\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 281, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/281\hfil Competitive population system]
{Dynamics of a competitive population system with impulsive reduction
 of  the invasive population}

\author[J. Jiao, S. Cai, W. Liu, L. Li \hfil EJDE-2017/281\hfilneg]
{Jianjun Jiao, Shaohong Cai, Wenjiang Liu, Limei Li}

\address{Jianjun Jiao (corresponding author) \newline
School of  Mathematics and  Statistics,
Guizhou University of Finance and Economics,
Guiyang 550004, China. \newline
College of  Mathematics and Systems Science,
Shandong University of Science and Technology,
Qingdao 266590,  China}
\email{jiaojianjun05@126.com}

\address{Shaohong Cai \newline
School of  Mathematics and  Statistics,
Guizhou University of Finance and Economics,
Guiyang 550004, China}
\email{caishaohong2014@126.com}

\address{Wenjiang Liu \newline
School of  Mathematics and  Statistics,
Guizhou University of Finance and Economics,
Guiyang 550004, China}
\email{1499353344@qq.com}

\address{Limei Li \newline
School of  Continuous Education,
Guizhou University of Finance and Economics,
Guiyang 550004,  China}
\email{lilimei05@126.com}

\dedicatory{Communicated by Goong Chen}

\thanks{Submitted July 16, 2017. Published November 10, 2017.}
\subjclass[2010]{34D23, 92B05}
\keywords{Competitive population system; impulsive invasion; extinction;
\hfill\break\indent impulsive reduction;  permanence}

\begin{abstract}
 Biological invasion refers to the phenomenon that some organisms have been
 accidentally or artificially introduced into the wild. The  invasive
 populations compete with the local population,and cause  damage to the local
 ecosystem. To protect the local ecosystem, the invasive populations should
 be artificially reduced. Such processes are seldom studied in dynamical models.
 In this work, we consider  a competitive population system  with
 impulsive reduction of the  invasive population.
 All solutions of  the investigated system are proved to be ultimately uniformly
 bounded. Sufficient conditions are obtained to guarantee the linear stability
 of the population $x(t)$-extinction periodic solution.
 This signifies that the alien species invade successfully, and cause  the
 extinction of  the native species. The permanency of the conditions is  also
 obtained, which shows that the alien species invade successfully, and they
 coexist with the native species.
 Numerical simulations are included to illustrate our results.
 Through such computation, we find that there exists a threshold of
 unsuccessful invasion, indicating  that   the native species cause  the
 extinction of the alien species. These results offer insights  that impulsive
 invasion plays an important role in the dynamics of ecosystem, and provide
 some reliable tactical analysis for  biological resources protection.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

 Biological invasion refers to the phenomenon that some organisms have been
accidentally or artificially introduced into the wild. The  invasive  
populations compete with the local populations, and cause  damage to the 
local ecosystem.  The impacts of biological invasions now rank among the most
pervasive threats to native ecosystems and human economies \cite{c1,s1}.
 Invasion by alien organisms is a common worldwide phenomenon.
Such  invasion by alien species is especially likely to occur on oceanic islands
\cite{s3}.
Biological invasions are rapidly producing planet-wide changes in
biodiversity and ecosystem function.  Experiments on the competitions  
are usually employed to investigate biological invasions \cite{e1}.
Biological invasions are seldom studied  in dynamical models \cite{z2}.

The theory of  differential equations  is used in mathematical ecology, 
and the  predator-prey,   competitive and cooperative
models have been studied by many authors \cite{c1,g1,j1,j2,j3,j4,s2,z1}. 
One of the famous models for population dynamics  is the Lotka-Volterra 
competitive system, as competition is an important biotic process that affects
the population dynamics of ecosystems. Many authors  \cite{b1,h1} have investigated 
the  population dynamics by the  theory of impulsive differential equations. 
Almost all domains of applied science \cite{j5,m1} have found the occurrence of  
impulsive phenomena.
  Liu and Chen \cite{l2} developed the Holling type II
Lotka-Volterra predator-prey system model, which may inherently oscillate,
by introducing periodic constant impulsive immigration of a predator.
Their results showed  that the dynamics of such a system is dependent on
the impulsive immigration amount of the predator.  Meng and Chen
\cite{m1} formulated a robust impulsive Lotka-Volterra n-species
competitive system with both discrete delays and continuous delays.
Theirs results indicated that under the appropriate linear bounded
impulsive perturbations, the impulsive delay Lotka-Volterra system
maintains  the original permanence and globally asymptotical stability
of the nonimpulsive delay Lotka-Volterra system. Jiao et al.~\cite{j6}
suggested a five-dimensional chemostat model with
impulsive diffusion and pulse input environmental toxicant. 
The results revealed that impulsive diffusion plays an important role 
on the outcome of the
chemostat. Jiao et al.~\cite{j7}  investigated the dynamics of a chemostat
model with impulsive input and effect of delayed response in growth. 
Their results indicated that the discrete time delay has an influence on 
the dynamical behaviors of the investigated
system, and provided a tactical basis for the experimenters to control 
the outcome of the chemostat.
Even though there already is plenty of work using impulsive differential 
equations to study predator-prey, chemostat and invasive population,
few papers can be  found to  combine impulsive  dynamical systems with 
biological invasions.

The organization of this paper is as follows. In the following section,
we introduce the model and background. In Section 3, some
important lemmas are presented. In Section 4, we give the linear stability  
conditions of  population $x(t)$-extinction periodic
solution of system \eqref{e2.1},  and the permanency condition of  system
\eqref{e2.1}. In Section 5,  a  brief discussion
is given to conclude this work.

