\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 31, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/31\hfil Periodic solutions]
{Periodic solutions for a delayed predator-prey system with
dispersal and  impulses}

\author[Q. Liu, S. Dong \hfil EJDE-2005/31\hfilneg]
{Qiming Liu, Shijie Dong}  

\address{Qiming Liu \hfill\break
Department of Mathematics,
Shijiazhuang Mechanical Engineering  College,
Shijiazhuang 050003, China}
\email{llqm2002@yahoo.com.cn}

\address{Shijie Dong \hfill\break
Department of Mathematics, Shijiazhuang Mechanical Engineering
College, Shijiazhuang 050003, China} 
\email{j\_ds@sina.com}


\date{}
\thanks{Submitted November 1, 2004. Published March 18, 2005.}
\subjclass{34K13, 34K45, 92D25} 
\keywords{Predator-prey system; impulse; dispersion; periodic solution; \hfill\break\indent
coincidence degree}

\begin{abstract}
 A delayed predator-prey system with prey dispersal in n-patch
 environments and impulse effects is investigated.
 By using Gaines and Mawhin's continuation theorem of coincidence
 degree theory, a set of easily verifiable sufficient conditions
 are derived for the existence of positive periodic solutions
 to the system.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] % theorems numbered with section #
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

 An important and ubiquitous problem in mathematical ecology
concerns the effect of environment change in the growth and
diffusion of a species in a heterogenous habitat. There have been
many studies in the literatures that investigate the population
dynamics with diffusion process \cite{k1,l1,s1,t1}.

In most of the
models considered so far, it has been assumed that all biological
and environmental parameters are constants in time. However, any
biological or environmental parameters are naturally subject to
fluctuation in time. The effects of a periodically varying
environment are important for evolutionary theory as the selective
forces on systems in a fluctuating environment differ from those
in a stable environment. Thus, the assumptions of periodicity of
the parameters are a way of incorporating the periodicity of the
environment (such as seasonal effects of weather, food supplies,
mating habits and so forth); on the other hand, it is generally
recognized that some kinds of time delays are inevitable in
population interactions. Time delay due to
   gestation is a common example, because generally the consumption of
   prey by the predator throughout its past history governs the
present birth rate of
    the predator. Therefore, more realistic models of population
interactions should take into account the seasonality of the
changing environment and the effect of time delays.

