ABSTRACT

In this paper, we consider an impulsive competitive system with infinite delay and diffusion. Firstly, on basis of inequality estimation techniques and comparison theorem of impulsive differential equations, we obtain some sufficient conditions for the permanence and extinction of the system. Then, we establish sufficient conditions for the globally attractive of the system by constructing appropriate Lyapunov function. Besides, under different impulsive conditions, we discuss the effect of time delay and diffusion on dynamic behavior of the competitive system.

Keywords:

Diffusion, Infinite Delay, Permanence, Extinction, Attractive

1. Introduction

In practical world, owing to many natural and man-made factors (e.g., fire, drought, flooding, crop-dusting, deforestation, hunting, harvesting, etc.), the biological species or ecological environments usually undergo some discrete changes of relatively short duration at some fixed times. Such sudden changes can often be characterized mathematically in the form of impulses. With the development of impulsive differential equations, many experts have adequate mathematical models to investigate the dynamical behaviors of such ecosystems with impulsive effects [1] [2] [3] [4] [5]. On the other hand, the Lotka-Volterra competition systems are very important and significant mathematical models in a non-autonomous environment. Many interesting results of the competitive systems on the existence of positive periodic solutions, permanence, extinction, global stability had been studied extensively (see [6] [7] [8] [9] [10] ). For example, Wang [10] investigated the following competitive system with impulsive effects:

$\{\begin{array}{l}\begin{array}{l}{u}^{\prime }\left(t\right)=u\left(t\right)\left(b_1\left(t\right)-a_{11}\left(t\right)u\left(t\right)-a_{12}\left(t\right)v\left(t\right)\right)\\ v^{\prime }\left(t\right)=u\left(t\right)\left(b_2\left(t\right)-a_{21}\left(t\right)u\left(t\right)-a_{22}\left(t\right)v\left(t\right)\right)\end{array}\},t\ne t_k\\ \begin{array}{l}u\left(t_k^{+}\right)=\left(1+h_k\right)u\left(t_k\right)\\ v\left(t_k^{+}\right)=\left(1+g_k\right)v\left(t_k\right)\end{array}\},t=t_k,k=1,2,\cdots .\end{array}$

The author obtained sufficient conditions on the uniform persistence and extinction of the system by applying the theorem of differential equations.

In ecological environment, because of the natural enemy, severe competition, deterioration of the patch environment, spatial heterogeneity and human activities, species dispersal in two or more patches becomes one of widespread phenomena of nature. It is an important subject to study the effects of dispersion on the dynamics of species living in patchy environments. Many works on population dynamics in patch environment have been investigated [11] [12] [13] [14]. Moreover, in real ecology environment, the existed number on the history will affect indirectly the number of the species at the moment. Therefore, in order to establish more realistic models, the past history of systems should be taken into account, which has led to the introduction of time-delays in differential equations. Such biological system with infinite delay can be found in [15] [16] [17].

Motivated by above arguments, we establish an impulsive competitive system with infinite delay and diffusion as follows：

$\{\begin{array}{l}\begin{array}{l}{x^{\prime }}_1\left(t\right)=x_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)-c_1\left(t\right){\displaystyle {\int }_{-\infty }^0{k}_1\left(s\right)y\left(t+s\right)\text{d}s}\right)\\ +D_{12}\left(t\right)\left(x_2\left(t\right)-x_1\left(t\right)\right)\\ {x^{\prime }}_2\left(t\right)=x_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)x_2\left(t\right)\right)+D_{21}\left(t\right)\left(x_1\left(t\right)-x_2\left(t\right)\right)\\ y^{\prime }\left(t\right)=y\left(t\right)\left(a_3\left(t\right)-b_3\left(t\right)y\left(t\right)-c_2\left(t\right){\displaystyle {\int }_{-\infty }^0{k}_2\left(s\right)x_1\left(t+s\right)\text{d}s}\right)\end{array}\},t\ne t_k\\ \begin{array}{l}{x}_1\left(t_k^{+}\right)=h_{1k}{x}_1\left(t_k\right)\\ x_2\left(t_k^{+}\right)=h_{2k}{x}_2\left(t_k\right)\\ y\left(t_k^{+}\right)=g_{k}y\left(t_k\right)\end{array}\},k=1,2,\cdots \end{array}$ (1.1)

where $x_i\left(t\right)\left(i=1,2\right)$ and $y\left(t\right)$ represent the population densities at time t, respectively. Let $0=t_0<t_k<t_{k+1},k=1,2,\cdots$ and $t_k\to \infty$ as $k\to \infty$. Species $x_1$ competes with y in patch 1, while $x_1$ can disperse between patch 1 and patch 2, and y is confined to patch 1. $D_{12}\left(t\right),D_{21}\left(t\right)$ denotes the diffusion coefficients of species x. $h_{1k},h_{2k}$ and $g_k$ are impulsive coefficients at time $t_k$, respectively.

