2}}. $$

Besides, in ecosystems, mutual interference between species is always present. The authors [3] proposed a mutual interference factor that tended to leave when the host or parasite met. A lot of articles studied the ecosystem with interference factors. Their obtained results showed that the effect of this factor should not be ignored [47]. For example, Wang et al. [6] concluded that mutual interference had great effect on the relative properties of predator-prey models.

In real life, time delay always exists. Food digestion time, resource regeneration time, mature time, pregnancy period and so on, these all can be expressed by time delay. Usually time delay plays a key role in many systems. For example, time delay can destroy the stability of the positive equilibrium. The obtained results showed that delayed differential equations exhibited more complex dynamical properties than ordinary differential equations [814]. Du et al. [10] gave the following model:

$$ \textstyle\begin{cases} \dot{x} = x ( t ) ( r_{1} ( t ) - b_{1} ( t ) x ( t - \tau ( t ) ) ) - \frac{c _{1} ( t ) x^{2} ( t ) }{x^{2} ( t ) + k ^{2}}y^{m} ( t ) , \\ \dot{y} = y ( t ) ( - r_{2} ( t ) - b_{2} ( t ) y ( t ) ) + \frac{c_{2} ( t ) x ^{2} ( t ) }{x^{2} ( t ) + k^{2}}y^{m} ( t ) , \end{cases} $$
(1.1)

where all parameter meanings can be seen in [10]. The time delay of system (1.1) made the system very unstable and led to more complex dynamical behaviors. At the same time, the research methods were also very different from other systems.

From the point of view of the interaction between biology and environment, Darwin thought that biological variation, heredity and natural selection could lead to the adaptive change of organisms. We know that natural environment is not a constant, and organisms can change their habits to adapt to the new environment, which is called adaptive control. In recent years, adaptive control has been widely used in biological control systems, aerospace systems, satellite tracking systems, and so on [15, 16].

On the other hand, in Ref. [10], the authors assume that the coefficients \(r_{1} ( t ) \), \(b_{1} ( t ) \), \(\tau ( t ) \), \(c _{1} ( t ) \), \(r_{2} ( t ) \), \(b_{2} ( t ) \), \(c _{2} ( t ) \) of system (1.1) are continuous positive almost periodic functions. It is well known that the assumption of almost periodicity of the coefficients in (1.1) is a way of incorporating the time-dependent variability of the environment, especially when the factors of the environment exhibit periodical changes with not necessarily commensurate periods, such as weather, food, mating habits, harvest, etc. In view of these factors, it is necessary to study the relevant properties of ecosystems by using almost periodic coefficients. Recently, many scholars have studied the almost periodic solution and got some nice results, which showed that the almost periodic solution of a population dynamical system with mutual interference and time delay had wider application value [10, 1719].

However, in the actual ecosystem, predator and prey always coexist, which is a common and widespread phenomenon. The dynamical property of a multispecies predator-prey system is much more complex than the system with only two or three species, and the analytical methods are very different [11, 2022].

Based on the above discussion, we establish a multispecies predator-prey model with almost periodic coefficients, mutual interference and time delays. The corresponding mathematical model is as follows:

$$ \textstyle\begin{cases} x_{i}^{\prime } ( t ) = x_{i} ( t ) [ r_{i} ( t ) - \sum_{k = 1}^{n} b_{ik} ( t ) x_{k} ( t - \tau_{k} ( t ) ) - \sum_{k = 1}^{m} \frac{c_{ik} ( t ) x_{i} ( t ) }{x_{i}^{2} ( t ) + f _{ik} ( t ) }y_{k}^{\alpha } ( t ) ] , \\ \quad i = 1,2,\ldots,n, \\ y_{j}^{\prime } ( t ) = y_{j} ( t ) [ - r _{j} ( t ) - \sum_{k = 1}^{m} p_{jk} ( t ) y_{k} ( t ) + \sum_{k = 1}^{n} \frac{q_{kj} ( t ) x_{k} ^{2} ( t ) }{x_{k}^{2} ( t ) + f_{kj} ( t ) }y _{j}^{\alpha - 1} ( t ) ] , \\ \quad j = 1,2,\ldots,m, \end{cases} $$
(1.2)

with the initial conditions

$$ x_{i} ( \chi ) = \phi_{i} ( \chi ) ,\quad\quad y_{j} ( \chi ) = \psi_{j} ( \chi ) ;\quad \phi_{i} ( \chi ) , \psi_{j} ( \chi ) \in C \bigl( [ - \tau ,0 ] ,R _{ +} \bigr),\chi \in [ - \tau ,0 ] , $$
(1.3)

where \(\tau = \max_{t \in R} \{ \tau_{k} ( t ) ,k = 1,2,\ldots,m \} \), \(\tau_{k} ( t ) \) is a nonnegative and continuously differentiable almost periodic function on R and \(\min_{t \in R} \{ 1 - \tau '_{k} ( t ) \} > 0\). \(r _{i} ( t ) \), \(b_{ik} ( t ) \), \(c_{ik} ( t ) \), \(f _{ik} ( t ) \), \(r_{j} ( t ) \), \(p_{jk} ( t ) \), \(q _{kj} ( t ) \), \(f_{kj} ( t ) \) are all continuous positive almost periodic functions on R and the brief description about other parameters used in system (1.2) is presented in Table 1.

Table 1 Notations used to denote parameters

In this article, we aim to investigate the dynamical properties of almost periodic system (1.2), which can greatly enrich the biological background.

The structure of the article as follows. In Section 2, we introduce several important definitions and lemmas. We discuss the permanence of the system in Section 3. Next, we prove the global attractivity of system (1.2) in Section 4. In Section 5, we give conditions of the existence and uniqueness of almost periodic solutions for the system. We put numerical simulations in Section 6. In Section 7, we give a brief conclusion to this paper.

Main descriptions

In this part, we give some definitions and lemmas.

For continuous and bounded f on R, we denote \(f^{u} = \sup_{t \in R}f ( t ) \), \(f^{l} = \inf_{t \in R}f ( t ) \).

Definition 2.1

The positive solution \(( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y_{1} ( t ) ,y _{2} ( t ) ,\ldots, y_{m} ( t ) ) ^{T}\) of system (1.2) is said to be globally attractive if, for any other positive solution \(( \bar{x} ( t ) ,\bar{y} ( t ) ) ^{T} = ( \bar{x}_{1} ( t ) ,\bar{x}_{2} ( t ) ,\ldots, \bar{x}_{n} ( t ) ,\bar{y}_{1} ( t ) ,\bar{y}_{2} ( t ) ,\ldots,\bar{y}_{m} ( t ) ) ^{T}\) of (1.2), the following condition holds:

$$ \lim_{t \to + \infty } \Biggl( \sum_{i = 1}^{n} \bigl\vert x_{i} ( t ) - \bar{x}_{i} ( t ) \bigr\vert + \sum_{j = 1}^{n} \bigl\vert y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr\vert \Biggr) = 0. $$

Definition 2.2

([23])

A function \(f ( t,x ) \) is said to be almost periodic in t uniformly with respect to \(x \in X\) if \(f ( t,x ) \) is continuous and, for \(\forall \varepsilon > 0\), it is possible to find a constant \(I ( \varepsilon ) > 0\) such that, for any interval of length \(I ( \varepsilon ) \), there exists τ such that

$$ \bigl\vert f ( t + \tau ,x ) - f ( t,x ) \bigr\vert < \varepsilon , $$

where the number τ is called an ε-translation number of \(f ( t,x ) \).

By the continuity of almost periodic functions, we obtain that the almost periodic coefficients satisfy \(\min_{ i = 1,2,\ldots,n ; j = 1,2,\ldots,m} \{ r_{i}^{l},r_{j}^{l},b_{ik}^{l},p_{jk}^{l},c_{ik}^{l},q_{kj} ^{l} \} > 0\) and \(\max_{i = 1,2,\ldots,n ;j = 1,2,\ldots,m}\{ r_{i}^{u},r_{j}^{u},b_{ik}^{u}, p_{jk}^{u},c_{ik} ^{u},q_{kj}^{u} \} < + \infty \). For, the characteristics and relevant definitions of almost periodic functions, the reader may refer to [10, 17, 24].

Definition 2.3

([25])

An almost periodic function \(f:R \to R\) is said to be asymptotic if there exist an almost periodic function \(q ( t ) \) and a continuous function \(r ( t ) \) such that

$$ f ( t ) = q ( t ) + r ( t ) ,\quad \quad r ( t ) \to 0 \quad \mbox{as } t \to \infty . $$

Lemma 2.1

([26])

If the function \(f ( t ) \) is nonnegative, integral and uniformly continuous on \([ 0, + \infty )\), then \(\lim_{t \to \infty } f ( t ) = 0\).

Lemma 2.2

The set \(\{ ( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y _{1} ( t ) ,y_{2} ( t ) ,\ldots,y_{m} ( t ) ) ^{T} \in R^{n + m}| x_{i} ( t_{0} ) > 0,i = 1,2,\ldots,n;y_{j} ( t_{0} ) > 0,j = 1,2,\ldots, m,\exists t_{0} \in R \}\) is positive invariant with respect to system (1.2).

Proof

For \(x_{i} ( t_{0} ) > 0\), \(y_{j} ( t_{0} ) > 0\), we have

$$\begin{aligned}& x_{i} ( t ) = x_{i} ( t_{0} ) \exp \Biggl\{ \int _{t_{0}}^{t} \Biggl[ r_{i} ( s ) - \sum _{k = 1}^{n} b_{ik} ( s ) x_{k} \bigl( s - \tau_{k} ( s ) \bigr) - \sum _{k = 1}^{m} \frac{c_{ik} ( s ) x_{i} ( s ) }{x _{i}^{2} ( s ) + f_{ik} ( s ) }y_{k}^{\alpha } ( s ) \Biggr] \,ds \Biggr\} > 0, \\& y_{j} ( t ) = y_{j} ( t_{0} ) \exp \Biggl\{ \int _{t_{0}}^{t} \Biggl[ - r_{j} ( s ) - \sum_{k = 1}^{m} p_{jk} ( s ) y_{k} ( s ) + \sum_{k = 1}^{n} \frac{q_{kj} ( s ) x_{k}^{2} ( s ) }{x_{k}^{2} ( s ) + f_{kj} ( s ) }y_{j}^{\alpha - 1} ( s ) \Biggr] \,ds \Biggr\} > 0. \end{aligned}$$

Then Lemma 2.2 is obtained. □

Lemma 2.3

([27])

Suppose that the continuous operator A maps the closed and bounded convex set \(Q \subset R^{n}\) onto itself, then the operator A has at least one fixed point in the set Q.