 \section{The model}

 In this work,  we consider a competitive population system  with
 impulsive reduction of  the invasive population
\[
\left.\begin{gathered}
\frac{dx(t)}{dt}=x(t)(a_1-b_1x(t))-k_1\beta x(t)y(t), \\
\frac{dy(t)}{dt}=-d_1y(t)-k_2\beta x(t)y(t),\\
\frac{dz(t)}{dt}=z(t)(a_2-b_2z(t)),
\end{gathered}
\right\}  \quad t\neq (n+l)\tau,\; t\neq (n+1)\tau,
\]
\begin{equation} \label{e2.1}
\left. \begin{gathered}
\Delta x(t)=0,\\
\Delta y(t)=\epsilon  d  z(t),\\
\Delta z(t)=-d z(t) ,
\end{gathered}
\right\} \quad t= (n+l)\tau,\; n= 1,2\dots,
\end{equation}
\[
\left.\begin{gathered}
\Delta x(t)=-p_1x(t),\\
\Delta y(t)=-p_2 y(t),\\
\Delta z(t)=0 ,
\end{gathered}
\right\}\quad  t= (n+1)\tau,\; n= 1,2\dots, 
 \]
where it is  assumed  that the above system 
 is composed of two patches connected by impulsive  invasion.
Populations  $x(t)$ and $y(t)$ inhabit in patch $1$, and they have 
a competitive relation  in  patch $1$.
 Population $z(t)$ inhabits in patch $2$.
  In patch $1$, the  intrinsic rate of natural increase and the density
dependence rate of  population $x(t)$ are denoted by $a_1, b_1$
respectively, and  $\frac{a_1}{ b_1}$ denotes the carrying capacity of  
population $x(t)$. Population $x(t)$ and population $y(t)$ are in  
competitive relation, where $\beta$ represents the competitive coefficient.
Constants $k_1$ and $k_2$ are competitive effects of population $x(t)$
and population $y(t)$,  respectively. Constant $d_1$ represents the death 
coefficient of population $y(t)$.
  In patch $2$, the intrinsic rate of natural increase and density
dependence rate of  population $z(t)$ are denoted by $a_2, b_2$
respectively, and  $\frac{a_2}{ b_2}$ denotes the carrying capacity
of  population $z(t)$. Impulsive invasion occurs every
$\tau $ period ($\tau$ is a positive constant). The system
evolves from its initial state without being further affected by
invasion until the next pulse appears.  We define the notation  
$\Delta y((n+l)\tau) =
y((n+l)\tau^{+})-y((n+l)\tau)(0<l<1)$, where $y((n+l)\tau^{+})$ represents
the density of population $y(t)$ in the first patch immediately after the
$(n+l)$th invasion pulse at time $ t = (n+l)\tau$, while $y((n+l)\tau)$
represents the density of population $y(t)$ in the first patch before the
$(n+l)$th invasion pulse at time $t = (n+l)\tau, n\in Z_{+}$.
 Constant $d(0 < d < 1)$ is the  impulsive invasion coefficient,  and
$1-\epsilon \hspace{0.2cm}  (0<\epsilon \leq 1)$ is the loss rate of population 
$z(t)$ in the invasion process.
The constant $p_1\hspace{0.2cm}(0 \leq p_1\leq 1 )$ represents the
reduction effect of population $x(t)$ accompanying with impulsive reduction of 
population $y(t)$ at $t=(n+1)\tau,    n\in Z_{+}$.
while the constant $p_2\hspace{0.2cm}(0 \leq p_2\leq 1) $
represents the impulsive reduction effect of population $y(t)$
at $t=(n+1)\tau, n\in Z_{+}$.


\section{Auxiliary lemmas}

 Before discussing the main results, we will give some
definitions, notation and lemmas.   Denote $f=(f_1,f_2,f_3)$
the map defined by the right hand of system \eqref{e2.1}.   The solution
of  system \eqref{e2.1}, denoted  by $Z(t)=(x(t),y(t),
z(t))^{T}$, is a
 piecewise continuous function $Z:R_{+}\to R_{+}^{3}$,
 where
$R_{+}=[0,\infty),R_{+}^{3}=\{Z\in R^{3}:Z>0\}$.
 $Z(t)$ is continuous on $(n\tau,(n+l)\tau]\times R^{3}_{+} $  and 
$((n+l)\tau, (n+1)\tau]\times R^{3}_{+}$.
 According to Reference  \cite{b1}, the global existence and uniqueness of 
solutions of  System
 \eqref{e2.1} is guaranteed by the smoothness properties of $f$, which
 denotes the mapping defined by the right-side of system \eqref{e2.1}.

Let $V:R_{+}\times R_{+}^{3} \to R_{+}$. Then $V$ is said to belong to class 
$V_0$, if
\begin{itemize}
\item[(i)]  $V$ is continuous on $(n\tau, (n+l)\tau]\times R^{3}_{+}$ and on
 $((n+l)\tau, (n+1)\tau]\times R^{3}_{+}$  for each $Z\in R^{3}_{+},n \in Z_{+}$.  
$\lim_{(t, Y)\to ((n+l)\tau^{+},Z)}V(t,Y)=V((n+l)\tau^{+},  Z)$ and  
$\lim_{(t, Y)\to ((n+1)\tau^{+},Z)} V(t,Y)=V((n+1)\tau^{+},  Z)$  exist.

\item[(ii)] $V$ is locally Lipschitzian in $Z$.
\end{itemize}

\begin{definition} \label{def3.1} \rm
Let $V \in V_0 $, then for $(t, Z) \in (n \tau,
(n+1)\tau]\times R^{3}_{+}$, the upper right derivative of $V(t,Z)$
 with respect to the impulsive differential system \eqref{e2.1} is defined as
$$ 
D^{+}V(t,Z)= \lim _{h \to  0} \sup \frac{1}{h}[V(t+h, Z+hf(t,Z))-V(t,Z))].
$$
\end{definition}

 \begin{lemma} \label{lem3.2} 
There exists a constant $M>0$ such that $x(t)\leq M$, $y(t)\leq M$  and 
$z(t)\leq M$ for each solution $(x(t),y(t),z(t) )$ of  system \eqref{e2.1} with 
$t$ large  enough.
\end{lemma}

\begin{proof} 
Define $V(t)=x(t)+y(t)+z(t)$. When $t\neq n\tau$ and $t\neq (n+l)\tau$, we have
\begin{align*}
&D^{+}V(t)+d_1V(t) \\
&=(a_1+d_1 )x(t)-b_1 x^{2}(t)-k_1\beta x(t)y(t)-k_2\beta x(t)y(t)
 +(a_2+d_1)z(t)-b_2z^{2}(t) \\
&\leq (a_1+d_1 )x(t)-b_1 x^{2}(t)+(a_2+d_1)z(t)-b_2z^{2}(t) \\
&=-b_1[x(t)-\frac{a_1+d_1}{2b_1}]^{2}+\frac{(a_1+d_1)^{2}}{4b_1}
 -b_2[z(t)-\frac{a_2+d_1}{2b_2}]^{2}+\frac{(a_2+d_1)^{2}}{4b_2} \\
&\leq \frac{(a_1+d_1)^{2}}{4b_1}+\frac{(a_2+d_1)^{2}}{4b_2}=:\xi.
\end{align*}
When $t=(n+l)\tau$, we  have
\begin{align*}
V((n+l)\tau^{+})
&= x((n+l)\tau)+y((n+l)\tau)+z((n+l)\tau)-(1-\epsilon)d z((n+l)\tau)\tau)\\
&=V((n+l)\tau)-(1-\epsilon)d z((n+l)\tau)\tau) \\
&\leq V((n+l)\tau).
\end{align*}
When $t=(n+1)\tau$, we  have
\begin{align*}
V((n+1)\tau^{+})
&=x((n+1)\tau)-p_1x((n+1)\tau)+y((n+1)\tau) \\
&\quad -p_2y((n+1)\tau)+z((n+1)\tau)\\
&=V(n\tau)-p_1x((n+1)\tau)-p_2y((n+1)\tau) \\
&\leq V((n+1)\tau).
\end{align*}
From \cite{l1}, for $t\in ((n\tau,(n+1)\tau]$, we have
$$
V(t) \leq V(0^{+})e^{-d_1t}+\frac{\xi}{d_1}(1-e^{-d_1t}) \to \frac{\xi}{d_1},
\quad\text{as } t\to \infty.
$$
So $V(t)$ is ultimately uniformly bounded. Hence, by the definition
of $V(t)$,   there exists a constant $M>0$ such that  $x(t)\leq M,y(t)\leq M$  
and $z(t)\leq M$ for $t$ large enough. The proof is complete.
\end{proof}