We consider system (1.1) with the following initial conditions

$\begin{array}{l}{x}_1\left(\theta \right)={\varphi }_1\left(\theta \right),x_2\left(\theta \right)={\varphi }_2\left(\theta \right),y\left(\theta \right)={\varphi }_3\left(\theta \right),\\ {\varphi }_i\in P{C}_{+}\left(R_{-},R_{+}\right),i=1,2,3,R_{-}=\left(-\infty ,0\right],R_{+}=\left[0,+\infty \right).\end{array}$ (1.2)

where $P{C}_{+}=\left\{\varphi =\left({\varphi }_1,{\varphi }_2,{\varphi }_3\right)\in PC\right\}$, ${\varphi }_i\left(\theta \right)\ge 0$ for all $\theta \in R_{-}$ and ${\varphi }_i\left(0^{+}\right)>0$ for $i=1,2,3$. $PC$ is the space of bounded function $\varphi \left(s\right)$ : $R_{-}\to R^3$ which is continuous everywhere expect at the point $t=t_k\in I$ and $\varphi \left(t_k^{+}\right),\varphi \left(t_k^{-}\right)$ exists with $\varphi \left(t_k^{-}\right)=\varphi \left(t_k\right)$ and with norm $\Vert \varphi \Vert =\underset{\theta \in R_{-}}{\sup }\Vert {\varphi }_{\theta }\Vert$.

In this paper, for any continuous function $f\left(t\right)$, we denote

$f\left(t\right)=\underset{i\in I}{\max }\left\{f_i\left(t\right)\right\}$, $\bar{f}\left(t\right)=\underset{i\in I}{\min }\left\{f_i\left(t\right)\right\}$.

Throughout this paper, we assume that the system (1.1) satisfies ((C1), (C2) see [12] , (C3) see [17] ):

(C1) all functions are positive, continuous and bounded defined on $R_{+}$, $0<D_{12}\left(t\right),D_{21}\left(t\right)\le D$.

(C2) $h_{1k},h_{2k}$ and $g_k$ are positive constants for all $k=1,2,\cdots$.

(C3) $k_i\left(s\right)\left(i=1,2\right)$ is a non-negative, piece-wise continuous function defined on $R_{-}$ and satisfy ${\displaystyle {\int }_0^{+\infty }{k}_i\left(s\right)\text{d}s=1}$.

Applying some inequality techniques, comparison theorem of impulsive differential equations and Lyapunov function, we study the dynamic behaviors of an impulsive competitive system with infinite delay and diffusion, included permanence, extinction and globally attractive. This paper is organized as follows. Section 2 contains some preliminaries and presents the proof of the lemma. In Section 3, we establish some sufficient conditions which guarantee the system is permanence. In finally section, we give some conditions on the extinction of the system. In Section 4, we study the globally attractive of system (1.1).

2. Preliminaries

We consider the following impulsive non-autonomous logistic model

$\{\begin{array}{l}{x}^{\prime }\left(t\right)=x\left(t\right)\left(\alpha \left(t\right)-\beta \left(t\right)x\left(t\right)\right),t\ne t_k\\ x\left(t_k^{+}\right)=h_{k}x\left(t_k\right),k=1,2,\cdots ,\end{array}$ (2.1)

where $\alpha \left(t\right)$ and $\beta \left(t\right)$ are bounded and continuous functions defined on $R_{+}$, $\beta \left(t\right)\ge 0$ for all $t\in R_{+}$ and impulsive coefficients $h_k$ are positive constants for any $k=1,2,\cdots$. Then we have the following Lemma 2.1.

Lemma 2.1. Suppose that there is a constant $\sigma >0$ such that

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\sigma }\beta \left(s\right)\text{d}s}\right)>0$ (2.2)

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\sigma }\alpha \left(s\right)\text{d}s+{\displaystyle \underset{t\le t_k<t+\sigma }{\sum }\ln f_k}}\right)>0$ (2.3)

and function

$h\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln h_k}$ (2.4)

is bounded on $t\in R_{+}$ and $\mu \in \left[0,\sigma \right)$. Then we have

1) There exist constant $M>0$ and $m>0$ such that

$m\le \underset{t\to \infty }{lim\inf }x\left(t\right)\underset{t\to \infty }{\le \lim \sup }x\left(t\right)\le M$ (2.5)

for any positive solution $x\left(t\right)$ of system (2.1).