Lemma 2.4

([28])

If \(x' \mathbin{{\ge} {( {\le} )}} x ( b - ax^{\alpha } ) \), where \(a > 0\), \(b > 0\) and α is a positive constant, then

$$ \lim_{t \to \infty } \sup x ( t ) \le \biggl( \frac{b}{a} \biggr) ^{\frac{1}{\alpha }}\quad\quad \biggl( \lim_{t \to \infty } \inf x ( t ) \ge \biggl( \frac{b}{a} \biggr) ^{\frac{1}{\alpha }} \biggr) . $$

Lemma 2.5

([29])

If \(x' \mathbin{{\ge} ( {\le} )} x^{m} ( t ) ( b - ax^{1 - m} ( t ) ) \), \(x ( 0 ) > 0\), \(a > 0\), \(b > 0\), then \(\forall t \ge 0\), we have

$$ x ( t ) \mathbin{{\ge} ( {\le} )} \biggl( \frac{b}{a} + \biggl( x ^{1 - m} ( 0 ) - \frac{b}{a} \biggr) e^{ - a ( 1 - m ) t} \biggr) ^{\frac{1}{1 - m}}. $$

Permanence of system (1.2)

Theorem 3.1

If the following condition holds:

$$[H _{1}]\quad \hat{g}_{i} = r_{i}^{l} - \sum_{k = 1,k \ne i}^{n} b_{ik} ^{u} ( t ) M_{k} - \sum_{k = 1}^{m} \frac{c_{ik}^{u}M_{i}N _{k}}{f_{ik}^{l}} > 0, $$

then system (1.2) is permanent, that is, there exists \(T>0\), for \(t > T > 0\), the solution \(( x ( t ) ,y ( t ) ) ^{T}\) of (1.2) satisfies \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n _{j} \le y_{j} ( t ) \le N_{j}\), where

$$\begin{aligned}& m_{i} = \frac{ \hat{g}_{i}}{b_{ii}^{u}}\exp \bigl( \bigl( \hat{g}_{i} - b_{ii}^{u}M _{i} \bigr) \tau \bigr) , \quad\quad M_{i} = \frac{r_{i}^{u}}{b_{ii}^{l}e^{ - r_{i}^{u}\tau }}, \\& n_{j} = \biggl( \frac{\sum_{k = 1}^{n} q_{kj}^{l}m_{k}^{2}}{2 ( \sum_{k = 1} ^{n} ( M_{k}^{2} + f_{kj}^{u} ) ) ( r_{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} ) } \biggr) ^{\frac{1}{1 - \alpha }}, \quad \quad N_{j} = \biggl( \frac{3\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{2r _{j}^{l}} \biggr) ^{\frac{1}{1 - \alpha }} \end{aligned}$$

for \(i = 1,2,\ldots,n\); \(j = 1,2,\ldots,m\). In this article, the values of i, j are no longer repeated.

Proof

By the first equation of (1.2), we get

$$ x'_{i} ( t ) \le x_{i} ( t ) r_{i} ( t ) . $$
(3.1)

Integrating (3.1), we have \(x_{i} ( t ) \le x_{i} ( t - \tau ) \exp ( r_{i}^{u}\tau ) \), \(t > \tau \), that is,

$$ x_{i} ( t - \tau ) \ge x_{i} ( t ) \exp \bigl( - r _{i}^{u}\tau \bigr) ,\quad t > \tau . $$
(3.2)

Combining (3.2) and the first equation of (1.2), we have

$$ x'_{i} ( t ) \le x_{i} ( t ) \bigl[ r_{i}^{u} - b _{ii}^{l}x_{i} ( t ) \exp \bigl( - r_{i}^{u}\tau \bigr) \bigr] ,\quad t > \tau . $$
(3.3)

By applying Lemma 2.4 to (3.3), we obtain

$$ \lim_{t \to + \infty } \sup x_{i} ( t ) \le \frac{r_{i}^{u}}{b _{ii}^{l}e^{ - r_{i}^{u}\tau }} \equiv M_{i}. $$
(3.4)

By (3.4), there exists \(T_{1} > \tau \), when \(t \ge T_{1}\) and \(T_{1} \to \infty \), then

$$ x_{i} ( t ) \le M_{i}. $$
(3.5)

By (3.5), there also exists \(T_{2} = T_{1} + \tau \), when \(t \ge T _{2}\), then

$$ x_{i} ( t - \tau ) \le M_{i}. $$
(3.6)

Combining (3.6) and the second equation of (1.2), we have

$$ \begin{aligned}[b] y'_{j} ( t ) & \le y_{j} ( t ) \Biggl[ \sum_{k = 1} ^{n} q_{kj}^{u} ( t ) y_{j}^{\alpha - 1} ( t ) - r _{j}^{l} ( t ) \Biggr] \\ &\le y_{j}^{\alpha } ( t ) \Biggl[ \sum _{k = 1}^{n} q_{kj} ^{u} ( t ) - r_{j}^{l} ( t ) y^{1 - \alpha } ( t ) \Biggr] ,\quad t \ge T_{2}. \end{aligned} $$
(3.7)

Using Lemma 2.5 to (3.7), then

$$ y_{j} ( t ) \le \biggl[ \frac{\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{r_{j}^{l} ( t ) } + \biggl( y^{1 - \alpha } ( 0 ) - \frac{\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{r _{j}^{l} ( t ) } \biggr) e^{ - r_{j}^{l} ( t ) ( 1 - \alpha ) t} \biggr] ^{\frac{1}{1 - \alpha }},\quad \forall t \ge 0. $$
(3.8)

Therefore, there exists \(T_{3} > 0\) such that

$$ y_{j} ( t ) \le \biggl( \frac{3\sum_{k = 1}^{n} q_{kj}^{u} ( t ) }{2r_{j}^{l}} \biggr) ^{\frac{1}{1 - \alpha }} \equiv N_{j},\quad t > T_{3}. $$
(3.9)

Combining (3.5), (3.6), (3.9) and the first equation of (1.2), we get

$$ x_{i}^{\prime } ( t ) \ge x_{i} ( t ) \Biggl[ r _{i}^{l} - \sum_{k = 1,k \ne i}^{n} b_{ik}^{u} ( t ) M_{k} - b_{ii}^{u}x_{i} \bigl( t - \tau_{i} ( t ) \bigr) - \sum_{k = 1}^{m} \frac{c_{ik}^{u}M_{i}N_{k}^{\alpha }}{f_{ik}^{l}} \Biggr] . $$
(3.10)

Suppose \(x_{i} ( \tilde{t} ) \) is any local minimal value of \(x_{i} ( t ) \), then we have

$$ 0 = x_{i}^{\prime } ( \tilde{t} ) \ge x_{i} ( \tilde{t} ) \Biggl[ r_{i}^{l} - \sum _{k = 1,k \ne i}^{n} b_{ik} ^{u} ( t ) M_{k} - b_{ii}^{u}x_{i} \bigl( \tilde{t} - \tau _{i} ( \tilde{t} ) \bigr) - \sum _{k = 1}^{m} \frac{c_{ik} ^{u}M_{i}N_{k}^{\alpha }}{f_{ik}^{l}} \Biggr] . $$
(3.11)

Let

$$ \hat{g}_{i} = r_{i}^{l} - \sum _{k = 1,k \ne i}^{n} b_{ik}^{u} ( t ) M_{k} - \sum_{k = 1}^{m} \frac{c_{ik}^{u}M_{i}N_{k}^{\alpha }}{f _{ik}^{l}}. $$
(3.12)

From (3.11) and (3.12), we have

$$ x_{i} \bigl( \tilde{t} - \tau_{i} ( \tilde{t} ) \bigr) \ge \frac{\hat{g}_{i}}{b_{ii}^{u}}. $$
(3.13)

Integrating (3.10) on \([ \tilde{t} - \tau_{i} ( \tilde{t} ) , \tilde{t} ] \) and noticing that \(\hat{g}_{i} - b_{ii}^{u}x_{i} ( \tilde{t} - \tau_{i} ( \tilde{t} ) ) \le 0\), we obtain

$$ \ln \biggl( \frac{x_{i} ( \tilde{t} ) }{x_{i} ( \tilde{t} - \tau_{i} ( \tilde{t} ) ) } \biggr) \ge \int_{\tilde{t} - \tau_{i} ( \tilde{t} ) }^{\tilde{t}} \bigl( \hat{g}_{i} - b_{ii}^{u}x_{i} \bigl( \tilde{t} - \tau_{i} ( \tilde{t} ) \bigr) \bigr) \,dt \ge \bigl( \hat{g}_{i} - b_{ii} ^{u}M_{i} \bigr) \tau . $$
(3.14)

From (3.13) and (3.14), then

$$ x_{i} ( \tilde{t} ) \ge \frac{\hat{g}_{i}}{b_{ii}^{u}}\exp \bigl( \bigl( \hat{g}_{i} - b_{ii}^{u}M_{i} \bigr) \tau \bigr) . $$
(3.15)

Hence, for \(T_{4} > 0\) and \(t > T_{4}\), we have

$$ x_{i} ( t ) \ge x_{i} ( \tilde{t} ) \ge \frac{ \hat{g}_{i}}{b_{ii}^{u}} \exp \bigl( \bigl( \hat{g}_{i} - b_{ii}^{u}M _{i} \bigr) \tau \bigr) \equiv m_{i}. $$
(3.16)

Combining (3.9), (3.16) and the second equation of (1.2), when \(T_{5} \ge \max \{ T_{3},T_{4} \} > 0\), for \(t > T_{5}\), we get

$$ \begin{aligned} y'_{j} ( t ) &\ge y_{j} ( t ) \Biggl[ - r_{j}^{u} - \sum_{k = 1}^{m} p_{jk}^{u}N_{k} + \sum _{k = 1}^{n} \frac{q_{kj}^{l}m _{k}^{2}}{M_{k}^{2} + f_{kj}^{u}}y_{j}^{\alpha - 1} ( t ) \Biggr] \\ &= y_{j}^{\alpha } ( t ) \Biggl[ \sum _{k = 1}^{n} \frac{q_{kj} ^{l}m_{k}^{2}}{M_{k}^{2} + f_{kj}^{u}} - \Biggl( r_{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} \Biggr) y_{j}^{1 - \alpha } ( t ) \Biggr] . \end{aligned} $$