 If $x(t)=0$, we have the following subsystem of  system \eqref{e2.1}:
\begin{equation}
\begin{gathered}
\left. \begin{gathered}
\frac{dy(t)}{dt}=-d_1y(t),\\
\frac{dz(t)}{dt}=z(t)(a_2-b_2z(t)),
\end{gathered}
\right\}\quad  t\neq n\tau,   \\
\left.\begin{gathered}
\Delta y(t)= \epsilon d z(t) ,\\
\Delta z(t)=-d z(t),
\end{gathered}
\right\} \quad t= n\tau,\\
\left.
\begin{gathered}
\Delta y(t)= -p_2 y(t) ,\\
\Delta z(t)=0,
\end{gathered}
\right\} \quad t= n\tau.
\end{gathered}
\label{e3.1}
\end{equation}
The analytic solution of system \eqref{e3.1} on $(n\tau,(n+l)\tau]$
 is obtained as follows:
\begin{equation} \label{e3.2}
\begin{gathered}
y(t)=y(n\tau^{+})e^{-d_1(t-n\tau)},\quad t\in(n\tau, (n+l)\tau],\\
z(t)=\frac{a_2e^{a_2(t-n\tau)}z(n\tau^{+})}{a_2+b_2[e^{a_2(t-n\tau)}-1]z(n\tau^{+})},
\quad t\in(n\tau, (n+l)\tau].
\end{gathered}
\end{equation}
Considering the third and fourth equations of system \eqref{e3.1}, we
have
\begin{equation} \label{e3.3}
\begin{gathered}
y((n+l)\tau^{+})=y(n\tau^{+})e^{-d_1l\tau}+\epsilon d  
\frac{a_2e^{a_2l\tau}z(n\tau^{+})}{a_2+b_2(e^{a_2l\tau}-1)z(n\tau^{+})},\\
z((n+l)\tau^{+})=(1- d)  \frac{a_2e^{a_2l\tau}z(n\tau^{+})}{a_2+b_2(e^{a_2l\tau}-1)
 z(n\tau^{+})}.
\end{gathered}
\end{equation}
The analytic solution of system \eqref{e3.1} on $((n+l)\tau,(n+1)\tau]$ 
are 
\begin{equation} \label{e3.4}
\begin{gathered}
y(t)=y((n+l)\tau^{+})e^{-d_1(t-(n+l)\tau)},\quad t\in((n+l)\tau, (n+1)\tau],\\
z(t)=\frac{a_2e^{a_2(t-(n+l)\tau)}z((n+l)\tau^{+})}{a_2+b_2[e^{a_2(t-(n+l)\tau)}-1]
z((n+l)\tau^{+})},\quad t\in((n+l)\tau, (n+1)\tau].
\end{gathered}
\end{equation}
Considering the fifth and sixth equations of system \eqref{e3.1}, we
have
\begin{equation} \label{e3.5}
\begin{gathered}
y((n+1)\tau^{+})=(1-p_2)y((n+1)\tau^{+}),\\
z((n+1)\tau^{+})=z((n+1)\tau^{+}).\\
\end{gathered}
\end{equation}
Then, we obtain the stroboscopic map of system \eqref{e3.1},
\begin{equation} \label{e3.6}
\begin{gathered}
y((n+1)\tau^{+})=(1-p_2)e^{-d_1\tau}y(n\tau^{+})
 +\frac{(1-p_2)\epsilon d a_2e^{[a_2l-d_1(1-l)]\tau}z(n\tau^{+})}{a_2+b_2(e^{a_2\tau}
-1)z(n\tau^{+})},\\
z((n+1)\tau^{+})=\frac{(1-d)a_2e^{a_2\tau}z(n\tau^{+})}{a_2+b_2(e^{a_2\tau}-1)
z(n\tau^{+})}.
\end{gathered} 
\end{equation}
Making notation as $A=(1-p_2)e^{-d_1\tau}$, 
$B_1=(1-p_2)\epsilon d a_2e^{[a_2l-d_1(1-l)]\tau}$, 
$C_1=e^{a_2\tau}-1$,
$B_2=(1-d)a_2e^{a_2\tau}$, $C_2=e^{a_2\tau}-1$, we can rewrite \eqref{e3.6} as
\begin{equation} \label{e3.7}
\begin{gathered}
y((n+1)\tau^{+})=Ay(n\tau^{+})+\frac{B_1z(n\tau^{+})}{a_2+b_2C_1z(n\tau^{+})},\\
z((n+1)\tau^{+})=\frac{B_2z(n\tau^{+})}{a_2+b_2C_2z(n\tau^{+})}.
\end{gathered}
\end{equation}
 There are two fixed points of \eqref{e3.7} are obtained  as
$G_1(0,0)$ and $G_2(y^{\ast},z^{\ast})$, where
\begin{equation} \label{e3.8}
\begin{gathered}
y^{\ast}=\frac{B_1(B_2-a_2)}{b_2[a_2C_2+(B_2-a_2)C_1]}, \quad  B_2>a_2,\\
z^{\ast}=\frac{B_2-a_2}{b_2C_2}, \quad B_2>a_2. \\
\end{gathered}
\end{equation}

\begin{lemma} \label{lem3.3} 
\begin{itemize}
\item[((i)]  If $(1-d)e^{a_2\tau}<1$, the fixed
point $G_1(0,0)$ of \eqref{e3.7} is globally asymptotically stable;


\item[(ii)]   If $(1-d)e^{a_2\tau}>1$, the fixed point
$G_2(y^{\ast}, z^{\ast})$ of \eqref{e3.7} is globally asymptotically
stable.
\end{itemize}
\end{lemma}

\begin{proof} For convenience, we make a notation as
$(y^{n},z^{n})=(y(n\tau^{+}),z(n\tau^{+}))$. The
linear form of \eqref{e3.7} can be written as
\begin{equation} \label{e3.9}
\begin{pmatrix}
y^{n+1} \\
z^{n+1}
\end{pmatrix}
= M \begin{pmatrix}
y^{n} \\
z^{n}
\end{pmatrix}.
\end{equation}
Obviously,  the local dynamics of $G_1(0,0)$ and
$G_2(y^{\ast}, z^{\ast})$  are determined by linear
system \eqref{e3.9}.   The stabilities  of $G_1(0,0)$ and
$G_2(y^{\ast}, z^{\ast})$  are determined by the
eigenvalue of $M$ less than $1$. If $M$ satisfies the  \emph{Jury
criteria}  \cite{j1}, we can know that the eigenvalue of $M$ is  less than $1$. That is
\begin{equation} \label{e3.10}
1- \operatorname{tr} M+ \det M>0.
\end{equation}

(i)  If $(1-d)e^{a_2\tau}<1$, namely, $B_2<a_2$, $G_1(0,0)$ is the unique
fixed point of System \eqref{e3.7}, we have
\begin{equation}
 M=\begin{pmatrix}
A& \frac{B_1}{a_2}\\
0&\frac{B_2}{a_2}
\end{pmatrix}.
\end{equation}
Obviously, $A<1$, calculations give
\begin{align*}
 1- \operatorname{tr}M+ \det M
&=1-(A+\frac{B_2}{a_2})+A \frac{B_2}{a_2} \\
&=(1-A)(1-\frac{B_2}{a_2})>0.
\end{align*}
From Jury  criteria,  $G_1(0,0)$ is locally stable, then  it is
globally asymptotically stable.