2) If all conditions of (1) hold, further if

$\underset{t\to \infty }{\lim \sup }\left({\displaystyle {\int }_t^{t+\sigma }{\alpha }_1\left(s\right)\text{d}s+{\displaystyle \underset{t\le t_k<t+\sigma }{\sum }\ln f_k}}\right)\le 0$ (2.6)

then we have $\underset{t\to \infty }{\lim }x\left(t\right)=0$ for any positive solution $x\left(t\right)$ of system (2.1).

The proof of this Lemma can be found in [18] , here we omit it.

Next we consider the following impulsive periodic single species logistic system with diffusion

$\{\begin{array}{l}{x^{\prime }}_i\left(t\right)=x_i\left(t\right)\left(r_i\left(t\right)-a_i\left(t\right)\right)+{\displaystyle \underset{j=1}{\overset{n}{\sum }}{D}_{ij}\left(x_j\left(t\right)-x_i\left(t\right)\right),t\ne t_k,}\\ x_i\left(t_k^{+}\right)=h_{ik}{x}_i\left(t_k\right),i=1,2,\cdots ,n,k=1,2,\cdots ,\end{array}$ (2.5)

Assume that $r_i\left(t\right),a_i\left(t\right),D_{ij}\left(t\right)\left(i,j\in I\right)$ are positive, continuous and bounded functions defined on $R_{+}$. $D_2\ge D_{ij}\ge D_1>0\left(i\ne j\right)$, $D_{ii}=0$ for all $i,j\in I\left(I=1,2,\cdots ,n\right)$ and $t\in R_{+}$, $h_{ik}>0$ for all $i\in I$, $k=1,2,\cdots$, then we have the following conclusions.

Lemma 2.2. Suppose that there is a positive constant $\bar{\sigma }$ such that

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\bar{\sigma }}\bar{a}\left(s\right)\text{d}s}\right)>0$ (2.6)

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\bar{\sigma }}\bar{r}\left(t\right)-{\displaystyle \underset{j=1}{\overset{n}{\sum }}{D}_{ij}}\left(t\right)\text{d}t}+{\displaystyle \underset{t\le t_k<t+\bar{\sigma }}{\sum }ln{\bar{h}}_k}\right)>0$ (2.7)

and function

$h\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln h_k}$, $\bar{h}\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln {\bar{h}}_k}$ (2.8)

is bounded on $t\in R_{+}$ and $\mu \in \left[0,\bar{\sigma }\right)$. Then we have

1) There are constants $M>0$ and $m>0$ such that

$m\le \underset{t\to \infty }{\lim \inf }{x}_i\left(t\right)\le \underset{t\to \infty }{\lim \sup }{x}_i\left(t\right)\le M$

for any positive solution $x_i\left(t\right)$ of system(2.1). $t\in R_{+},i\in I$.

2) If all conditions of (1) hold, further if

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\bar{\sigma }}{\beta }_1\left(t\right)\text{d}t}\right)>0$, (2.9)

where ${\beta }_1\left(t\right)=\underset{i\in I}{\min }\left\{a_i\left(t\right)-{\displaystyle \underset{j=1}{\overset{n}{\sum }}\frac{D_{ij}\left(t\right)}{m}}\right\}\ge 0$ for all $t\in R_{+},i,j\in I$, then system (2.2) is globally attractive.

Proof: Firstly, we prove system (2.5) is permanent. Let $\left(x_1\left(t\right),\cdots ,x_i\left(t\right)\right)$ be any solution of system (2.5). Define the function $V_1\left(t\right)=\underset{i\in I}{\max }\left\{x_i\left(t\right)\right\}$, when $t\ne t_k$ calculating the upper-right derivative of $V_1\left(t\right)$, we have

$D^{+}{V}_1\left(t\right)\le x_i\left(t\right)\left(r_i\left(t\right)-a_i\left(t\right)x_i\left(t\right)\right)\le V_1\left(t\right)(r\left(t\right)-\bar{a}\left(t\right)V_1(t))$

when $t=t_k$, we have

$V_1\left(t_k^{+}\right)=\underset{i\in I}{\max }\left\{x_i\left(t_k^{+}\right)\right\}=\underset{i\in I}{\max }\left\{h_{ik}{x}_i\left(t_k\right)\right\}\le \underset{i\in I}{\max }\left\{h_{ik}\right\}\underset{i\in I}{\max }\left\{x_i\left(t_k\right)\right\}=h_k{V}_1(tk)$

Consider the following auxiliary system

$\{\begin{array}{l}{D}^{+}w\left(t\right)=w\left(t\right)\left(r\left(t\right)-\bar{a}\left(t\right)w\left(t\right)\right),t\ne t_k\\ w\left(t_k^{+}\right)=h_{k}w\left(t_k\right),k=1,2,\cdots \end{array}$ (2.10)

with initial condition $w\left(0\right)=V_1\left(0\right)$. Obviously, from condition (2.6), (2.8) and Lemma 2.1, there exists a constant $M>0$ such that $\underset{t\to \infty }{\lim \sup }w\left(t\right)\le M$. Then according to comparison theorem of impulsive differential equations, we derive