It follows from Lemma 2.5 that there exists \(T_{6} > 0\) such that

$$ \begin{aligned} y_{j} ( t ) &\ge \biggl( \frac{\sum_{k = 1}^{n} \frac{q_{kj} ^{l}m_{k}^{2}}{M_{k}^{2} + f_{kj}^{u}}}{r_{j}^{u} + \sum_{k = 1}^{m} p _{jk}^{u}N_{k}} + \biggl( y_{j}^{1 - \alpha } ( 0 ) - \frac{ \sum_{k = 1}^{n} \frac{q_{kj}^{l}m_{k}^{2}}{M_{k}^{2} + f_{kj}^{u}}}{r _{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k}} \biggr) e^{ - ( r _{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} ) ( 1 - \alpha ) t} \biggr) ^{\frac{1}{1 - \alpha }} \\ &\ge \biggl( \frac{\sum_{k = 1}^{n} q_{kj}^{l}m_{k}^{2}}{2 ( \sum_{k = 1}^{n} ( M_{k}^{2} + f_{kj}^{u} ) ) ( r _{j}^{u} + \sum_{k = 1}^{m} p_{jk}^{u}N_{k} ) } \biggr) ^{\frac{1}{1 - \alpha }} \equiv n_{j}. \end{aligned} $$

Make \(T \ge \max \{ T_{2},T_{5},T_{6} \} > 0\), for \(t > T\), we get \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n_{j} \le y_{j} ( t ) \le N_{j}\).

Therefore system (1.2) is permanent.

Next, we prove that system (1.2) has at least one bounded positive solution for \(t \ge 0\). Define \(\Omega = \{ ( x ( t ) ,y ( t ) ) ^{T} = ( x_{1} ( t ) ,x_{2} ( t ) ,\ldots,x_{n} ( t ) ,y _{1} ( t ) ,y_{2} ( t ) ,\ldots,y_{m} ( t ) ) ^{T}\in R^{n + m}| ( x ( t ) ,y ( t ) ) ^{T}\text{ is the solution of system (1.2), satisfying }m_{i} \le x_{i} ( t ) \le M_{i},n_{j} \le y_{j} ( t ) \le N_{j},t \in R \}\). □

Theorem 3.2

For system (1.2), the set \(\Omega \ne \emptyset\).

Proof

According to the characteristics of an almost periodic function, for a sequence of \(\{ t_{\gamma } \} \), \(t_{\gamma } \to \infty \) as \(\gamma \to \infty \), then \(r_{i} ( t + t_{\gamma } ) \to r_{i} ( t ) \), \(r_{j} ( t + t_{\gamma } ) \to r_{j} ( t ) \), \(b_{il} ( t + t_{\gamma } ) \to b_{il} ( t ) \), \(p_{jk} ( t + t_{\gamma } ) \to p_{jk} ( t ) \), \(c_{ik} ( t + t_{\gamma } ) \to c_{ik} ( t ) \), \(q_{lj} ( t + t_{\gamma } ) \to q_{lj} ( t ) \), \(\tau_{i} ( t + t_{\gamma } ) \to \tau_{i} ( t ) \), \(f_{ij} ( t + t_{\gamma } ) \to f_{ij} ( t ) \) (\(i,l = 1,2,\ldots,n\); \(j,k = 1,2,\ldots,m\)) uniformly on R as \(\gamma \to \infty \). By Lemma 2.3, system (1.2) has at least one solution \(z ( t ) = ( x ( t ) ,y ( t ) ) ^{T}\) satisfying \(m_{i} \le x_{i} ( t ) \le M_{i}\), \(n_{j} \le y_{j} ( t ) \le N_{j}\) when \(t > T\).

Obviously, the sequence \(z ( t + t_{\gamma } ) \) is uniformly bounded and equi-continuous on any bounded subset of R. By the Ascoli theorem, we know there exists a subsequence \(z ( t + t_{\lambda } ) \) which converges to a continuous function

$$ g ( t ) = \bigl( g_{1} ( t ) ,g_{2} ( t ) \bigr) ^{T} = \bigl( g_{11} ( t ) ,g_{21} ( t ) ,\ldots,g _{n1} ( t ) ,g_{12} ( t ) ,g_{22} ( t ) ,\ldots,g _{m2} ( t ) \bigr) ^{T} $$

as \(\lambda \to \infty \) uniformly on any bounded subset of R.

Make \(T_{7} \in R\), suppose \(T_{7} + t_{\lambda } \ge T\) for all λ. When \(t \ge 0\), we obtain

$$\begin{aligned}& \begin{aligned}[b] &x_{i} ( t + t_{\lambda } + T_{7} ) - x_{i} ( t_{\lambda } + T_{7} ) \\ &\quad = \int_{T_{7}}^{t + T_{7}} x_{i} ( s + t_{ \lambda } ) \Biggl( r_{i} ( s + t_{\lambda } ) - \sum _{k = 1}^{n} b_{ik} ( s + t_{\lambda } ) x_{k} \bigl( ( s + t_{\lambda } ) - \tau_{k} ( s + t_{\lambda } ) \bigr) \\ &\quad\quad{} - \sum_{k = 1}^{m} \frac{c_{ik} ( s + t_{\lambda } ) x_{i} ( s + t_{\lambda } ) }{x_{i}^{2} ( s + t_{\lambda } ) + f_{ik} ( s + t_{\lambda } ) }y_{k}^{\alpha } ( s + t_{\lambda } ) \Biggr) \,ds, \end{aligned} \end{aligned}$$
(3.17)
$$\begin{aligned}& \begin{aligned}[b] & y_{j} ( t + t_{\lambda } + T_{7} ) - y_{j} ( t_{\lambda } + T_{7} ) \\ &\quad = \int_{T_{7}}^{t + T_{7}} y_{j} ( t_{\lambda } + s ) \Biggl( - r_{j} ( s + t_{\lambda } ) - \sum _{k = 1}^{m} p_{jk} ( s + t_{\lambda } ) y_{k} ( s + t_{\lambda } ) \\ &\quad\quad{} + \sum_{k = 1}^{n} \frac{q_{kj} ( s + t_{\lambda } ) x_{k} ^{2} ( s + t_{\lambda } ) }{x_{k}^{2} ( s + t_{\lambda } ) + f_{kj} ( s + t_{\lambda } ) }y_{j}^{\alpha - 1} ( s + t_{\lambda } ) \Biggr) \,ds. \end{aligned} \end{aligned}$$
(3.18)

Letting \(\lambda \to \infty \) in (3.17) and (3.18), for \(\forall t \ge 0\), by the Lebesgue dominated convergence theorem, we get

$$\begin{aligned}& g_{i1} ( t + T_{7} ) - g_{i1} ( T_{7} ) \\& \quad = \int _{T_{7}}^{t + T_{7}} g_{i1} ( s ) \Biggl( r_{i} ( s ) - \sum_{k = 1}^{n} b_{ik} ( s ) g_{k1} \bigl( s - \tau_{k} ( s ) \bigr) - \sum_{k = 1}^{m} \frac{c_{ik} ( s ) g _{i1} ( s ) }{g_{i1}^{2} ( s ) + f_{ik} ( s ) }g _{k2}^{\alpha } ( s ) \Biggr) \,ds, \\& g_{j2} ( t + T_{7} ) - g_{j2} ( T_{7} ) \\& \quad = \int _{T_{7}}^{t + T_{7}} g_{j2} ( s ) \Biggl( - r_{j} ( s ) - \sum_{k = 1}^{m} p_{jk} ( s ) g_{k2} ( s ) + \sum _{k = 1}^{n} \frac{q_{kj} ( s ) g_{k1}^{2} ( s ) }{g _{k1}^{2} ( s ) + f_{kj} ( s ) }g_{j2}^{\alpha - 1} ( s ) \Biggr) \,ds. \end{aligned}$$

Since \(T_{7} \in R\) is arbitrarily given, \(g ( t ) \) is a solution of system (1.2) on R.

It is easy to know \(m_{i} \le g_{i1} ( t ) \le M_{i}\), \(n_{j} \le g_{j2} ( t ) \le N_{j}\) for any \(t \in R\). Thus, the set \(\Omega \ne \emptyset \), that is, system (1.2) has at least one bounded positive solution. □

Global attractivity of system (1.2)

Theorem 4.1

If the parameters of system (1.2) satisfy condition \([H_{1}]\) and the following conditions:

$$\begin{aligned}& [H _{2} ]\quad \lim_{t \to + \infty } \inf A_{i} ( t ) > 0, \\& [H _{3}] \quad \lim_{t \to + \infty } \inf B_{j} ( t ) > 0, \end{aligned}$$

where

$$\begin{aligned}& \begin{aligned} A_{i} ( t ) &= - \sum_{k = 1}^{m} c_{ik} ( t ) E _{ik} + \sum_{k = 1}^{n} b_{ik} ( t ) \\ &\quad{} - \sum_{k = 1}^{n} \Biggl[ r_{k} ( s ) + \sum_{j = 1}^{n} b_{kj} ( s ) M _{j} + \sum_{j = 1}^{m} \frac{q_{kj} ( s ) M_{k}N_{j}^{ \alpha }}{f_{kj}^{l} + m_{k}^{2}} \Biggr] \int_{t}^{\varphi_{k}^{ - 1} ( t ) } b_{ik} ( u ) \,du \\ &\quad{} - \sum_{k = 1}^{n} \sum _{j = 1}^{m} c_{ik} ( t ) M_{k}E_{kj} \int_{t}^{\varphi_{k}^{ - 1} ( t ) } b_{ik} ( u ) \,du \\ &\quad{} - \sum _{k = 1}^{n} \sum_{j = 1}^{n} \frac{M_{k}b_{kj} ( \varphi _{j}^{ - 1} ( t ) ) }{1 - \tau_{j}^{\prime } ( \varphi_{j}^{ - 1} ( l ) ) } \int_{\varphi_{j}^{ - 1} ( t ) }^{\varphi_{k}^{ - 1} ( \varphi_{j}^{ - 1} ( t ) ) } b_{ik} ( u ) \,du - \sum_{k = 1}^{m} q_{kj} ( t ) \frac{2M_{i}N_{j}^{\alpha - 1}}{f_{ik} ( t ) + m_{i}^{2}}, \end{aligned} \\& B_{j} ( t ) = - \sum_{k = 1}^{m} c_{ik} ( t ) F _{ik} - \sum_{k = 1}^{n} \sum_{j = 1}^{m} q_{kj} ( t ) M_{k}E _{kj} \int_{t}^{\varphi_{k}^{ - 1} ( t ) } b_{ik} ( u ) \,du + \sum _{k = 1}^{m} p_{jk} ( t ) , \\& E_{ik} = \frac{f_{ik}^{u}N_{k}^{\alpha } + 3M_{i}^{2}N_{k}^{\alpha }}{ ( f_{ik} + m_{i}^{2} ) ^{2}},\quad\quad F_{ik} = \frac{f_{ik}^{u}M_{i} + M_{i}^{3}}{ ( f_{ik} + m_{i}^{2} ) ^{2}} \end{aligned}$$

and \(\varphi_{i} ^{ - 1}\) is the inverse function of \(\varphi_{i} ( t ) = t - \tau_{i} ( t ) \), then the solution of system (1.2) is globally attractive.