(ii) If $(1-d)e^{a_2\tau}>1$, namely,  $B_2>a_2$, $G_1(0,0)$ is unstable,
 and $ G_2(y^{\ast},z^{\ast})$ exists,  and
 \begin{equation}
M=\begin{pmatrix}
A & \frac{a_2b_2B_1C_2}{ a_2}\\
0&  \frac{a_2}{B_2}
\end{pmatrix}.
\end{equation}
For
\begin{align*}
1- \operatorname{tr}M+ \det M
&=1-(A+\frac{a_2}{B_2})+A \times \frac{a_2}{B_2} \\
=(1-A)(1-\frac{a_2}{B_2})>0.
\end{align*}
From Jury criteria,  $G_2(y^{\ast}, z^{\ast})$
 is locally stable,  then it is globally
asymptotically stable. This
 completes the proof.
\end{proof}

The following lemma can  be proved easily, so we omit its proof.

\begin{lemma} \label{lem3.4}  
\begin{itemize}
\item[(i)] If $(1-d)e^{a_2\tau}<1$, the trivial
periodic solution $(0,0)$ of System \eqref{e3.1} is globally
asymptotically stable;


\item[(ii)] If $(1-d)e^{a_2\tau}>1$, the periodic solution
$(\tilde{y}(t),\tilde{z}(t) )$ of System \eqref{e3.1} is
globally asymptotically stable, where
$(\widetilde{y_1(t)},\widetilde{y_2(t)})$ can be expressed as
\begin{equation} \label{e3.13}
\begin{gathered}
\tilde{y}(t)= \begin{cases}
y^{\ast}e^{-d_1(t-n\tau)}, & t\in(n\tau,(n+l)\tau],\\
y^{\ast\ast}e^{-d_1(t-(n+l)\tau)}, & t\in((n+l)\tau, (n+1)\tau],
\end{cases}\\
\tilde{z}(t)=\begin{cases}
\frac{a_2z^{\ast}e^{a_2(t-(n+l)\tau)}}{a_2+b_2z^{\ast}(e^{a_2(t-(n+l)\tau)}-1)}, 
& t\in(n\tau,(n+l)\tau],\\[4pt]
\frac{a_2z^{\ast\ast}e^{a_2(t-(n+l)\tau)}}{a_2+b_2z^{\ast\ast}
(e^{a_2(t-(n+l)\tau)}-1)}, & t\in((n+l)\tau, (n+1)\tau],
\end{cases} 
\end{gathered}
\end{equation}
where $y^{\ast}$ and $z^{\ast}$ is determined in \eqref{e3.8}, and 
$y^{\ast\ast}$ and $z^{\ast\ast}$ are determined by 
\begin{equation} \label{e3.14}
\begin{gathered}
y^{\ast\ast}=y^{\ast}e^{-d_1l\tau}+\epsilon d 
 \frac{a_2e^{a_2l\tau}z^{\ast\ast}}{a_2+b_2(e^{a_2l\tau}-1)z^{\ast}},\\
z^{\ast\ast}=(1- d) \times \frac{a_2e^{a_2l\tau}z^{\ast}}{a_2+b_2(e^{a_2l\tau}-1)
z^{\ast}}.
\end{gathered}
\end{equation}
\end{itemize}
\end{lemma}

\section{The dynamics}

 In this section, we easily find that there exist trivial periodic solution 
$(0,0,0)$ and population $x(t)$-extinction boundary periodic solution 
$(0,\tilde{y}(t),\widetilde{z (t)})$ of system \eqref{e2.1}.
We will prove that  the trivial
 periodic solution $(0,0,0)$ of system \eqref{e2.1} is linear unstable, and prove
the population $x(t)$-extinction boundary periodic solution
$(0,\tilde{y}(t),\widetilde{z (t)})$ of system \eqref{e2.1}
 is linearly stable/unstable. Then,  we will prove that system \eqref{e2.1}
 is permanent.

\begin{theorem} \label{thm4.1}
\begin{itemize}

\item[(i)] If $(1-d)e^{a_2\tau}<1$, and $(1-p_1)e^{a_1\tau}<1$, then
the trivial periodic solution $(0,0,0)$ of \eqref{e2.1} is linear stable.

\item[(ii)] If $(1-d)e^{a_2\tau}>1$, or $(1-p_1)e^{a_1\tau}>1$, 
the trivial periodic solution $(0,0,0)$ of system \eqref{e2.1} is linear unstable.


\item[(iii)]  If
\begin{gather*}
\ln \frac{1}{1-p_1}>a_1\tau-\frac{k_1\beta (1-e^{-d_1\tau})}{d_1}y^{\ast}
-\frac{k_1\beta (1-e^{-d_1(1-l)\tau})}{d_1}y^{\ast\ast}, \text{ and} \\
\ln\frac{1}{1-d}>a_2\tau-2\ln[1+\frac{b_2(e^{a_2\tau}-1)}{a_2} z^{\ast}]
 -2\ln[1+\frac{b_2(e^{a_2\tau}-1)}{a_2} z^{\ast\ast}],
\end{gather*}
hold, then the population $x(t)$-extinction boundary periodic solution \\
$(0,\tilde{y}(t),\widetilde{z (t)})$ of system \eqref{e2.1} is linearly
stable.  Where $y^{\ast}$ is  defined in \eqref{e3.8}.