$\underset{t\to \infty }{\lim \sup }{x}_i\left(t\right)\le \underset{t\to \infty }{\lim \sup }{V}_1\left(t\right)\le \underset{t\to \infty }{\lim \sup }w\left(t\right)\le M$ for $i\in I$. (2.11)

Now we prove there is a constant $m>0$ such that $\underset{t\to \infty }{\lim \inf }{x}_i\left(t\right)\ge m$. Defined $V_2\left(t\right)=\underset{i\in I}{\min }\left\{x_i\left(t\right)\right\}$, calculating the right-lower derivative of $V_2\left(t\right)$ when $t\ne t_k$, similar to above conclusion, we can obtain

$D^{+}{V}_2\left(t\right)\ge x_i\left(t\right)\left(r_i\left(t\right)-a_i\left(t\right)x_i\left(t\right)\right)\ge V_2\left(t\right)\left(\bar{r}\left(t\right)-a\left(t\right)V_2(t)\right)$

when $t=t_k$, we have

$V_2\left(t_k^{+}\right)=\underset{i\text{=}\left\{\text{1,2}\right\}}{\min }\left\{x_i\left(t_k^{+}\right)\right\}=\underset{i\text{=}\left\{\text{1,2}\right\}}{\min }\left\{h_{ik}{x}_i\left(t_k\right)\right\}\ge \underset{i\text{=}\left\{\text{1,2}\right\}}{\min }\left\{h_{ik}\right\}\underset{i\text{=}\left\{\text{1,2}\right\}}{\min }\left\{x_i\left(t_k\right)\right\}={\bar{h}}_k{V}_2(tk)$

by comparison theorem of impulsive differential equations, we derive $V_2\left(t\right)\ge v\left(t\right)$ for all $t\in R_{+}$, where $v\left(t\right)$ is the solution of the auxiliary equation

$\{\begin{array}{l}{D}^{+}v\left(t\right)=v\left(t\right)\left(\bar{r}\left(t\right)-a\left(t\right)v_2\left(t\right)\right),t\ne t_k\\ v\left(t_k^{+}\right)={\bar{h}}_{k}v\left(t_k\right),k=1,2,\cdots ,\end{array}$ (2.12)

with initial condition $V_2\left(0\right)=v\left(0\right)$. Clearly, from condition (2.7), (2.8) and Lemma 2.1, there is a constant $m>0$ such that $\underset{t\to \infty }{\lim \inf }v\left(t\right)\ge m$. Therefore we have

$\underset{t\to \infty }{\lim \inf }{x}_i\left(t\right)\ge \underset{t\to \infty }{\lim \inf }{V}_2\left(t\right)\ge \underset{t\to \infty }{\lim \inf }v\left(t\right)\ge m$ for $i\in I$.

Next we consider the globally attractive of system (2.5). Construct a Lyapunov function

$V\left(t\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|\ln x_i\left(t\right)-\ln {\tilde{x}}_i\left(t\right)\right|}$

when $t=t_k$, we always have

$V\left(t_k\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|\ln h_{ik}{x}_i\left(t_k\right)-\ln h_{ik}{\tilde{x}}_i\left(t_k\right)\right|}=V\left(t_k^{+}\right)$

then $V\left(t\right)$ is continuous for all $t\in R_{+}$. In addition, when $t\in R_{+}$ and $t\ne t_k$ we have

$\frac{1}{B}\left|x_i\left(t\right)-{\tilde{x}}_i\left(t\right)\right|\le \left|\ln x_i\left(t\right)-\ln {\tilde{x}}_i\left(t\right)\right|\le \frac{1}{A}\left|x_i\left(t\right)-{\tilde{x}}_i\left(t\right)\right|$ (2.13)

Moreover, calculating the derivative of $V\left(t\right)$, we also have

$\begin{array}{c}{D}^{+}V\left(t\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\mathrm{sgn}\left(x_1\left(t\right)-{\tilde{x}}_1\left(t\right)\right)\left(\frac{{x^{\prime }}_i\left(t\right)}{x_i\left(t\right)}-\frac{{{\tilde{x}}^{\prime }}_i\left(t\right)}{{\tilde{x}}_i\left(t\right)}\right)}\\ \le {\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(-a_i\left(t\right)\left|x_i\left(t\right)-{\tilde{x}}_i\left(t\right)\right|\right)}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{D}}_{ij}\left(t\right)}}\end{array}$ (2.14)

where ${\bar{D}}_{ij}\left(t\right)=\{\begin{array}{l}{D}_{ij}\left(t\right)\left(\frac{x_j\left(t\right)}{x_i\left(t\right)}-\frac{{\tilde{x}}_j\left(t\right)}{{\tilde{x}}_i\left(t\right)}\right),x_i\left(t\right)>{\tilde{x}}_i\left(t\right),\\ D_{ij}\left(t\right)\left(\frac{{\tilde{x}}_j\left(t\right)}{{\tilde{x}}_i\left(t\right)}-\frac{x_j\left(t\right)}{x_i\left(t\right)}\right),x_i\left(t\right)<{\tilde{x}}_i\left(t\right).\end{array}$