Proof

Let \(( x ( t ) ,y ( t ) ) ^{T}\), \(( \bar{x} ( t ) ,\bar{y} ( t ) ) ^{T}\) be any two solutions of system (1.2). From Theorem 3.1, for \(\forall t > T\), we get

$$ m_{i} \le x_{i} ( t ) ,\bar{x}_{i} ( t ) \le M_{i};\quad\quad n _{j} \le y_{j} ( t ) , \bar{y}_{j} ( t ) \le N_{j}. $$
(4.1)

Next, we set up several Lyapunov functions. Let

$$ V_{i1} ( t ) = \bigl\vert \ln \bar{x}_{i} ( t ) - \ln x_{i} ( t ) \bigr\vert . $$
(4.2)

By calculating the upper right derivative of \(V_{i1} ( t ) \) along system (1.2), we have

$$\begin{aligned} D^{ +} V_{i1} ( t ) &= \operatorname {sign}\bigl( \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr) \biggl( \frac{\bar{x}_{i}^{\prime } ( t ) }{\bar{x}_{i} ( t ) } - \frac{x_{i}^{\prime } ( t ) }{x_{i} ( t ) } \biggr) \\ &= \operatorname {sign}\bigl( \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr) \Biggl\{ - \sum_{k = 1}^{n} b_{ik} ( t ) \bigl[ \bar{x}_{k} \bigl( t - \tau_{k} ( t ) \bigr) - x_{k} \bigl( t - \tau_{k} ( t ) \bigr) \bigr] \\ &\quad{} - \sum_{k = 1}^{m} \biggl[ \frac{c_{ik} ( t ) \bar{x}_{i} ( t ) \bar{y}_{k}^{\alpha } ( t ) }{\bar{x}_{i} ^{2} ( t ) + f_{ik} ( t ) } - \frac{c_{ik} ( t ) x_{i} ( t ) y_{k}^{\alpha } ( t ) }{x_{i} ^{2} ( t ) + f_{ik} ( t ) } \biggr] \Biggr\} \\ &= \operatorname {sign}\bigl( \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr) \Biggl\{ - \sum_{k = 1}^{n} b_{ik} ( t ) \bigl[ \bar{x}_{k} \bigl( t - \tau_{k} ( t ) \bigr) - x_{k} \bigl( t - \tau_{k} ( t ) \bigr) \bigr] \\ &\quad{} - \sum _{k = 1}^{m} c_{ik} ( t ) \biggl[ \frac{f_{ik} ( t ) \bar{y}_{k}^{\alpha } ( t ) + \bar{y}_{k}^{\alpha } ( t ) x_{i}^{2} ( t ) - x _{i} ( t ) y_{k}^{\alpha } ( t ) ( x_{i} ( t ) + \bar{x}_{i} ( t ) ) }{ ( \bar{x}_{i} ^{2} ( t ) + f_{ik} ( t ) ) ( x_{i}^{2} ( t ) + f_{ik} ( t ) ) } \bigl( \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr) \\ &\quad{} + \frac{f_{ik} ( t ) x_{i} ( t ) + x_{i}^{3} ( t ) }{ ( \bar{x}_{i}^{2} ( t ) + f_{ik} ( t ) ) ( x_{i}^{2} ( t ) + f_{ik} ( t ) ) } \bigl( \bar{y}_{k}^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr) \biggr] \Biggr\} \\ &\le \sum_{k = 1}^{m} c_{ik} ( t ) E_{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1}^{m} c _{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert \\ &\quad{} - \sum _{k = 1}^{n} b_{ik} \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert + \sum _{k = 1}^{n} b_{ik} \biggl\vert \int_{t - \tau_{k} ( t ) }^{t} \bar{x}_{k}^{\prime } ( s ) - x_{k}^{\prime } ( s ) \,ds \biggr\vert , \end{aligned}$$
(4.3)

where

$$ E_{ik} = \frac{f_{ik}^{u}N_{k}^{\alpha } + 3M_{i}^{2}N_{k}^{\alpha }}{ ( f_{ik} + m_{i}^{2} ) ^{2}}, \quad\quad F_{ik} = \frac{f_{ik}^{u}M_{i} + M_{i}^{3}}{ ( f_{ik} + m_{i}^{2} ) ^{2}}. $$

Substituting (1.2) into (4.3), we get

$$\begin{aligned} & D^{ +} V_{i1} ( t ) \\ &\quad \le \sum_{k = 1}^{m} c_{ik} ( t ) E _{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1}^{m} c_{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{ \alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ &\quad \quad{} + \sum_{k = 1}^{n} b_{ik} ( t ) \Biggl\vert \int_{t - \tau_{k} ( t ) }^{t} \Biggl\{ \bar{x}_{k} ( s ) \Biggl[ r_{k} ( s ) - \sum_{j = 1} ^{n} b_{kj} ( s ) \bar{x}_{j} \bigl( s - \tau_{j} ( s ) \bigr) - \sum_{j = 1}^{m} \frac{c_{kj} ( s ) \bar{x}_{k} ( s ) \bar{y}_{j}^{\alpha } ( s ) }{f _{kj} ( s ) + \bar{x}_{k}^{2} ( s ) } \Biggr] \\ &\quad\quad{} - x_{k} ( s ) \Biggl[ r_{k} ( s ) - \sum_{j = 1} ^{n} b_{kj} ( s ) x_{j} \bigl( s - \tau_{j} ( s ) \bigr) - \sum _{j = 1}^{m} \frac{c_{kj} ( s ) x_{k} ( s ) y _{j}^{\alpha } ( s ) }{f_{kj} ( s ) + x_{k}^{2} ( s ) } \Biggr] \Biggr\} \,ds \Biggr\vert \\ &\quad = \sum_{k = 1}^{m} c_{ik} ( t ) E_{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1}^{m} c _{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) E_{ik} \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ &\quad \quad{} + \sum_{k = 1}^{n} b_{ik} ( t ) \Biggl\vert \int_{t - \tau_{k} ( t ) }^{t} \Biggl\{ \Biggl[ r_{k} ( s ) - \sum_{j = 1}^{n} b_{kj} ( s ) \bar{x}_{j} \bigl( s - \tau_{j} ( s ) \bigr) - \sum _{j = 1} ^{m} \frac{c_{kj} ( s ) \bar{x}_{k} ( s ) \bar{y} _{j}^{\alpha } ( s ) }{f_{kj} ( s ) + \bar{x}_{k} ^{2} ( s ) } \Biggr] \\ &\quad\quad{}\times \bigl( \bar{x}_{k} ( s ) - x _{k} ( s ) \bigr) - x_{k} ( s ) \sum_{j = 1}^{n} b_{kj} ( s ) \bigl( \bar{x}_{j} \bigl( s - \tau_{j} ( s ) \bigr) - x_{j} \bigl( s - \tau_{j} ( s ) \bigr) \bigr) \\ &\quad\quad{} - x_{k} ( s ) \sum _{j = 1}^{m} c_{kj} ( s ) \frac{f _{kj} ( s ) \bar{y}_{j}^{\alpha } ( s ) + \bar{y} _{j}^{\alpha } ( s ) x_{k}^{2} ( s ) - y_{j}^{ \alpha } ( s ) x_{k} ( s ) ( x_{k} ( s ) + \bar{x}_{k} ( s ) ) }{ ( f_{kj} ( s ) + x_{k}^{2} ( s ) ) ( f_{kj} ( s ) + \bar{x}_{k}^{2} ( s ) ) } \bigl( \bar{x}_{k} ( s ) - x_{k} ( s ) \bigr) \\ &\quad\quad{} - x_{k} ( s ) \sum_{j = 1}^{m} c_{kj} ( s ) \frac{f _{kj} ( s ) x_{k} ( s ) + x_{k}^{3} ( s ) }{ ( f_{kj} ( s ) + x_{k}^{2} ( s ) ) ( f_{kj} ( s ) + \bar{x}_{k}^{2} ( s ) ) } \bigl( \bar{y}_{j}^{\alpha } ( s ) - y_{j}^{\alpha } ( s ) \bigr) \Biggr\} \,ds \Biggr\vert \\ & \quad \le \sum_{k = 1}^{m} c_{ik} ( t ) E_{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1}^{m} c _{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ &\quad \quad{} + \sum_{k = 1}^{n} b_{ik} ( t ) \int_{t - \tau_{k} ( t ) }^{t} \Biggl\{ \Biggl[ r_{k} ( s ) + \sum_{j = 1}^{n} b_{kj} ( s ) \bar{x}_{j} \bigl( s - \tau_{j} ( s ) \bigr) + \sum _{j = 1} ^{m} \frac{c_{kj} ( s ) \bar{x}_{k} ( s ) \bar{y} _{j}^{\alpha } ( s ) }{f_{kj} ( s ) + \bar{x}_{k} ^{2} ( s ) } \Biggr] \\ &\quad\quad{}\times \bigl\vert \bar{x}_{k} ( s ) - x _{k} ( s ) \bigr\vert + x_{k} ( s ) \sum_{j = 1}^{n} b_{kj} ( s ) \bigl\vert \bigl( \bar{x}_{j} \bigl( s - \tau_{j} ( s ) \bigr) - x_{j} \bigl( s - \tau_{j} ( s ) \bigr) \bigr) \bigr\vert \\ &\quad\quad{} + x_{k} ( s ) \sum _{j = 1}^{m} c_{kj} ( s ) E_{kj} \bigl\vert \bar{x}_{k} ( s ) - x_{k} ( s ) \bigr\vert + x _{k} ( s ) \sum_{j = 1}^{m} c_{kj} ( s ) F_{kj} \bigl\vert \bar{y}_{j}^{\alpha } ( s ) - y_{j}^{\alpha } ( s ) \bigr\vert \Biggr\} \,ds. \end{aligned}$$
(4.4)