\item[(iv)]  If
$$\
ln \frac{1}{1-p_1}<a_1\tau-\frac{k_1\beta (1-e^{-d_1\tau})}{d_1}y^{\ast}
-\frac{k_1\beta (1-e^{-d_1(1-l)\tau})}{d_1}y^{\ast\ast},
$$
or
$$
\ln\frac{1}{1-d}<a_2\tau-2\ln[1+\frac{b_2(e^{a_2\tau}-1)}{a_2} z^{\ast}]
-2\ln[1+\frac{b_2(e^{a_2\tau}-1)}{a_2} z^{\ast\ast}],
$$
hold, then the population $x(t)$-extinction boundary periodic solution \\
$(0,\tilde{y}(t),\widetilde{z (t)})$ of system \eqref{e2.1} is linearly
unstable.
 Where $y^{\ast}$ and $z^{\ast}$  are defined in \eqref{e3.8}, and 
$y^{\ast\ast}$ and $z^{\ast\ast}$  are defined in \eqref{e3.14}.
\end{itemize}
\end{theorem}

\begin{proof}  
Define $x_1(t)=x(t), y_1(t)=y(t)-\tilde{y}(t),
z_1(t)=z(t)-\tilde{z}(t)$, we have the following
linearly similar system  of system \eqref{e2.1}
$$ 
\begin{pmatrix}
\frac{dx_1(t)}{dt} \\
\frac{dy_1(t)}{dt}\\
\frac{dz_1(t)}{dt}
\end{pmatrix}
= \begin{pmatrix}
a_1- k_1\beta \tilde{y}(t) & 0&0 \\
k_2\beta \tilde{y}(t) & -d_1 & 0  \\
0 & 0 & a_2-2b_2\tilde{z}(t) 
\end{pmatrix} 
\begin{pmatrix}
x_1(t) \\
y_1(t)\\
z_1(t)
\end{pmatrix} . 
$$ 
 It is easy to to obtain the fundamental solution matrix 
 $$
\Phi (t)=
\begin{pmatrix}
\exp(\int^{t}_0(a_1-k_1\beta \tilde{y}(s))ds) & 0&0 \\
\ast & \exp[-d_1t]&0 \\
0 & 0 &\exp(\int^{t}_0(a_2-2b_2\widetilde{z(s)})ds) \\
\end{pmatrix}. 
$$
There is no need to calculate the exact form of $\ast$ as it is  
 not required in the analysis that
follows. The linearization of the
 fourth,  fifth and sixth equations of system \eqref{e2.1} is
 $$ 
\begin{pmatrix}
x_1((n+l)\tau^{+})\\
y_1((n+l)\tau^{+})\\
z_1((n+l)\tau^{+})
\end{pmatrix}
= \begin{pmatrix}
1 & 0&0 \\
0& 1 & \epsilon d \\
0& 0&1-d \\
\end{pmatrix}
\begin{pmatrix}
x_1((n+l)\tau) \\
y_1((n+l)\tau)\\
z_1((n+l)\tau)
\end{pmatrix}. 
$$
The linearization of the
 seventh, eighth and ninth equations of system \eqref{e2.1} is
 $$ 
\begin{pmatrix}
x_1((n+1)\tau^{+})\\
y_1((n+1)\tau^{+})\\
z_1((n+1)\tau^{+})
\end{pmatrix}
= \begin{pmatrix}
1-p_1 & 0&0 \\
0& 1-p_2 & 0 \\
0& 0&1
\end{pmatrix}
\begin{pmatrix}
x_1((n+l)\tau) \\
y_1((n+l)\tau)\\
z_1((n+l)\tau)
\end{pmatrix}. 
$$
The stability of the population $x(t)$-extinction periodic solution
$(0,\tilde{y}(t),\tilde{z}(t))$ is determined by the
eigenvalues of
$$
M=\begin{pmatrix}
1 & 0&0\\
0& 1&\epsilon d\\
0& 0&1-d
\end{pmatrix}
\begin{pmatrix}
1-p_1 & 0&0\\
0& 1-p_2&0\\
0& 0&1
\end{pmatrix}
\Phi(\tau),
$$
which are
\begin{gather*}
\lambda_1=(1-p_1)e^{\int^{\tau}_0(a_1-k_1\beta \tilde{y}(s))ds},\\
\lambda_2=(1-p_2)e^{-d_1\tau}<1,\\
\lambda_3=(1-d)e^{\int^{\tau}_0(a_2-2b_2\widetilde{z(s)})ds}.
\end{gather*}

(i) For the trivial periodic solution $(0,0,0)$ of system \eqref{e2.1}.
According the first condition  of this theorem, we easily know that 
$\lambda_1=(1-p_1)e^{\int^{\tau}_0(a_1-k_1\beta \tilde{y}(s))ds}
=(1-p_1)e^{a_1\tau}<1$,
   and $\lambda_3=(1-d)e^{\int^{\tau}_0(a_2-2b_2\widetilde{z(s)})ds}
=(1-d)e^{-a_2\tau}<1$. From the Floquet theory \cite{s2},  the trivial 
periodic solution  $(0,0,0)$ is linearly  stable.

(ii) For the trivial periodic solution $(0,0,0)$ of system \eqref{e2.1}.
According the second condition  of this theorem, we easily know that 
$\lambda_1=(1-p_1)e^{\int^{\tau}_0(a_1-k_1\beta \tilde{y}(s))ds}
=(1-p_1)e^{a_1\tau}>1$,
   or $\lambda_3=(1-d)e^{\int^{\tau}_0(a_2-2b_2\widetilde{z(s)})ds}
=(1-d)e^{a_2\tau}>1$. From the Floquet theory \cite{s2},  the trivial periodic 
solution  $(0,0,0)$ is linearly  unstable.

\item[(iii)] For the boundary periodic solution 
$(0,\tilde{y}(t),\tilde{z}(t))$ of system \eqref{e2.1}.
According the third conditions of this theorem, we easily know that 
$\lambda_1=e^{\int^{\tau}_0(a_1-k_1\beta \tilde{y}(s))ds}<1$,
 and $(1-d)e^{\int^{\tau}_0(a_2-2b_2\widetilde{z(s)})ds}<1$, then 
$\lambda_1<1$, and $\lambda_3<1$. From
the Floquet theory \cite{s2},  the population $x(t)$-extinction boundary periodic 
solution
$(0,\tilde{y}(t),\tilde{z}(t))$ is linearly stable.


(iv)  Its proof is similar to (iii).
The proof is complete.
\end{proof}


The next task is to investigate the  permanence of system \eqref{e2.1}.
Before starting this  work, we  should  give the following
definition.