For all $t\in R_{+}$, we think about under the following two cases:

Case 1: if $x_i\left(t\right)>{\tilde{x}}_i\left(t\right)$, then

${\bar{D}}_{ij}\left(t\right)\le \frac{D_{ij}\left(t\right)}{x_i\left(t\right)}\left(x_j\left(t\right)-{\tilde{x}}_j\left(t\right)\right)\le \frac{D_{ij}\left(t\right)}{A}\left|x_j\left(t\right)-{\tilde{x}}_j\left(t\right)\right|$

Case 2: if $x_i\left(t\right)<{\tilde{x}}_i\left(t\right)$, then

${\bar{D}}_{ij}\left(t\right)\le \frac{D_{ij}\left(t\right)}{{\tilde{x}}_i\left(t\right)}\left({\tilde{x}}_j\left(t\right)-x_j\left(t\right)\right)\le \frac{D_{ij}\left(t\right)}{A}\left|x_j\left(t\right)-{\tilde{x}}_j\left(t\right)\right|$

From Case 1, Case 2 and (2.14), we can obtain

$\begin{array}{c}{D}^{+}V\left(t\right)\le {\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(-a_i\left(t\right)\left|x_i\left(t\right)-{\tilde{x}}_i\left(t\right)\right|\right)}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j=1}{\overset{n}{\sum }}\frac{D_{ij}}{A}\left|x_j\left(t\right)-{\tilde{x}}_j\left(t\right)\right|}}\\ \le -{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(a_i\left(t\right)-{\displaystyle \underset{j=1}{\overset{n}{\sum }}\frac{D_{ij}}{A}}\right)\left|x_i\left(t\right)-{\tilde{x}}_i\left(t\right)\right|}\\ \le -\beta \left(t\right)AV\left(t\right)\end{array}$ (2.15)

By (2.15) and condition (2.9), we have $V\left(t\right)\le V\left(0\right)\exp \left(-A{\displaystyle {\int }_0^t\beta \left(s\right)\text{d}s}\right)\to 0$ as $t\to \infty$. Furthermore, by (2.13), we have $\underset{t\to \infty }{\lim }\left(x_i\left(t\right)-{\tilde{x}}_i\left(t\right)\right)=0$, that is, system (2.5) has globally attractive positive solution. This completes the proof of Lemma 2.2.

3. Permanence

Note that system (1.1) always has a positive solution for all $t\in R_{+}$ if it has a positive initial condition. Here we state and prove the permanent of system (1.1).

Theorem 3.1. There exists a constant $M>0$ such that $\underset{t\to \infty }{\lim \sup }{x}_i\left(t\right)\le M$ and $\underset{t\to \infty }{\lim \sup }y\left(t\right)\le M$ for any positive solution of system (1.1) if there exists a constant $\omega >0$ such that

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\omega }\bar{b}\left(s\right)\text{d}s}\right)>0$ (3.1)

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\omega }{b}_3\left(s\right)\text{d}s}\right)>0$ (3.2)

and function

$h\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+v}{\sum }\ln h_k}$, $g\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln g_k}$ (3.3)

are bounded on $t\in R_{+}$ and $\mu \in \left[0,\omega \right)$ for all $t\in R_{+},k=1,2,\cdots$.

Proof. Firstly, we prove that there is a constant $M_1>0$ such that $\underset{t\to \infty }{\lim \sup }{x}_i\left(t\right)\le M_1\left(i=1,2\right)$. Define the function $V\left(t\right)=\max \left\{x_{\text{1}}\left(t\right),x_2\left(t\right)\right\}$, we have two cases when $t\ne t_k$.