Considering (4.1) and (4.4), for \(t \ge T + \tau \), we get

$$\begin{aligned} &D^{ +} V_{i1} ( t ) \\ &\quad \le \sum_{k = 1}^{m} c_{ik} ( t ) E _{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1}^{m} c_{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{ \alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ &\quad \quad{} + \sum_{k = 1}^{n} b_{ik} ( t ) \int_{t - \tau_{k} ( t ) }^{t} \Biggl\{ \Biggl[ r_{k} ( s ) + \sum_{j = 1}^{n} b_{kj} ( s ) M _{j} + \sum_{j = 1}^{m} \frac{c_{kj} ( s ) M_{k}N_{j}^{ \alpha }}{f_{kj}^{l} + m_{k}^{2}} \Biggr] \bigl\vert \bar{x}_{k} ( s ) - x_{k} ( s ) \bigr\vert \\ &\quad\quad{} + M_{k}\sum _{j = 1}^{n} b_{kj} ( s ) \bigl\vert \bigl( \bar{x} _{j} \bigl( s - \tau_{j} ( s ) \bigr) - x_{j} \bigl( s - \tau _{j} ( s ) \bigr) \bigr) \bigr\vert \\ &\quad \quad{} + M_{k}\sum_{j = 1}^{n} c_{kj} ( s ) E_{kj} \bigl\vert \bar{x} _{k} ( s ) - x_{k} ( s ) \bigr\vert + M_{k}\sum _{j = 1}^{m} c_{kj} ( s ) F_{kj} \bigl\vert \bar{y} _{j}^{\alpha } ( s ) - y_{j}^{\alpha } ( s ) \bigr\vert \Biggr\} \,ds \\ &\quad \equiv \sum_{k = 1}^{m} c_{ik} ( t ) E_{ik} \bigl\vert \bar{x} _{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1} ^{m} c_{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ &\quad \quad{} + \sum_{k = 1}^{n} b_{ik} ( t ) \int_{t - \tau_{k} ( t ) }^{t} G_{k} ( s ) \,ds. \end{aligned}$$
(4.5)

Define

$$ V_{i2} ( t ) = \sum_{k = 1}^{n} \int_{t}^{\varphi_{k}^{ - 1} ( t ) } \int_{\tau_{k} ( u ) }^{t} b_{ik} ( u ) G _{k} ( s ) \,ds\,du. $$
(4.6)

Combining (4.5) and (4.6), for \(t \ge T + \tau \), we get

$$ \begin{aligned}[b] D^{ +} V_{i1} ( t ) + V_{i2}^{\prime } ( t ) &\le \sum_{k = 1}^{m} c_{ik} ( t ) E_{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert \\ &\quad{} + \sum_{k = 1}^{m} c _{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k}^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ &\quad{} + \sum_{k = 1}^{n} \int_{t}^{\varphi_{k}^{ - 1} ( t ) } b _{ik} ( u ) \,du G_{k} ( t ) . \end{aligned} $$
(4.7)

Next, we define

$$ V_{i} ( t ) = V_{i1} ( t ) + V_{i2} ( t ) + V_{i3} ( t ) , $$
(4.8)

where

$$ V_{i3} ( t ) = \sum_{k = 1}^{n} \sum_{j = 1}^{n} M_{k} \int_{t - \tau_{j} ( t ) }^{t} \int_{\varphi_{j}^{ - 1} ( l ) }^{\varphi_{k}^{ - 1} ( \varphi_{j}^{ - 1} ( l ) ) } \frac{b_{ik} ( u ) b _{kj} ( \varphi_{j}^{ - 1} ( l ) ) }{1 - \tau_{j} ^{\prime } ( \varphi_{j}^{ - 1} ( l ) ) } \bigl\vert \bar{x}_{j} ( l ) - x_{j} ( l ) \bigr\vert \,du\,dl. $$
(4.9)

Considering (4.7)-(4.9), for \(t \ge T + \tau \), we get

$$\begin{aligned} & D^{ +} V_{i} ( t ) \\ &\quad \le \sum_{k = 1}^{m} c_{ik} ( t ) E _{ik} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{k = 1}^{m} c_{ik} ( t ) F_{ik} \bigl\vert \bar{y}_{k} ^{\alpha } ( t ) - y_{k}^{\alpha } ( t ) \bigr\vert - \sum _{k = 1}^{n} b_{ik} ( t ) \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ & \quad \quad {} + \sum_{k = 1}^{n} \Biggl[ r_{k} ( s ) + \sum_{j = 1}^{n} b _{kj} ( s ) M_{j} + \sum_{j = 1}^{m} \frac{c_{kj} ( s ) M _{k}N_{j}^{\alpha }}{f_{kj}^{l} + m_{k}^{2}} \Biggr] \int_{t}^{\varphi _{k}^{ - 1} ( t ) } b_{ik} ( u ) \,du \bigl\vert \bar{x} _{k} ( t ) - x_{k} ( t ) \bigr\vert \\ & \quad \quad {} + \sum_{k = 1}^{n} \sum _{j = 1}^{m} c_{ik} ( t ) M_{k}E_{kj} \int_{t}^{\varphi_{k}^{ - 1} ( t ) } b_{ik} ( u ) \,du \bigl\vert \bar{x}_{k} ( t ) - x_{k} ( t ) \bigr\vert \\ & \quad \quad {} + \sum_{k = 1}^{n} \sum _{j = 1}^{m} c_{kj} ( t ) M_{k}F_{kj} \int_{t}^{\varphi_{k}^{ - 1} ( t ) } b_{ik} ( u ) \,du \bigl\vert \bar{y}_{j}^{\alpha } ( t ) - y_{j}^{\alpha } ( t ) \bigr\vert \\ & \quad \quad {} + \sum_{k = 1}^{n} \sum _{j = 1}^{n} \frac{M_{k}b_{kj} ( \varphi _{j}^{ - 1} ( t ) ) }{1 - \tau_{j}^{\prime } ( \varphi_{j}^{ - 1} ( l ) ) } \int_{\varphi_{j}^{ - 1} ( t ) }^{\varphi_{k}^{ - 1} ( \varphi_{j}^{ - 1} ( t ) ) } b_{ik} ( u ) \,du \bigl\vert \bar{x}_{j} ( t ) - x_{j} ( t ) \bigr\vert . \end{aligned}$$
(4.10)

Define

$$ V_{j} ( t ) = \bigl\vert \ln y_{j} ( t ) - \ln \bar{y}_{j} ( t ) \bigr\vert . $$
(4.11)

Calculating its Dini derivative along system (1.2), we get

$$\begin{aligned} D^{ +} V_{j} ( t ) &= \operatorname {sign}\bigl( y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr) \biggl( \frac{y_{j}^{\prime } ( t ) }{y_{j} ( t ) } - \frac{\bar{y}_{j}^{\prime } ( t ) }{\bar{y}_{j} ( t ) } \biggr) \\ &= \operatorname {sign}\bigl( y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr) \Biggl[ - \sum_{k = 1}^{m} p_{jk} ( t ) \bigl( y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr) \\ &\quad{}+ \sum _{k = 1}^{m} q _{kj} ( t ) \biggl( \frac{x_{k}^{2} ( t ) y_{j}^{ \alpha - 1} ( t ) }{f_{ik} ( t ) + x_{k}^{2} ( t ) } - \frac{\bar{x}_{k}^{2} ( t ) \bar{y}_{j}^{\alpha - 1} ( t ) }{f_{ik} ( t ) + \bar{x}_{k}^{2} ( t ) } \biggr) \Biggr] \\ &= - \sum_{k = 1}^{m} p_{jk} ( t ) \bigl\vert y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr\vert \\ &\quad{}+ \sum_{k = 1}^{m} q_{kj} ( t ) \operatorname {sign}\bigl( y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr) \biggl( \frac{x_{k}^{2} ( t ) y_{j}^{\alpha - 1} ( t ) }{f_{ik} ( t ) + x_{k}^{2} ( t ) } - \frac{ \bar{x}_{k}^{2} ( t ) \bar{y}_{j}^{\alpha - 1} ( t ) }{f _{ik} ( t ) + \bar{x}_{k}^{2} ( t ) } \biggr) \\ &= - \sum_{k = 1}^{m} p_{jk} ( t ) \bigl\vert y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr\vert + \sum_{k = 1}^{m} q_{kj} ( t ) \operatorname {sign}\bigl( y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr) \\ &\quad{}\times \biggl( \frac{x_{k}^{2} ( t ) y_{j}^{\alpha - 1} ( t ) }{f _{ik} ( t ) + x_{k}^{2} ( t ) } - \frac{x_{k}^{2} ( t ) \bar{y}_{j}^{\alpha - 1} ( t ) }{f_{ik} ( t ) + x_{k}^{2} ( t ) } + \frac{x_{k}^{2} ( t ) \bar{y}_{j}^{\alpha - 1} ( t ) }{f _{ik} ( t ) + x_{k}^{2} ( t ) } - \frac{\bar{x} _{k}^{2} ( t ) \bar{y}_{j}^{\alpha - 1} ( t ) }{f _{ik} ( t ) + \bar{x}_{k}^{2} ( t ) } \biggr) \\ &= - \sum_{k = 1}^{m} p_{jk} ( t ) \bigl\vert y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr\vert + \sum_{k = 1}^{m} q_{kj} ( t ) \operatorname {sign}\bigl( y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr) \\ &\quad{}\times \biggl( \frac{x_{k}^{2} ( t ) }{f_{ik} ( t ) + x_{k}^{2} ( t ) } \bigl( y_{j}^{\alpha - 1} ( t ) - \bar{y} _{j}^{\alpha - 1} ( t ) \bigr) \\ &\quad{} + \frac{f_{ik} ( t ) ( x_{k} ( t ) + \bar{x} _{k} ( t ) ) \bar{y}_{j}^{\alpha - 1} ( t ) }{ ( f_{ik} ( t ) + x_{k}^{2} ( t ) ) ( f_{ik} ( t ) + \bar{x}_{k}^{2} ( t ) ) } \bigl( x_{k} ( t ) - \bar{x}_{k} ( t ) \bigr) \biggr) \\ &\le - \sum_{k = 1}^{m} p_{jk} ( t ) \bigl\vert y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr\vert + \sum_{k = 1}^{m} q _{kj} ( t ) \frac{ ( x_{k} ( t ) + \bar{x} _{k} ( t ) ) \bar{y}_{j}^{\alpha - 1} ( t ) }{f _{ik} ( t ) + x_{k}^{2} ( t ) } \bigl\vert x_{k} ( t ) - \bar{x}_{k} ( t ) \bigr\vert \\ &\le - \sum_{k = 1}^{m} p_{jk} ( t ) \bigl\vert y_{j} ( t ) - \bar{y}_{j} ( t ) \bigr\vert + \sum_{k = 1}^{m} q _{kj} ( t ) \frac{2M_{i}N_{j}^{\alpha - 1}}{f_{ik} ( t ) + m_{i}^{2}} \bigl\vert x_{k} ( t ) - \bar{x}_{k} ( t ) \bigr\vert . \end{aligned}$$
(4.12)