There are still some other possible exterior effects under which
the population densities change very rapidly. For example, impulse
reduction of the population density of a given species is possible
after its partial destruction by catching or by poisoning with
chemical used at some transitory slots in fishing or
agriculture \cite{b1,l2,t2}.
 Recently, many authors studies the existence of
positive periodic solution in population models by using power and
effective method of coincidence degree \cite{l3,z1}, In the present
paper, we are concerned with the study on the combined effects of
dispersion, periodicity of environment, time delays and impulses
on the dynamics of predator-prey system. To do so we are devoted
to the study of the following  delayed periodic predator-prey
system with prey dispersal in n-patch environments and impulses
\begin{equation}
\begin{gathered}
\begin{aligned}
\dot x_i(t)=&x_i(t)[r_i(t)-a_{ii}(t)x_i(t)-a_{i,n+1}(t)x_{n+1}(t)]\\
&+\sum_{j=1,j\neq i}^nD_j(t)[x_j(t)-x_i(t)],\quad i=1,\dots,n,
\end{aligned}\\
\begin{aligned}
 \dot  x_{n+1}(t)=&x_{n+1}(t)\Big[-r_{n+1}(t)+\sum_{j=1}^na_{n+1,j}
 (t)x_j(t-\sigma_j)\\
 &-a_{n+1,n+1}(t)x_{n+1}(t-\sigma_{n+1})\Big],\quad
  t\neq \tau_k,\; k\in Z_+ ,
 \end{aligned} \\
\Delta (x_i(\tau_k))=-c_{ik}x_i(\tau_k),\quad t=\tau_k, \; k\in
Z_+, \quad i=1,2,\dots, n+1.
\end{gathered} \label{e1.1}
\end{equation}
with initial conditions
\begin{equation}
 x_i(s)=\phi_i(s),\quad s\in [-\sigma, 0],\quad  \phi_i(0)>0,\quad
 i=1,\dots,n+1, \label{e1.2}
\end{equation}
where $x_i(t)$ $(i=1,\dots,n)$ denote the densities of prey
species in  patch $i$, $x_{n+1}$ denote the densities of all
predator for all patches. $r_i(t)$ is the intrinsic growth rate of
the prey, $a_{ii}(t)$ is the density-dependent coefficient of the
prey species. $a_{i,n+1}(t)$ are the capturing rates of the
predator and $a_{n+1,i}/a_{i,n+1}$ are the conversion rates of
nutrients into the reproduction of predator, $r_{n+1}(t)$ are the
death rates of the predator. $D_i(t)$ $(i=1,2,\dots,n)$ is
dispersal rate of prey species, $\sigma_{n+1}\geq 0$ denotes the
delay due to negative feedback of the predator species $\sigma_i$
$(i=1,2,\dots,n)$ are the time
   delays due to gestation, that is, mature adult predators
   can only contribute to the production of predator  biomass.
$a_{ii}(t)$, $a_{i,n+1}(t)$, $a_{n+1,i}$ $(i,j=1,\dots,n)$,
 $r_i(t)$ $(i=1,\dots, n+1)$ and $D_i(t)$ $(i=1,2,\dots,n)$ are
 continuously positive periodic  functions with
period $\omega >0$. $c_{ik}$ are positive constants and
$0<c_{ik}<1$, $Z_+$ is the set of all positive integers and there
exists an integer $p>0$ such that
$c_{i(k+p)}=c_{ik}$, $\tau_{k+p}=\tau_k+\omega$.

 It is well known that by the fundamental theory of
functional differential equations \cite{h1}, system \eqref{e1.1} has a unique
solution $x(t)=(x_1(t),\dots,x_{n+1}(t))$ satisfying initial
conditions \eqref{e1.2}. It is easy to verify that solutions of
\eqref{e1.1} corresponding to initial conditions \eqref{e1.2} are defined on
$[0,+\infty)$ and remain positive for all $t\geq 0$. In this
paper, the solution of system \eqref{e1.1} satisfying
 initial conditions \eqref{e1.2} is said to be positive.

We shall use the following notation:
 Let $J\subset R$. Denote by $PC(J,R)$ the set of function
 $\psi: J\to R$ which are continuous for $t\in J, t\neq \tau_k$,
 are continuous from the left for $t\in J$ and have
 discontinuities of the first kind at the points $\tau_k\in J$.
 Denote the Banach space of $\omega$-periodic functions by
$PC_\omega=\{\psi \in PC[0,\omega], R\}| \psi (0)=\psi (\omega)\}$
and we denote
$$
\bar{f}=\frac{1}{\omega}\int_{0}^{\omega}f(t)dt,\quad
f^L=\min_{t\in [0, \omega]}f(t),\quad f^M=\max_{[0, \omega]}f(t),
$$
where $f\in PC_\omega$.

 The organization of this paper is as follows. In the next section,
by using Gaines and Mawhin's continuation theorem of coincidence
degree theory, sufficient conditions are derived for the existence
of positive periodic solutions of system \eqref{e1.1} with initial
conditions \eqref{e1.2}.


\section{Main result}

In this section, by using Gaines and Mawhin's continuation theorem
of coincidence degree theory, we show the existence of positive
$\omega$-periodic solutions of \eqref{e1.1}-\eqref{e1.2}. To this end, we first
introduce the following notations.