1) If $V\left(t\right)=x_1\left(t\right)$, we have

$\begin{array}{c}{D}^{+}V\left(t\right)=x_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)-c_1\left(t\right){\displaystyle {\int }_{-\infty }^0{k}_1\left(s\right)y\left(t+s\right)\text{d}s}\right)\\ +D_{12}\left(t\right)\left(x_2\left(t\right)-x_1\left(t\right)\right)\\ \le x_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)\right)\\ \le V\left(t\right)\left(a\left(t\right)-\bar{b}\left(t\right)V\left(t\right)\right)\end{array}$ (3.4)

2) If $V\left(t\right)=x_2\left(t\right)$, we have

$\begin{array}{c}{D}^{+}V\left(t\right)=x_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)x_2\left(t\right)\right)+D_{21}\left(t\right)\left(x_1\left(t\right)-x_2\left(t\right)\right)\\ \le x_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)x_2\left(t\right)\right)\\ \le V\left(t\right)\left(a\left(t\right)-\bar{b}\left(t\right)V\left(t\right)\right)\end{array}$ (3.5)

clearly, from (3.4) and (3.5), we get

$D^{+}V\left(t\right)\le V\left(t\right)\left(a\left(t\right)-\bar{b}\left(t\right)V\left(t\right)\right)$ for all $t\in R_{+}$

On the other hand, when $t=t_k$, we have

$V\left(t_k^{+}\right)=\underset{i=\left\{1,2\right\}}{\max }\left\{x_i\left(t_k^{+}\right)\right\}=\underset{i=\left\{1,2\right\}}{\max }\left\{h_{ik}{x}_i\left(t_k\right)\right\}\le \underset{i=\left\{1,2\right\}}{\max }\left\{h_{ik}\right\}\underset{i=\left\{1,2\right\}}{\max }\left\{x_i\left(t_k\right)\right\}=h_{k}V\left(t_k\right)$.

Consider the following auxiliary equation

$\{\begin{array}{l}{D}^{+}v\left(t\right)=v\left(t\right)\left(a\left(t\right)-\bar{b}\left(t\right)v\left(t\right)\right),t\ne t_k\\ v\left(t_k^{+}\right)=h_{k}v\left(t_k\right),k=1,2,\cdots \end{array}$ (3.6)

with initial condition $v\left(0\right)=V\left(0\right)$. Since the condition of Lemma 2.1 holds from (3.1) and (3.3), using Lemma 2.1, we get that there exists a constant $M_1>0$ such that $\underset{t\to \infty }{\lim \sup }v\left(t\right)\le M_1$. Applying the comparison theorem of impulsive differential equation, we obtain $V\left(t\right)\le v\left(t\right)$ for all $t\in R_{+}$. Finally, we have

$\underset{t\to \infty }{\lim \sup }{x}_i\left(t\right)\le \underset{t\to \infty }{\lim \sup }V\left(t\right)\le \underset{t\to \infty }{\lim \sup }v\left(t\right)\le M_1$ for $i=1,2$ (3.7)

Then we prove that there is a constant $M_2>0$ such that $\underset{t\to \infty }{\lim \sup }y\left(t\right)\le M_2$. From the third and sixth equations of system (1.1), we obtain

$\{\begin{array}{l}{y}^{\prime }\left(t\right)=y\left(t\right)\left(a_3\left(t\right)-b_3\left(t\right)y\left(t\right)-c_2\left(t\right){\displaystyle {\int }_{-\infty }^0{k}_2\left(s\right)x_1\left(t-s\right)\text{d}s}\right)\\ \le y\left(t\right)\left(a_3\left(t\right)-b_3\left(t\right)y\left(t\right)\right),t\ne t_k\\ y\left(t_k^{+}\right)=g_{k}y\left(t_k\right),k=1,2,\cdots \end{array}$

considering the following subsystem

$\{\begin{array}{l}{u}^{\prime }\left(t\right)=u\left(t\right)\left(a_3\left(t\right)-b_3\left(t\right)u\left(t\right)\right),t\ne t_k\\ u\left(t_k^{+}\right)=g_{k}u\left(t\right),k=1,2,\cdots \end{array}$ (3.8)

with initial condition $u\left(0\right)=y\left(0\right)$. Obviously, the condition of Lemma 2.1 holds from (3.2) and (3.3), we obtain that there is a constant $M_2>0$ such that $\underset{t\to \infty }{\lim \sup }u\left(t\right)\le M_2$. Similar to the prove process of the bounded of species x, we have $\underset{t\to \infty }{\lim \sup }y\left(t\right)\le M_2$.

Let $M=\max \left\{M_1,M_2\right\}$, evidently, we have

$\underset{t\to \infty }{\lim \sup }{x}_i\left(t\right)\le M\left(i=1,2\right)$, $\underset{t\to \infty }{\lim \sup }y\left(t\right)\le M$

The proof of Theorem 3.1 is completed.