Define the Lyapunov functional \(V ( t ) \) as follows:

$$ V ( t ) = \sum_{i = 1}^{n} V_{i} ( t ) + \sum_{j = 1}^{m} V_{j} ( t ) . $$
(4.13)

Considering (4.10), (4.11) and (4.12), for \(t \ge T + \tau \), we have

$$ D^{ +} V ( t ) \le - \sum_{i = 1}^{n} A_{i} ( t ) \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert - \sum_{j = 1}^{m} B_{j} ( t ) \bigl\vert \bar{y}_{j} ( t ) - y_{j} ( t ) \bigr\vert , $$
(4.14)

where \(A_{i} ( t ) \), \(B_{j} ( t ) \) are given in Theorem 4.1.

From conditions \([H _{2} ]\) and \([H _{3} ]\), there exist \(\alpha_{i}, \beta_{j} > 0\) and \(T_{0} \ge T + \tau \) such that

$$ 0 < \alpha_{i} \le A_{i} ( t ) ,\quad\quad 0 < \beta_{j} \le B_{j} ( t ) \quad \mbox{for } t \ge T_{0}. $$
(4.15)

Let \(\alpha_{0} = \min \{ \alpha_{1},\alpha_{2},\ldots,\alpha_{n}; \beta_{1},\beta_{2},\ldots,\beta_{m} \} \), combining (4.14) and (4.15), then

$$ D^{ +} V ( t ) \le - \alpha_{0} \Biggl( \sum _{i = 1}^{n} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{j = 1}^{m} \bigl\vert \bar{y}_{j} ( t ) - y_{j} ( t ) \bigr\vert \Biggr) . $$
(4.16)

Integrating (4.16) on \([ T_{0},t ] \), we get

$$ V ( t ) + \alpha_{0} \int_{T_{0}}^{t} \Biggl( \sum _{i = 1} ^{n} \bigl\vert \bar{x}_{i} ( u ) - x_{i} ( u ) \bigr\vert + \sum_{j = 1}^{m} \bigl\vert \bar{y}_{j} ( u ) - y_{j} ( u ) \bigr\vert \Biggr) \,du \le V ( T_{0} ) ,\quad t \ge T_{0}. $$
(4.17)

So, \(\int_{T_{0}}^{ + \infty } ( \sum_{i = 1}^{n} \vert \bar{x}_{i} ( u ) - x_{i} ( u ) \vert + \sum_{j = 1}^{m} \vert \bar{y}_{j} ( u ) - y_{j} ( u ) \vert ) \,du < + \infty \) and \(V ( t ) \) is bounded on the interval \([ T_{0}, + \infty ) \). Combining Theorem 3.1 and (1.2), we get \(\bar{x}_{i} ( t ) - x_{i} ( t ) \), \(\bar{y}_{j} ( t ) - y_{j} ( t ) \) and \(( \bar{x}_{i} ( t ) - x_{i} ( t ) ) ^{\prime }\), \(( \bar{y}_{j} ( t ) - y_{j} ( t ) ) ^{\prime }\) are bounded on the interval \([ T_{0}, + \infty ) \). Then \(\sum_{i = 1}^{n} \vert \bar{x}_{i} ( u ) - x_{i} ( u ) \vert + \sum_{j = 1}^{m} \vert \bar{y}_{j} ( u ) - y_{j} ( u ) \vert \) is uniformly continuous.

Using Lemma 2.1, we get

$$ \begin{gathered} \lim_{t \to + \infty } \Biggl( \sum _{i = 1}^{n} \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert + \sum_{j = 1}^{m} \bigl\vert \bar{y}_{j} ( t ) - y_{j} ( t ) \bigr\vert \Biggr) = 0 \quad \mbox{and} \\ \lim_{t \to + \infty } \bigl\vert \bar{x}_{i} ( t ) - x_{i} ( t ) \bigr\vert = 0,\quad\quad \lim_{t \to + \infty } \bigl\vert \bar{y} _{j} ( t ) - y_{j} ( t ) \bigr\vert = 0. \end{gathered} $$
(4.18)

Therefore, system (1.2) is globally attractive. □

Existence of almost periodic solution

Theorem 5.1

Suppose \([H_{1}]\), \([H_{2}]\) and \([H_{3}]\) hold, then system (1.2) has a unique almost periodic solution.

Proof

From Theorem 3.2, we know \(( x ( t ) ,y ( t ) ) ^{T} \), \(t \in R\) is a bounded positive solution. Then there exists a sequence \(\{ t'_{\lambda } \} \), \(t'_{ \lambda } \to \infty \) as \(\lambda \to + \infty \) such that \(( x ( t + t'_{\lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T}\) is a solution of the following system (5.1):

$$\begin{aligned} \textstyle\begin{cases} x'_{i} ( t ) = x_{i} ( t ) [ r_{i} ( t + t'_{ \lambda } ) - \sum_{k = 1}^{n} b_{ik} ( t + t'_{\lambda } ) x _{k} ( t - \tau_{k} ( t ) ) - \sum_{k = 1}^{m} \frac{c _{ik} ( t + t'_{\lambda } ) x_{i} ( t ) }{x_{i}^{2} ( t ) + f_{ik} ( t + t'_{\lambda } ) }y_{k}^{ \alpha } ( t ) ] , \\ y'_{j} ( t ) = y_{j} ( t ) [ - r_{j} ( t + t'_{\lambda } ) - \sum_{k = 1}^{m} p_{jk} ( t + t'_{\lambda } ) y_{k} ( t ) + \sum_{k = 1}^{n} \frac{q_{kj} ( t + t'_{\lambda } ) x_{k}^{2} ( t ) }{x_{k}^{2} ( t ) + f_{kj} ( t + t'_{\lambda } ) }y_{j}^{\alpha - 1} ( t ) ] . \end{cases}\displaystyle \end{aligned}$$
(5.1)

From the above and Theorem 3.1, we know \(( x ( t + t'_{ \lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T}\) and \(( x' ( t + t'_{\lambda } ) ,y' ( t + t'_{\lambda } ) ) ^{T}\) are uniformly bounded. Clearly, the sequence \(( x ( t + t'_{\lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T}\) is equi-continuous. By the Ascoli theorem, there exists a uniformly convergent subsequence \(\{ ( x ( t + t_{\lambda } ) ,y ( t + t_{\lambda } ) ) ^{T} \} \subseteq \{ ( x ( t + t'_{\lambda } ) ,y ( t + t'_{\lambda } ) ) ^{T} \} \) such that, for any \(\forall \varepsilon > 0\), there exists \(\lambda_{0} ( \varepsilon ) > 0\) with the property that if λ, \(\varpi > \lambda_{0} ( \varepsilon ) \), then

$$ \bigl\vert x_{i} ( t + t_{\varpi } ) - x_{i} ( t + t_{ \lambda } ) \bigr\vert < \varepsilon , \quad\quad \bigl\vert y_{j} ( t + t_{ \varpi } ) - y_{j} ( t + t_{\lambda } ) \bigr\vert < \varepsilon , $$

which indicates that \(( x ( t + t_{\lambda } ) ,y ( t + t_{\lambda } ) ) ^{T}\) is an almost periodic and asymptotic function. Then there exist functions \(g_{i1} ( t ) \), \(g_{j2} ( t ) \), \(h_{i1} ( t ) \), \(h_{j2} ( t ) \) such that

$$ x_{i} ( t ) = g_{i1} ( t ) + h_{i1} ( t ) , \quad\quad y _{j} ( t ) = g_{j2} ( t ) + h_{j2} ( t ) , $$

where

$$\begin{aligned}& \lim_{\lambda \to + \infty } g_{i1} ( t + t_{\lambda } ) = g _{i1} ( t ) ,\quad\quad \lim_{\lambda \to + \infty } g_{j2} ( t + t _{\lambda } ) = g_{j2} ( t ) , \\& \lim_{\lambda \to + \infty } h_{i1} ( t + t_{\lambda } ) = 0,\quad\quad \lim_{\lambda \to + \infty } h_{j2} ( t + t_{\lambda } ) = 0, \end{aligned}$$

\(g_{i1} ( t ) \), \(g_{j2} ( t ) \) are almost periodic functions. It shows that

$$ \lim_{\lambda \to + \infty } x_{i} ( t + t_{\lambda } ) = g _{i1} ( t ) ,\quad\quad \lim_{\lambda \to + \infty } y_{j} ( t + t _{\lambda } ) = g_{j2} ( t ) . $$

Besides, we still have

$$ \lim_{\lambda \to + \infty } x_{i}^{\prime } ( t + t_{\lambda } ) = \lim_{\lambda \to + \infty } \lim_{\bar{h} \to 0} \frac{x _{i} ( t + t_{\lambda } + \bar{h} ) - x_{i} ( t + t_{ \lambda } ) }{\bar{h}} = \lim_{\bar{h} \to 0}\frac{g_{i1} ( t + \bar{h} ) - g_{i1} ( t ) }{\bar{h}}, $$
(5.2)
$$ \lim_{\lambda \to + \infty } y_{j}^{\prime } ( t + t_{\lambda } ) = \lim_{\lambda \to + \infty } \lim_{\bar{h} \to 0} \frac{y _{j} ( t + t_{\lambda } + \bar{h} ) - y_{j} ( t + t_{ \lambda } ) }{\bar{h}} = \lim_{\bar{h} \to 0}\frac{g_{j2} ( t + \bar{h} ) - g_{j2} ( t ) }{\bar{h}}. $$
(5.3)

Therefore, the derivatives \(g_{i1}^{\prime } ( t ) \), \(g_{j2} ^{\prime } ( t ) \) exist.