Let $X, Z$ be real Banach spaces, let $L$: $\mathop{\rm
Dom}L\subset X\to Z$ be a linear mapping, and $N:X\to Z$ be a
continuous mapping. The mapping $L$ is called a Fredholm mapping
of index zero if $\mathop{\rm dim }\ker L=\mathop{\rm
codim}\mathop{\rm Im}L<+\infty$ and $\mathop{\rm Im}L$ is closed
in $Z$. If $L$ is a Fredholm mapping of index zero and there exist
continuous projectors $P: X\to X$, and $Q: Z\to Z$ such that
$\mathop{\rm Im}P=\ker L$,
$\ker Q=\mathop{\rm Im}L=\mathop{\rm Im}(I-Q)$, then the restriction
$L_P$ of $L\to \mathop{\rm Dom}L\cap \ker P: (I-P)X\to \mathop{\rm Im}L$
is invertible. Denote the inverse of $L_P$ by $K_P$. If $\Omega$
is an open bounded subset of $X$, the mapping $N$ will be called
$L$-compact on
$\bar{\Omega}$ if $QN(\bar{\Omega})$ is bounded and
$K_P(I-Q)N:\bar{\Omega}\to X$ is compact. Since $\mathop{\rm Im}Q$
is isomorphic to $\ker L$, there exists isomorphism
$J:\mathop{\rm Im}Q\to \ker L$.

For convenience, we introduce the continuation theorem of
coincidence degree theory \cite{g1} and compactness criterion for set
$F \subset PC_\omega$ \cite{b2} as follows.

\begin{lemma} \label{lem1}
Let $\Omega\subset X$ be an open bounded set. Let $L$ be a Fredholm mapping
of index zero and $N$ be $L$-compact on $\bar\Omega$. Assume
\begin{itemize}
\item[(a)] For each $\lambda\in (0,1)$, $x\in \partial\Omega\cap\mathop{\rm Dom}L$,
$Lx\neq \lambda Nx$

\item[(b)] For each $x\in \partial \Omega\cap \ker L$, $QNx\neq 0$

\item[(c)] $\deg\{JQN, \Omega\cap\ker L, 0\}\neq 0$.
\end{itemize}
Then $Lx=Nx$ has at least one solution in $\bar\Omega\cap \mathop{\rm Dom}L$.
\end{lemma}

\begin{lemma}[Compactness criterion] \label{lem2}
A set $F \subset PC_\omega $ is relative compact if and only if
\begin{itemize}
\item[(a)]  $F$ is bounded, that is,
$\|\psi\|=sup\{|\psi|:t\in J\}\leq M$ for each $x\in F$  and some $ M>0$

\item[(b)]  $F$ is quasiequicontinuous in $J$.
\end{itemize}
\end{lemma}

 We are now in a position to state our
main result on the existence of a positive periodic solution to
system \eqref{e1.1}.

\begin{theorem} \label{thm1}
System \eqref{e1.1} with initial conditions \eqref{e1.2} has at least
 one strictly positive $\omega$-periodic
 solution provided that
 \begin{itemize}
\item[(H1)]
$$
\sum_{j=1}^na_{n+1,j}^M(\overline{r_j}-\sum_{k=1,k\neq
j}^n\overline{D_k}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{jk}})/a_{jj}^M>\overline{r_{n+1}}
+\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
$$
\item[(H2)] For $i=1,2,\dots,n$,
\begin{align*}
&\overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\\
&>a_{i,n+1}^M\frac{A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}-\frac{1}{\omega}
\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}{a_{n+1,n+1}^L},
\end{align*}
\end{itemize}
where
 $A=\max_{1\leq i\leq n}\{[(r_i-\sum_{j=1,j\neq
i}^nD_j)^M+\sum_{j=1,j\neq i}^nD_j^M]/a_{ii}^L\}$.
\end{theorem}