Theorem 3.2. Assume that all conditions of Theorem 3.1 are satisfied. In addition, there is a constant $\bar{\omega }>0$ such that

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\bar{\omega }}\bar{a}\left(s\right)\text{d}s+{\displaystyle \underset{t\le t_k<t+\bar{\omega }}{\sum }\ln {\bar{h}}_k}}\right)>0$ (3.9)

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\bar{\omega }}\left(a_3\left(t\right)-c_2\left(t\right){\displaystyle {\int }_{-\tau }^0{k}_2\left(s\right)}\left(x_1^{*}\left(t+s\right)\right)\text{d}s\right)\text{d}s+{\displaystyle \underset{t\le t_k<t+\bar{\omega }}{\sum }\ln g_k}}\right)>0$ (3.10)

$\underset{t\to \infty }{\lim \inf }\left({\displaystyle {\int }_t^{t+\bar{\omega }}{\beta }_2\left(t\right)\text{d}t}\right)>0$ (3.11)

and function

$\bar{h}\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln {\bar{h}}_k}$, $g\left(t,\mu \right)={\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln g_k}$ (3.12)

are bounded on $t\in R_{+}$ and $\mu \in \left[0,\bar{\omega }\right)$, where ${\beta }_2\left(t\right)=\underset{i\text{=}1,2}{\min }\left\{a_i\left(t\right)-{\displaystyle \underset{j=1}{\overset{2}{\sum }}\frac{D_{ij}\left(t\right)}{p}}\right\}\ge 0\left(i\ne j\right)$ for all $t\in R_{+}$.

Then the system is permanent.

Proof. Above all, we must prove that there exists a constant $m>0$ such that $\underset{t\to \infty }{\lim \inf }{x}_i\left(t\right)\ge m\left(i=1,2\right)$, $\underset{t\to \infty }{\lim \inf }y\left(t\right)\ge m$.

Firstly, we prove $\underset{t\to \infty }{\lim \inf }{x}_i\left(t\right)\ge m\left(i=1,2\right)$. Defined $V\left(t\right)=\min \left\{x_1\left(t\right),x_2\left(t\right)\right\}$ and

$H=\sup \left\{\left|x_i\left(t+s\right),y\left(t+s\right)\right|:t\in R_{+},s\in R_{-},i=1,2\right\}$ (3.13)

When $t\ne t_k$, consider the following two cases.

Case 1: If $V\left(t\right)=x_1\left(t\right)$, we can choose a constant $\tau >0$ such that $H{\displaystyle {\int }_{-\infty }^{-\tau }{k}_1\left(s\right)\text{d}s<M}$, then we have

$\begin{array}{c}{V}^{\prime }\left(t\right)=x_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)-c_1\left(t\right){\displaystyle {\int }_{-\infty }^0{k}_1\left(s\right)y\left(t+s\right)\text{d}s}\right)\\ +D_{12}\left(t\right)\left(x_2\left(t\right)-x_1\left(t\right)\right)\\ =x_1\left(t\right)(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)-c_1\left(t\right)({\displaystyle {\int }_{-\infty }^{-\tau }{k}_1\left(s\right)y\left(t+s\right)\text{d}s}\\ +{\displaystyle {\int }_{-\tau }^0{k}_1\left(s\right)y\left(t+s\right)\text{d}s}))+D_{12}\left(t\right)\left(x_2\left(t\right)-x_1(t)\right)\end{array}$

$\begin{array}{c}\ge x_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)-c_1\left(t\right)\left(H{\displaystyle {\int }_{-\infty }^{-\tau }{k}_1\left(s\right)\text{d}s}+{\displaystyle {\int }_{-\tau }^0{k}_1\left(s\right)y\left(t+s\right)\text{d}s}\right)\right)\\ \ge x_1\left(t\right)\left(a_1\left(t\right)-2c_1\left(t\right)M-b_1\left(t\right)x_1\left(t\right)\right)\\ \ge V\left(t\right)\left(\bar{a}\left(t\right)-2c_1\left(t\right)M-b\left(t\right)V\left(t\right)\right),\end{array}$

Case 2: If $V\left(t\right)=x_2\left(t\right)$, we have

$\begin{array}{c}{V}^{\prime }\left(t\right)=x_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)x_2\left(t\right)\right)+D_{21}\left(t\right)\left(x_1\left(t\right)-x_2\left(t\right)\right)\\ \ge x_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)x_2\left(t\right)\right)\\ \ge V\left(t\right)\left(\bar{a}\left(t\right)-b\left(t\right)V\left(t\right)\right).\end{array}$

By Case 1 and Case 2, we derive $V\left(t\right)\ge V\left(t\right)\left(\bar{a}\left(t\right)-b\left(t\right)V\left(t\right)\right)$.

When $t=t_k$, we can obtain

$V\left(t_k^{+}\right)=\underset{i=\left\{1,2\right\}}{\min }\left\{x_i\left(t_k^{+}\right)\right\}=\underset{i=\left\{1,2\right\}}{\min }\left\{h_{ik}{x}_i\left(t_k\right)\right\}\ge \underset{i=\left\{1,2\right\}}{\min }\left\{h_{ik}\right\}\underset{i=\left\{1,2\right\}}{\min }\left\{x_i\left(t_k\right)\right\}={\bar{h}}_{k}V\left(t_k\right)$.