Next, we prove \(g ( t ) = ( g_{1} ( t ) ,g _{2} ( t ) ) ^{T}\) is an almost periodic solution of system (1.2).

By the characteristics of almost periodic solution, there exists a sequence \(\{ t_{\gamma } \} \), \(t_{\gamma } \to \infty \) as \(\gamma \to + \infty \) such that \(r_{i} ( t + t_{\gamma } ) \to r_{i} ( t ) \), \(r_{j} ( t + t_{\gamma } ) \to r_{j} ( t ) \), \(b_{il} ( t + t_{\gamma } ) \to b_{il} ( t ) \), \(p_{jk} ( t + t_{\gamma } ) \to p_{jk} ( t ) \), \(c_{ik} ( t + t_{\gamma } ) \to c_{ik} ( t ) \), \(q_{lj} ( t + t_{\gamma } ) \to q_{lj} ( t ) \), \(\tau_{i} ( t + t_{\gamma } ) \to \tau_{i} ( t ) \), \(f_{ij} ( t + t_{\gamma } ) \to f_{ij} ( t ) \) (\(i,l = 1,2,\ldots,n\); \(j,k = 1,2,\ldots,m\)) as \(\gamma \to \infty \) uniformly on R. Obviously, \(\lim_{\gamma \to + \infty } x_{i} ( t + t_{\gamma } ) = g _{i1} ( t ) \), \(\lim_{\gamma \to + \infty } y_{j} ( t + t _{\gamma } ) = g_{j2} ( t ) \). So we have

$$\begin{aligned}& \begin{aligned} g_{i1}^{\prime } ( t ) &= \lim _{\gamma \to + \infty } x_{i} ^{\prime } ( t + t_{\gamma } ) \\ &= \lim_{\gamma \to + \infty } x_{i} ( t + t_{\gamma } ) \Biggl[ r_{i} ( t + t_{\gamma } ) - \sum _{k = 1}^{n} b_{ik} ( t + t_{\gamma } ) x_{k} \bigl( ( t + t_{\gamma } ) - \tau_{k} ( t + t_{\gamma } ) \bigr) \\ &\quad{} - \sum_{k = 1}^{m} \frac{c _{ik} ( t + t_{\gamma } ) x_{i} ( t + t_{\gamma } ) y _{k}^{\alpha } ( t + t_{\gamma } ) }{x_{i}^{2} ( t + t _{\gamma } ) + f_{ik} ( t + t_{\gamma } ) } \Biggr] \\ &= g_{i1} ( t ) \Biggl[ r_{i} ( t ) - \sum _{k = 1} ^{n} b_{ik} ( t ) g_{k1} \bigl( t - \tau_{k} ( t ) \bigr) - \sum_{k = 1}^{m} \frac{c_{ik} ( t ) g_{i1} ( t ) g_{k2}^{\alpha } ( t ) }{g_{i1}^{2} ( t ) + f_{ik} ( t ) } \Biggr] , \end{aligned} \\& \begin{aligned} g_{j2}^{\prime } ( t ) &= \lim _{\gamma \to + \infty } y_{j} ^{\prime } ( t + t_{\gamma } ) \\ &= \lim_{\gamma \to + \infty } y_{j} ( t + t_{\gamma } ) \Biggl[ - r_{j} ( t + t_{\gamma } ) - \sum _{k = 1}^{m} p_{jk} ( t + t_{\gamma } ) x_{k} \bigl( ( t + t_{\gamma } ) - \tau_{k} ( t + t_{\gamma } ) \bigr) \\ &\quad{}+ \sum_{k = 1}^{n} \frac{q _{ik} ( t + t_{\gamma } ) x_{k}^{2} ( t + t_{\gamma } ) y _{k}^{\alpha - 1} ( t + t_{\gamma } ) }{x_{k}^{2} ( t + t _{\gamma } ) + f_{kj} ( t + t_{\gamma } ) } \Biggr] \\ &= g_{j2} ( t ) \Biggl[ - r_{j} ( t ) - \sum _{k = 1} ^{m} b_{ik} ( t ) g_{k2} ( t ) + \sum_{k = 1}^{n} \frac{q_{kj} ( t ) g_{k1}^{2} ( t ) g_{k2}^{\alpha - 1} ( t ) }{g_{k1}^{2} ( t ) + f_{kj} ( t ) } \Biggr] . \end{aligned} \end{aligned}$$

From the above, we know \(g ( t ) \) satisfies (1.2), that is, it is a positive almost periodic solution.

Next, we prove that the positive almost periodic solution of system (1.2) is unique.

Let \(g ( t ) > 0\) and \(\bar{g} ( t ) > 0\) be any two almost periodic solutions of system (1.2), then we claim that \(g_{1} ( t ) \equiv \bar{g}_{1} ( t ) \) and \(g_{2} ( t ) \equiv \bar{g}_{2} ( t ) \) for \(\forall t \in R\). Otherwise, there is at least one \(\xi \in R\) such that \(g _{i1} ( \xi ) \ne \bar{g}_{i1} ( \xi ) \), that is, \(\vert g_{i1} ( \xi ) - \bar{g}_{i1} ( \xi ) \vert : = \delta > 0\). Then

$$ \delta = \bigl\vert g_{i1} ( \xi ) - \bar{g}_{i1} ( \xi ) \bigr\vert = \lim_{\gamma \to + \infty } \bigl\vert x_{i} ( \xi + t_{\gamma } ) - \bar{x}_{i} ( \xi + t_{\gamma } ) \bigr\vert = \lim_{t \to + \infty } \bigl\vert x_{i} ( t ) - \bar{x}_{i} ( t ) \bigr\vert > 0, $$

which is a contradiction to (4.18). Thus \(\forall t \in R\), \(g_{1} ( t ) \equiv \bar{g}_{1} ( t ) \) holds. By the same method, we can prove that \(\forall t \in R\), \(g_{2} ( t ) \equiv \bar{g}_{2} ( t ) \). □

Remark 5.1

If \(\tau_{i} ( t ) \equiv \tau_{i}\), where \(\tau_{i}\) (\(i = 1,2,\ldots,n \)) is a nonnegative constant, then assumptions \([H _{2} ] \) and \([H _{3} ] \) can be redefined. So, we give the following Corollary 5.1.

Corollary 5.1

Make \(\tau_{i} ( t ) \equiv \tau _{i}\), where \(\tau_{i} \ge 0\). If system (1.2) satisfies both \([H _{1} ]\) and the following two conditions:

$$\begin{aligned}& \lim_{t \to + \infty } \inf \Biggl\{ - \sum_{k = 1}^{m} c_{ik} ( t ) E_{ik} + \sum_{k = 1}^{n} b _{ik} ( t ) \\& \quad{} - \sum_{k = 1}^{n} \Biggl[ r_{k} ( s ) + \sum_{j = 1}^{n} b_{kj} ( s ) M_{j} + \sum_{j = 1}^{m} \frac{q _{kj} ( s ) M_{k}N_{j}}{f_{kj}^{l} + m_{k}^{2}} \Biggr] \int _{t}^{t + \tau_{i}} b_{ik} ( u ) \,du \\& \quad{} - \sum_{k = 1}^{n} \sum _{j = 1}^{m} c_{ik} ( t ) M_{k}E_{kj} \int_{t}^{t + \tau_{i}} b_{ik} ( u ) \,du - \sum _{k = 1}^{n} \sum_{j = 1}^{n} M_{k}p_{kj} ( t + \tau_{i} ) \int_{t + \tau_{i}}^{t + \tau_{k} + \tau_{i}} b_{ik} ( u ) \,du \\& \quad{} - \sum_{k = 1}^{m} q_{kj} ( t ) \frac{2M_{i}N_{j}^{\alpha - 1}}{f_{ik} ( t ) + m_{i}^{2}} \Biggr\} > 0, \\& \lim_{t \to + \infty } \inf \Biggl\{ - \sum_{k = 1}^{m} c _{ik} ( t ) F_{ik} - \sum_{k = 1}^{n} \sum_{j = 1}^{m} q_{kj} ( t ) M_{k}E_{kj} \int_{t}^{t + \tau_{i}} b_{ik} ( u ) \,du + \sum _{k = 1}^{m} q_{jk} ( t ) \Biggr\} > 0, \end{aligned}$$

then system (1.2) has a unique positive almost periodic solution which is globally attractive.

Model simulation

We give examples to verify the correctness of our theoretical results in this part.

Example 6.1

Consider the following system:

$$ \textstyle\begin{cases} x' ( t ) = x ( t ) [ ( 3 + 0.1\sin \sqrt{7} t ) - ( 2 - 0.1\sin t ) x ( t - 0.01 ) \\ \hphantom{x' ( t ) = }{}- \frac{ ( 0.05 - 0.01\sin t ) x ( t ) }{x ^{2} ( t ) + 1}y^{0.2} ( t ) ] , \\ y' ( t ) = y ( t ) [ - ( 0.02 + 0.01 \sin \sqrt{3} t ) - ( 0.05 - 0.01\sin t ) y ( t ) \\ \hphantom{y' ( t ) =}{}+ \frac{ ( 0.2 + 0.01\sin t ) x^{2} ( t ) }{x ^{2} ( t ) + 1}y^{ - 0.8} ( t ) ] , \end{cases} $$
(6.1)

with the initial conditions \(( \phi ( 0 ) ,\psi ( 0 ) ) = ( 1,1 ) \) and \(( \phi ( 0 ) , \psi ( 0 ) ) = ( 2.5,2 ) \).