\begin{definition} \label{def4.2} \rm
System \eqref{e2.1} is said to be permanent if there are
constants $m,M >0 $ (independent of the initial value) and a finite
time  $T_0$ such that for all solutions  $(x(t), y(t), z(t))$
with all initial values  $x(0^{+})>0$, $y(0^{+})>0$,
$z(0^{+})>0$,  $m\leq x(t)\leq M,m\leq y(t)\leq M$,
$m\leq z(t)\leq M $ holds for all $t\geq T_0$. Here  $T_0$ may
depend on the initial values  $(x(0^{+}), y(0^{+}),z(0^{+}))$.
\end{definition}

\begin{theorem} \label{thm4.3}  
If 
\begin{gather*}
\ln \frac{1}{1-p_1}<a_1\tau-\frac{k_1\beta (1-e^{-d_1\tau})}{d_1}y^{\ast}
-\frac{k_1\beta (1-e^{-d_1(1-l)\tau})}{d_1}y^{\ast\ast}\quad\text{and}\\
\ln\frac{1}{1-d}<a_2\tau-2\ln[1+\frac{b_2(e^{a_2\tau}-1)}{a_2} z^{\ast}]
-2\ln[1+\frac{b_2(e^{a_2\tau}-1)}{a_2} z^{\ast\ast}],
\end{gather*}
hold, system \eqref{e2.1} is permanent,  where $y^{\ast}$ and $z^{\ast}$ are defined 
in \eqref{e3.8}, and $y^{\ast\ast}$ and $z^{\ast\ast}$  are defined in \eqref{e3.14}.
\end{theorem}

\begin{proof} 
 Let $(x(t), y(t), z(t))$ be  a solution of \eqref{e2.1} with
 $x(0)>0$, $y(0)>0$, $z(0)>0$. By Lemma \ref{lem3.2}, we have proved there exists a
 constant $M >0$ such that $x(t)\leq M, y_1(t)\leq M, y_2(t)\leq M$ 
for $t$ large  enough.   We  may  assume $x(t)\leq M, y(t)\leq M, z(t)\leq M$, 
for $t>0$.

Firstly, we need to find a $m_1>0$ such that $x(t)\geq m_1$ for $t$ large enough.
Otherwise, there must exist  a $m_2>0$ small enough such that $x(t)<m_2$.
 By the condition $a_1\tau>\frac{k_1\beta (1-e^{-d_1\tau})}{d}y^{\ast}$,
 we can also select an $\varepsilon>0$ small enough  such that
\begin{equation*} %\label{e4.1}
a_1\tau-b_1m_2\tau-k_1\beta \tau \varepsilon-\frac{k_1\beta y^{\ast}
(1-e^{-d_1l\tau})}{d}-\frac{k_1\beta y^{\ast\ast}(1-e^{-d_1(1-l)\tau})}{d} >0,
\end{equation*}
and use the notation
\begin{equation}
\begin{aligned}
\sigma &:= a_1\tau-b_1m_2\tau-k_1\beta \tau \varepsilon
-\frac{k_1\beta y^{\ast}(1-e^{-d_1l\tau})}{d} \\
&\quad -\frac{k_1\beta y^{\ast\ast}(1-e^{-d_1(1-l)\tau})}{d} >0.
\end{aligned}
\end{equation}
 Considering the second equation of system \eqref{e2.1}, we obtain
 $$
\frac{dy(t)}{dt}\leq -d_1y(t).
$$
 Then, we have the following comparative differential equation
\begin{equation} \label{e4.3}
\begin{gathered}
\left. \begin{gathered}
\frac{dy_2(t)}{dt}=-d_1y_2(t),\\
\frac{dz_2(t)}{dt}=z_2(t)(a_2-b_2z_2(t)),
\end{gathered}
 \right\}\quad  t\neq (n+l)\tau,\; t\neq (n+1)\tau,  \\
\left. \begin{gathered}
\Delta y_2(t)= \epsilon d z_2(t) ,\\
\Delta z_2(t)=-d z_2(t),\\
\end{gathered}
\right\}\quad t= (n+l)\tau,\\
\left.\begin{gathered}
\Delta y_2(t)= -p_2 y_2(t) ,\\
\Delta z_2(t)=0,
\end{gathered}
\right\} \quad t= (n+1)\tau.\\
\end{gathered}
\end{equation}
From Lemma \ref{lem3.4},  we know that
 $y_2(t)\leq \tilde{y}(t)+\varepsilon, z_2(t)\leq \tilde{z}(t)+\varepsilon$ 
for all $t$ large enough,
 and $\varepsilon> 0$ is small enough.   
From the comparative theorem of  impulsive differential equation  \cite{s2},
 there exists a $T$, such that for $t>T$,
 \begin{gather} \label{e4.4}
y(t)\leq y_2(t)\leq \tilde{y}(t)+\varepsilon,\\
\label{e4.5}
z(t)\leq z_2(t)\leq \tilde{z}(t)+\varepsilon.\\
\end{gather}
Substituting \eqref{e4.4} in the first equation of system \eqref{e2.1}, we obtain
\begin{equation}  \label{e4.6}
x(t)\geq x(t)(a_1-b_1x(t))-k_1\beta(\tilde{y}(t)+\varepsilon)x(t).
\end{equation}
which can be written as
\begin{equation} \label{e4.7}
x(t)\geq x(t)[(a_1-k_1\beta(\tilde{y}(t)+\varepsilon)-b_1x(t)].
\end{equation}
Substituting  $x(t)<m_2$ into \eqref{e4.7}, we have
\begin{equation} \label{e4.8}
x(t)\geq x(t)[(a_1-b_1m_2)-k_1\beta(\tilde{y}(t)+\varepsilon)],\\
\end{equation}
for $t>T$, for some $T>0$. Let $N_1\in N$ and $N_1\tau>T$, 
integrating  on $(n\tau,(n+1)\tau]$, and $n>N_1$, we have
 \begin{equation} \label{e4.9}
x((n+1)\tau)\geq (1-p_1)x(n\tau^{+}) e^{\int^{(n+1)\tau}_{n\tau}[(a_1-b_1m_2)
-k_1\beta(\tilde{y}(t)+\varepsilon)]dt}.
\end{equation}
So we obtain
\begin{equation} \label{e4.10}
x((N_1+k)\tau)\geq (1-p_1)^{k}x(N_1\tau^{+}) e^{k\sigma}.\\
\end{equation}
Then, $x((N_1+k)\tau)\to +\infty$ as $k\to +\infty$,
which is a contradiction to the boundedness of $x(t)$. 
Therefore, there exists a $t_1>0$ such that $x(t)\geq m_1$.