Research the following equation with impulsive

$\{\begin{array}{l}{v}^{\prime }\left(t\right)=v\left(t\right)\left(\bar{a}\left(t\right)-b\left(t\right)v\left(t\right)\right),t\ne t_k\\ v\left(t_k^{+}\right)={\bar{h}}_{k}v\left(t\right),k=1,2,\cdots \end{array}$ (3.14)

By condition (3.9) and (3.12), we know that condition of Lemma 2.1 is satisfied. Consequently, there exists a constant $m_1>0$ such that

$\underset{t\to \infty }{\lim \inf }{x}_i\left(t\right)\ge \underset{t\to \infty }{\lim \inf }V\left(t\right)\ge \underset{t\to \infty }{\lim \inf }v\left(t\right)\ge m_1$ for $i=1,2$. (3.15)

Then we investigate the following system:

$\{\begin{array}{l}\begin{array}{l}{u^{\prime }}_1\left(t\right)=u_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)u_1\left(t\right)\right)+D_{12}\left(t\right)\left(u_2\left(t\right)-u_1\left(t\right)\right)\\ {u^{\prime }}_2\left(t\right)=u_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)u_2\left(t\right)\right)+D_{21}\left(t\right)\left(u_1\left(t\right)-u_2\left(t\right)\right)\end{array}\},t\ne t_k\\ \begin{array}{l}{u}_1\left(t_k^{+}\right)=h_{1k}{u}_1\left(t_k\right)\\ u_2\left(t_k^{+}\right)=h_{2k}{u}_2\left(t_k\right)\end{array}\},k=1,2,\cdots ,\end{array}$ (3.16)

From Lemma 2.2 and condition (3.1), (3.10), (3.12) and (3.13), we can know that there are positive constants p and P such that

$p\le x_i^{*}\left(t\right)\le P$

where $x_i^{*}\left(t\right)\left(i=1,2\right)$ is globally attractive for the system (3.16). In addition, we assume that $\left(u_1\left(t\right),u_2\left(t\right)\right)$ is a positive solution of system (3.16) with initial condition $u_i\left(0\right)=x_i\left(0\right)$. Evidently, we obtain that there exists a constant ${\epsilon }_0>0$ small enough such that

$x_i^{*}\left(t\right)-{\epsilon }_0\le u_i\left(t\right)\le x_i^{*}\left(t\right)+{\epsilon }_0$ (3.17)

Similar to the discussion in [18] , we obtain that condition (3.10) is independent of the choice of $x_i^{*}\left(t\right)$.

From condition (3.10), there are constant ${\epsilon }_0>0$ small enough and $T>0$ large enough such that

$\begin{array}{l}{\displaystyle {\int }_t^{t+\bar{\omega }}\left(a_3\left(t\right)-b_3\left(t\right){\epsilon }_0-2c_2\left(t\right){\epsilon }_0-c_2\left(t\right){\displaystyle {\int }_{-\tau }^0{k}_2\left(s\right)}\left(x_1^{*}\left(t+s\right)\right)\text{d}s\right)\text{d}s}\\ +{\displaystyle \underset{t\le t_k<t+\bar{\omega }}{\sum }\ln g_k}>{\epsilon }_0\end{array}$ (3.18)

for all $t\ge T$. By (3.12), we can get a positive constant G such that

$\left|g\left(t,\mu \right)\right|=\left|{\displaystyle \underset{t\le t_k<t+\mu }{\sum }\ln g_k}\right|\le G$. (3.19)

From system (1.1), we consider the following subsystem,

$\{\begin{array}{l}\begin{array}{l}{x^{\prime }}_1\left(t\right)=x_1\left(t\right)\left(a_1\left(t\right)-b_1\left(t\right)x_1\left(t\right)-c_1\left(t\right){\displaystyle {\int }_{-\infty }^0{k}_1\left(s\right)y\left(t+s\right)\text{d}s}\right)\\ +D_{12}\left(t\right)\left(x_2\left(t\right)-x_1\left(t\right)\right)\\ {x^{\prime }}_2\left(t\right)=x_2\left(t\right)\left(a_2\left(t\right)-b_2\left(t\right)x_2\left(t\right)\right)+D_{21}\left(t\right)\left(x_1\left(t\right)-x_2\left(t\right)\right)\end{array}\},t\ne t_k\\ \begin{array}{l}{x}_1\left(t_k^{+}\right)=h_{1k}{x}_1\left(t_k\right)\\ x_2\left(t_k^{+}\right)=h_{2k}{x}_2\left(t_k\right)\end{array}\},k=1,2,\cdots .\end{array}$