By calculation, the parameters of (6.1) meet the conditions of Theorem 3.1 and Corollary 5.1. Using MATLAB, by simulation, time series diagrams of (6.1) are shown in Figure 1. Figure 1 indicates that (6.1) is persistent and has a unique positive almost periodic solution which is globally attractive.

Figure 1
figure1

(a), (b) The time series diagrams with two initial values of prey and predator, respectively. (c) Two-dimensional periodic diagram of predator-prey. (d) Three-dimensional periodic diagram of predator-prey-time.

In order to demonstrate the dynamical behaviors of a multispecies predator-prey system, we give the time series diagrams with only three species in system (1.2).

Example 6.2

Consider the following system:

$$\begin{aligned} \textstyle\begin{cases} x_{1}^{\prime } ( t ) = x_{1} ( t ) [ ( 1 + 0.1\sin \sqrt{7} t ) - ( 1.3 - 1.1\sin t ) x _{1} ( t - 0.01 ) \\ \hphantom{x_{1}^{\prime } ( t ) = }{}- ( 1.4 - 1.2\sin t ) x_{2} ( t - 0.01 ) - \frac{ ( 0.05 - 0.01\sin t ) x_{1} ( t ) }{x_{1}^{2} ( t ) + 30}y^{0.5} ( t ) ] , \\ x_{2}^{\prime } ( t ) = x_{2} ( t ) [ ( 1 + 0.1\sin \sqrt{7} t ) - ( 1.3 - 1.1\sin t ) x _{1} ( t - 0.01 ) \\ \hphantom{x_{2}^{\prime } ( t ) =}{}- ( 1.4 - 1.2\sin t ) x_{2} ( t - 0.01 ) - \frac{ ( 0.05 - 0.01\sin t ) x_{2} ( t ) }{x_{2}^{2} ( t ) + 30}y^{0.5} ( t ) ] , \\ y' ( t ) = y ( t ) [ - ( 0.02 + 0.01 \sin \sqrt{3} t ) - ( 0.05 - 0.01\sin t ) y ( t ) \\ \hphantom{y' ( t ) =}{} + \frac{ ( 0.5 + 0.01\sin t ) x_{1} ( t ) y ^{ - 0.5} ( t ) }{x_{1}^{2} ( t ) + 30} + \frac{ ( 0.5 + 0.01\sin t ) x_{2} ( t ) y^{ - 0.5} ( t ) }{x_{2}^{2} ( t ) + 30} ] , \end{cases}\displaystyle \end{aligned}$$
(6.2)

with the initial conditions \(( \phi_{1} ( 0 ) ,\phi_{2} ( 0 ) ,\psi ( 0 ) ) = ( 2,2,2 ) \) and \(( \phi_{1} ( 0 ) ,\phi_{2} ( 0 ) ,\psi ( 0 ) ) = ( 0.1, 0.1,0.1 ) \).

By calculation, these parameters of (6.2) meet the conditions of Theorem 3.1 and Corollary 5.1. Using MATLAB, by simulation, time series diagrams of (6.2) are shown in Figure 2. Figure 2 shows that (6.2) is persistent and has a unique positive almost periodic solution which is globally attractive.

Figure 2
figure2

(a), (b) Time series diagrams with two initial values of prey \(x_{i}\) (\(i = 1,2 \)), respectively. (c) The time series diagram with two initial values of predator y. (d) Three-dimensional periodic diagram.

Conclusion

We construct a multispecies predator-prey model with mutual interference and time delays in this article. We obtain the conditions of permanence, global attractivity and uniqueness of positive almost periodic solutions of the system by using the Ascoli theorem, Lebesgue dominated convergence theorem, Lyapunov functions and comparison theorem. Finally, simulation results indicate the correctness of the theoretical results and demonstrate the complex dynamical behaviors of the system.

Compared with Ref. [10], Du only considered the basic dynamics of two species, which can not be effectively promoted and applied in actual production and life. However, we comprehensively integrate the universal phenomenon of multispecies coexistence in the real ecosystem. By studying the dynamics of multispecies predator-prey system, we can better protect the ecosystem and practice the concept of green development.

References

  1. 1.

    Holling, CS: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 97(S45), 5-60 (1965)

    Article  Google Scholar 

  2. 2.

    Pei, Y, Chen, L, Zhang, Q, Li, C: Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control. J. Theor. Biol. 235(4), 495-503 (2005)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hassel, MP: Density dependence in single-species population. J. Anim. Ecol. 44(1), 283-295 (1975)

    Article  Google Scholar 

  4. 4.

    Guo, H, Chen, X: Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response. Appl. Math. Comput. 217, 5830-5837 (2011)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Lin, X, Chen, F: Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response. Appl. Math. Comput. 214(2), 548-556 (2009)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Wang, K: Permanence and global asymptotical stability of a predator-prey model with mutual interference. Nonlinear Anal., Real World Appl. 12(2), 1062-1071 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Lv, Y, Du, Z: Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response. Nonlinear Anal., Real World Appl. 12(6), 3654-3664 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Zeng, G, Wang, F, Nieto, JJ: Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response. Adv. Complex Syst. 11(1), 77-97 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Wang, K, Zhu, Y: Permanence and global asymptotic stability of a delayed predator-prey model with Hassell-Varley type functional response. Bull. Iran. Math. Soc. 37(3), 197-215 (2011)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Du, Z, Lv, Y: Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time delays. Appl. Math. Model. 37(3), 1054-1068 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Shao, Y, Dai, B, Luo, Z: The dynamics of an impulsive one-prey multi-predators system with delay and Holling-type II functional response. Appl. Math. Comput. 217(6), 2414-2424 (2010)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bahaa, GM: Fractional optimal control problem for differential system with delay argument. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-017-1121-6

    MathSciNet  Google Scholar 

  13. 13.

    Wang, B, Cheng, J, Al-Barakati, A, Fardoun, HM: A mismatched membership function approach to sampled-data stabilization for T-S fuzzy systems with time-varying delayed signals. Signal Process. 140, 161-170 (2017)

    Article  Google Scholar 

  14. 14.

    Cheng, J, Park, JH, Zhang, L, Zhu, Y: An asynchronous operation approach to event-triggered control for fuzzy Markovian jump systems with general switching policies. IEEE Trans. Fuzzy Syst. (2016). https://doi.org/10.1109/TFUZZ.2016.2633325

    Google Scholar 

  15. 15.

    Meng, X, Zhao, S, Zhang, W: Adaptive dynamics analysis of a predator-prey model with selective disturbance. Appl. Math. Comput. 266, 946-958 (2015)

    MathSciNet  Google Scholar 

  16. 16.

    Aghajanzadeh, O, Sharif, M, Tashakori, S, Zohoor, H: Nonlinear adaptive control method for treatment of uncertain hepatitis B virus infection. Biomed. Signal Process. Control 38, 174-181 (2017)

    Article  Google Scholar 

  17. 17.

    Rui, J, Liu, B: Almost-periodic solutions of an almost-periodically forced wave equation. J. Math. Anal. Appl. 451(2), 629-658 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Meng, X, Chen, L: Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays. J. Theor. Biol. 243(4), 562-574 (2006)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Chen, X: Almost periodic solutions of nonlinear delay population equation with feedback control. Nonlinear Anal., Real World Appl. 8(1), 62-72 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Zhou, Z: Global attractivity and periodic solution of a discrete multispecies cooperation and competition predator-prey system. Discrete Dyn. Nat. Soc. 2011, Article ID 835321 (2011)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Qiao, Y, Zhou, Z: Existence of positive solutions of singular fractional differential equations with infinite-point boundary conditions. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-016-1042-9

    MathSciNet  Google Scholar 

  22. 22.

    Miao, Z, Chen, F, Liu, J, Pu, L: Dynamic behaviors of a discrete Lotka-Volterra competitive system with the effect of toxic substances and feedback controls. Adv. Differ. Equ. (2017). https://doi.org/10.1186/s13662-017-1130-5

    MathSciNet  Google Scholar 

  23. 23.

    Chen, F: Almost periodic solution of the non-autonomous two-species competitive model with stage structure. Appl. Math. Comput. 181(1), 685-693 (2006)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Fink, AM: Almost Periodic Differential Equations. Springer, Berlin (1974)

    Google Scholar 

  25. 25.

    Qi, W, Dai, B: Almost periodic solution for n-species Lotka-Volterra competitive system with delay and feedback controls. Appl. Math. Comput. 200(1), 133-146 (2008)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Gopalsamy, K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Springer, Berlin (1992)

    Google Scholar 

  27. 27.

    Zeng, G, Chen, L, Sun, L, Liu, Y: Permanence and the existence of the periodic solution of the non-autonomous two-species competitive model with stage structure. Adv. Complex Syst. (2004). https://doi.org/10.1142/S0219525904000238

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Shen, C: Permanence and global attractivity of the food-chain system with Holling IV type functional response. Appl. Math. Comput. 194(1), 179-185 (2007)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Zhu, Y: Uniformly persistence of a predator-prey model with Holling III type functional response. Int. J. Comput. Math. Sci. 4(7), 355-357 (2010)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper is supported by the Natural Science Foundation of Guangxi (2016GXNSFAA380194), National Natural Science Foundation of China (11161015).

Authors’ information

QL is now studying for the M.S. degree at Guilin University of Technology, Guilin, China. His research interest is the study of dynamical behaviors and simulation analysis of periodic impulsive differential equations. YS received Ph.D. degree in mathematics from Central South University, Changsha, China. He is a professor of School of Science, Guilin University of Technology, Guilin, China. His research interests are differential equations and dynamical systems. He is mainly engaged in qualitative studies of complex dynamical systems. SZ, ZW and HC are now studying for the M.S. degree at Guilin University of Technology, Guilin, China. Their research interests are stability analysis and numerical simulation of impulsive systems.

Author information

Affiliations

Authors

Contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Corresponding author

Correspondence to Yuanfu Shao.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, Q., Shao, Y., Zhou, S. et al. Dynamical analysis of almost periodic solution for a multispecies predator-prey model with mutual interference and time delays. Adv Differ Equ 2017, 393 (2017). https://doi.org/10.1186/s13662-017-1447-0

Download citation

Keywords

  • almost periodic solution
  • global attractivity
  • mutual interference
  • numerical simulation