In the next step, we intend to prove the boundedness of $y(t)$ and $z(t)$. 
From Lemma \ref{lem3.2}, we know $x(t)<M$  for $t>0$.
Substituting $x(t)<M$ into the second equation of system \eqref{e2.1}, we have
\begin{equation} \label{e4.11}
\begin{gathered}
\left.
\begin{gathered}
\frac{dy(t)}{dt}>-(d_1+k_1\beta M)y(t),\\
\frac{dz(t)}{dt}=z(t)(a_2-b_2z(t)),\\
\end{gathered}
\right\} \quad t\neq (n+l)\tau, t\neq (n+1)\tau,  \\
\left. \begin{gathered}
\Delta y(t)= \epsilon d z(t) ,\\
\Delta z(t)=-d z(t),
\end{gathered}
\right\} \quad t= (n+l)\tau,\\
 \left.\begin{gathered}
\Delta y(t)= -p_2 y(t) ,\\
\Delta z(t)=0,
\end{gathered}
\right\} \quad t= (n+1)\tau.
\end{gathered}
\end{equation}
The comparative differential equation of \eqref{e4.11} can be written as
\begin{equation}
\begin{gathered}
\left.\begin{gathered}
\frac{dy_3(t)}{dt}=-(d_1+k_1\beta M)y_3(t),\\
\frac{dz_3(t)}{dt}=z_3(t)(a_2-b_2z_3(t)),
\end{gathered}
\right\} \quad t\neq n\tau,   \\
 \left. \begin{gathered}
\Delta y_3(t)= \epsilon d z_3(t) ,\\
\Delta z_3(t)=-d z_3(t),
\end{gathered}
\right\} \quad t= n\tau,\\
 \left.\begin{gathered}
\Delta y_3(t)= -p_2z_3(t) ,\\
\Delta z_3(t)=0,
\end{gathered}
\right\} \quad t= n\tau.
\end{gathered}
\end{equation}
As in Lemma \ref{lem3.4}., we can obtain a globally asymptotically 
stable periodic solution 
\begin{equation} \label{e4.13}
\begin{gathered}
 \widetilde{y_3(t)}=\begin{cases}
y_3^{\ast}e^{-(d_1+k_1\beta M)(t-n\tau)},& t\in(n\tau,(n+l)\tau],\\
y_3^{\ast\ast}e^{-(d_1+k_1\beta M)(t-(n+l)\tau)}, & t\in((n+l)\tau, (n+1)\tau],
\end{cases} \\
 \tilde{z_3}(t)=\begin{cases}
\frac{a_2z_3^{\ast}e^{a_2(t-(n+l)\tau)}}{a_2+b_2z_3^{\ast}(e^{a_2(t-(n+l)\tau)}-1)},
& t\in(n\tau,(n+l)\tau],\\[4pt]
\frac{a_2z_3^{\ast\ast}e^{a_2(t-(n+l)\tau)}}{a_2
+b_2z_3^{\ast\ast}(e^{a_2(t-(n+l)\tau)}-1)}, & t\in((n+l)\tau, (n+1)\tau],
\end{cases} 
\end{gathered}
\end{equation}
where $y_3^{\ast}$ and $z_3^{\ast}$ are determined as
\begin{equation}
\begin{gathered}
y_3^{\ast}=\frac{B_3(B_4-a_2)}{b_2[a_2C_4+(B_4-a_2)C_3]},  \quad
  B_4>a_2,\\
 z_3^{\ast}=\frac{B_4-a_2}{b_2C_4},\quad B_4>a_2,
\end{gathered} \label{a}
\end{equation}
here  $B_3=(1-p_2)\epsilon d a_2e^{[a_2l-(d_1+k_1\beta M)(1-l)]\tau}$, 
$C_3=e^{a_2\tau}-1$,
$B_4=(1-d)a_2e^{a_2\tau}$, $C_4=e^{a_2\tau}-1$,
and $y_3^{\ast\ast}$ and $z_3^{\ast\ast}$  are  determined as follows:
\begin{equation} %4.15
\begin{gathered}
y_3^{\ast\ast}=y_3^{\ast}e^{-(d_1+k_1\beta M)l\tau}+\epsilon d 
 \frac{a_2e^{a_2l\tau}z_3^{\ast\ast}}{a_2+b_2(e^{a_2l\tau}-1)z^{\ast}},\\
z_3^{\ast\ast}=(1- d)  \frac{a_2e^{a_2l\tau}z_3^{\ast}}
{a_2+b_2(e^{a_2l\tau}-1)z_3^{\ast}}.
\end{gathered}
\end{equation}
There exists a $t_1>0$, for $t>t_1$, and  exists a  $\varepsilon_1>0$ 
small enough such that
\begin{align*}
y(t)&>y_3(t)\geq \widetilde{y_3(t)}-\varepsilon_1 \\
&\geq [y_3^{\ast}e^{-(d_1+k_1\beta M)l\tau}
 +y_3^{\ast\ast}e^{-(d_1+k_1\beta M)(1-l)\tau}]
 -\varepsilon_1=:m_3,\\
\end{align*}
and
\[
z(t)>z_3(t)
\geq  \tilde{z_3}(t)-\varepsilon_1\geq [z_3^{\ast}+z_3^{\ast\ast}]-\varepsilon_1
=:m_4.
\]
That is to say, $y(t)>m_3$ and $z(t)>m_4$  for $t>t_1$. 
This completes the proof.
\end{proof}


\section{Discussion}

 In this work,  we considered  a competitive predator-prey system
 with   impulsive reduction of  the invasive  population.
 We have proved that all solutions of  system \eqref{e2.1}
 are uniformly ultimately bounded.  The  stability of the  conditions of  
population $x(t)$-extinction  boundary periodic solution
$(0,\tilde{y}(t),\tilde{z}(t))$ of system \eqref{e2.1} is  obtained. 
The permanent conditions of
system \eqref{e2.1} are also obtained.  From Theorem \ref{thm4.1}
and Theorem \ref{thm4.3}, we know that there exists a threshold of 
 impulsive invasion parameter, which can make notation as $d_0$. If
$d>d_0$, the population $x(t)$-extinction solution
$(0,\tilde{y}(t), \tilde{z}(t))$ of System \eqref{e2.1} is
 stable. If $d<d_0$, System \eqref{e2.1} is
permanent. That is, if $d>d_0$, the population $y(t)$ invades from patch $1$ 
to patch $2$ successfully,  and  causes the native species $x(t)$ to extinct.
The invasive behaviors do harm to the native  biodiversity.
  If the $d<d_0$,  the population $y(t)$ invades from patch $1$ to patch 
$2$ successfully, then the alien species $y(t)$ coexist with the native 
species$x(t)$.  We can reach the optimal
invasion effect  by controlling the threshold $d_0$.


 If it is  assumed  that $x(0)=2$, $y(0)=2$, $z(0)=2$, $a_1=0.2$,
$b_1=1$, $k_1=0.5$, $k_2=0.5$, $\beta=0.6$, $d_1=1$, $a_2=2$, $b_2=1$,
$d=0.8$, $\tau=1$, $\epsilon=0.9$, $l=0.5$, then  the
population $x(t)$-extinction periodic solution $(0,\tilde{y}(t),\tilde{z}(t))$ 
of system \eqref{e2.1} is  stable, see Figure \ref{fig1}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig1a} % 1.eps
\includegraphics[width=0.48\textwidth]{fig1b} \\ %2.eps
 (a) \hfil (b) \\
\includegraphics[width=0.48\textwidth]{fig1c} % 3.eps
\includegraphics[width=0.48\textwidth]{fig1d} \\ % 4.eps
 (c) \hfil (d)
\end{center} 
\caption{The population $x(t)$-extinction
periodic solution $(0,\tilde{y}(t),\tilde{z}(t))$ of system \eqref{e2.1}
is stable  with
$x(0)=2$, $y(0)=2$, $z(0)=2$, $a_1=0.2$, $b_1=1$, $k_1=0.5$, $k_2=0.5$, 
$\beta=0.6$, $d_1=1$, $a_2=2$, $b_2=1$, $d=0.8$, $\tau=1$, $\epsilon=0.9$,
$l=0.5$. 
(a) Time-series of $x(t)$; 
(b) time-series of $y(t)$; 
(c) time-series of $z(t)$; 
(d) the phase portrait of the  stable population $x(t)$-extinction
periodic solution $(0,\tilde{y}(t),\tilde{z}(t))$ of  system \eqref{e2.1}.}
\label{fig1}
\end{figure}

If it is  also assumed that
$x(0)=2$, $y(0)=2$, $z(0)=2$, $a_1=1$, $b_1=0.5$, $k_1=0.5$, $k_2=0.5$, 
$\beta=0.6$, $d_1=1$, $a_2=2$, $b_2=1$, $d=0.5$, $\tau=1$, $\epsilon=0.7$, $l=0.5$, 
then  system \eqref{e2.1} is permanent, see Figure \ref{fig2}.

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.48\textwidth]{fig2a}  %5.eps
 \includegraphics[width=0.48\textwidth]{fig2b} \\ % 6.eps
 (a)\hfil (b) \\
 \includegraphics[width=0.48\textwidth]{fig2c} % 7.eps
 \includegraphics[width=0.48\textwidth]{fig2d}\\  % 8.eps
 (c) \hfil (d)
\end{center}
\caption{Permanence
of system \eqref{e2.1} with $x(0)=2$, $y(0)=2$, $z(0)=2$, $a_1=1$, 
$b_1=0.5$, $k_1=0.5$, $k_2=0.5$, $\beta=0.6$, $d_1=1$, $a_2=2$, $b_2=1$,
$d=0.5$, $\tau=1$, $\epsilon=0.7$, $l=0.5$.
(a) Time-series of $x(t)$; 
(b) time-series of $y(t)$; 
(c)  time-series of $z(t)$; 
(d) the phase portrait of the permanence of system
\eqref{e2.1}.}
\label{fig2}
\end{figure}

If it is   also assumed that
$x(0)=2$, $y(0)=2$, $z(0)=2$, $a_1=1$, $b_1=0.5$, $k_1=0.5$, $k_2=0.5$, 
$\beta=0.6$, $d_1=1$, $a_2=2$, $b_2=1$,
$d=0.01$, $\tau=1$, $\epsilon=0.7$, $l=0.5$, then   the
population $y(t)$-extinction solution of system \eqref{e2.1} is  stable,
see Figure \ref{fig3}.

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.48\textwidth]{fig3a} % 9.eps 
 \includegraphics[width=0.48\textwidth]{fig3b} \\ % 10.eps
 (a) \hfil (b) \\
 \includegraphics[width=0.48\textwidth]{fig3c} % 11.eps
 \includegraphics[width=0.48\textwidth]{fig3d} \\ % 12.eps
 (c) \hfil (d) 
\end{center} 
\caption{The population $y(t)$-extinction periodic solution  of system \eqref{e2.1}
 is stable  with $x(0)=2$, $y(0)=2$, $z(0)=2$, $a_1=1$, $b_1=0.5$,
$k_1=0.5$, $k_2=0.5$, $\beta=0.6$, $d_1=1$, $a_2=2$, $b_2=1$,
$d=0.01$, $\tau=1$, $\epsilon=0.7$, $l=0.5$.
 (a) Time-series of $x(t)$; (b) time-series of $y(t)$;
 (c)  time-series of $z(t)$; (d) the phase portrait of  the
population $y(t)$-extinction periodic solution  of System \eqref{e2.1}.}
\label{fig3}
\end{figure}

From the numerical analysis, we can further guess that there are two thresholds 
on parameter $d$, which can be written as $d^{\ast}$ and $d^{\ast\ast}$ with 
assumption $d^{\ast}>d^{\ast\ast}$.
When $1>d>d^{\ast}$, the population $x(t)$-extinction solution
$(0,\tilde{y}(t), \tilde{z}(t))$ of system \eqref{e2.1} is  stable. 
That is, if $1>d>d^{\ast}$, the population $y(t)$ invades from patch 
$1$ to patch $2$ successfully, and  exclude the native species $x(t)$ 
to extinct, The invasional behaviors do harm to the native  biodiversity. 
When $d^{\ast\ast}<d<d^{\ast}$, system \eqref{e2.1} is
permanent. That is to say, if $d^{\ast\ast}<d<d^{\ast}$, 
the population $y(t)$ invades from patch $1$ to patch $2$ successfully, 
and the alien species $y(t)$ coexist with the native species $x(t)$, 
and  the invasional behaviors will do no harm to the native  biodiversity.  
When $0<d<d^{\ast\ast}$, the population $y(t)$-extinction solution of 
system \eqref{e2.1} is  stable. That is to say, if $0<d<d^{\ast\ast}$, 
the population $y(t)$ invades from patch $1$ to patch $2$ unsuccessfully, 
and the alien species $y(t)$ will be excluded to extinction by the 
native species $x(t)$.

Combining the above numerical computation with  Theorems \ref{thm4.1}
and \ref{thm4.3},
we can also guess that there are two thresholds about parameter $\tau$,
which can be written as $\tau^{\ast}$ and $\tau^{\ast\ast}$ where  
$\tau^{\ast}>\tau^{\ast\ast}>0$.
When $\tau>\tau^{\ast}$, the population $y(t)$-extinction solution
 of system \eqref{e2.1} is stable. That is, if $\tau>\tau^{\ast}$, 
the population $y(t)$ invades from patch $1$ to patch 2 unsuccessfully, 
and the alien species $y(t)$ is excluded to extinction by the native species
 $x(t)$. The invasional behaviors do no harm to the native  biodiversity.
 When  $\tau^{\ast}>\tau>\tau^{\ast\ast}$, system \eqref{e2.1} is
permanent. That is to say, if $\tau^{\ast}>\tau>\tau^{\ast\ast}$, 
the population $y(t)$ invades from patch $1$ to patch $2$ successfully, 
and the alien species $y(t)$ coexist with the native species $x(t)$. 
The invasional behaviors also do no  harm to the native  biodiversity. 
When $0<\tau<\tau^{\ast\ast}$, the population $y(t)$-extinction solution
 of system \eqref{e2.1} is stable. That is, if $0<\tau<\tau^{\ast\ast}$, 
the population $y(t)$ invades from patch $1$ to patch $2$ successfully, 
and  exclude the native species $x(t)$ to extinction. 
The invasional behaviors will do harm to the native  biodiversity.
Our  results show that the  impulsive invasion amount and invasion period  
play  important roles for the the dynamics of system \eqref{e2.1}. Our
results also  provide reliable tactic basis for the practical  biodiversity
management.

\subsection*{Acknowledgments}
This research was supported by National Natural Science Foundation of China
(No. 11761019, 11361014), by the Development Project of Natural Science Research
 of Guizhou Province Department (No.2010027), by the Project of High Level
Creative Talents in Guizhou Province (No. 20164035), by and the Science Technology
Foundation of Guizhou (No. 2010J2130).

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