\begin{proof}
Since solutions of \eqref{e1.1}-\eqref{e1.2} remain
positive for all $t\geq 0$, we let
\begin{equation}
\quad y_i(t)=\ln [x_i(t)],~i=1, \dots, n+1. \label{e2.1}
\end{equation}
Substituting \eqref{e2.1} into system \eqref{e1.1}, we derive
\begin{equation}
\begin{gathered}
\begin{aligned}
 \dot y_i(t)=& r_i(t)-\sum_{j=1,j\neq i}^nD_j(t)-a_{ii}(t)e^{y_i(t)}
 -a_{i,n+1}(t)e^{y_{n+1}(t)}]\\
&+\sum_{j=1,j\neq i}^nD_j(t)e^{y_j(t)-y_i(t)},\quad i=1,\dots, n,
\end{aligned}\\
\begin{aligned}
 \dot  y_{n+1}(t)=&-r_{n+1}(t)+\sum_{j=1}^na_{n+1,j}(t)e^{y_j(t-\sigma_j)}
 -a_{n+1,n+1}(t)e^{y_{n+1}(t-\sigma_{n+1})},\\
 & t\neq\tau_k, \; k\in Z_+,
 \end{aligned}\\
 \Delta (y_i(\tau_k))=\ln(1-c_{ik}),\quad t=\tau_k, ~k\in Z_+, \quad
 i=1,2,\dots,n+1.
 \end{gathered} \label{e2.2}
\end{equation}
It is easy to see that if  \eqref{e2.2} has one $\omega$-periodic
solution $(y_1^*(t),\dots,y_{n+1}^*(t))^T,$ then
$x^*(t)=(x_1^*(t),\dots,x_{n+1}^*(t))^T=(\exp[y_1^*(t)],\dots,\exp[y_{n+1}^*(t)])^T$
is a positive $\omega$-periodic solution of system \eqref{e1.1}.
Therefore, to complete the proof, it suffices to show that system
\eqref{e2.2} has one $\omega$-periodic solution.
Let
 $$
 X=\{x\in PC(R,R^{n+1}): y(t+\omega)=y(t)\},
$$
with the norm
$\| x\| =\sup_{t\in [0,\omega]}\sum_{i=1}^{n+1}|y_i(t)|$,
where $x=(y_1,y_2,\dots,y_{n+1})^T$.
Let
 $Z=X\times R^{np}$ with the norm
$\| (x,a_1,\dots,a_p)\|=(\|x\|^2+|a_1|^2+\dots+|a_p|^2)^{1/2}$\,.
Then $X$ and $Z$ are Banach spaces.
Let
$$L: \mathop{\rm Dom}L\to Z \quad L(x)(t)=(\dot x, \Delta
(y(\tau_1),\dots,\Delta (y(\tau_p))
$$
where $\mathop{\rm Dom}L$ consist of functions $x\in X$ such that
$x$ is continuous for $t\neq \tau_k$, $x$  is  continuous from the left for
$t=\tau_k$, and $\dot x(\tau_k)$  exists.
Let $N: X\to Z$, be defined as
$$
(Nx)(t)=(f(t,x(t)), C_1,C_2,\dots,C_p),
$$
where
$f(t,x)$ equals
$$
\begin{pmatrix}
r_1(t)-\sum_{j=2}^nD_j(t)-a_{11}(t)e^{y_1(t)}-a_{1,n+1}(t)e^{y_{n+1}(t)}]
+\sum_{j=2}^nD_j(t)e^{y_j(t)-y_1(t)}\\
\vdots\\
r_n(t)-\sum_{j=1}^{n-1}D_j(t)-a_{nn}(t)e^{y_n(t)}-a_{n,n+1}(t)e^{y_{n+1}(t)}]
+\sum_{j=1}^{n-1}D_j(t)e^{y_j(t)-y_n(t)}\\
 -r_{n+1}(t)+\sum_{j=1}^na_{n+1,j}(t)e^{y_j(t-\sigma_j)}-a_{n+1,n+1}(t)e^{y_{n+1}(t-\sigma_{n+1})},
\end{pmatrix}
$$
and
 $$
C_k=(\ln(1-c_{1k}),\ln(1-c_{2k}),\dots,\ln(1-c_{n+1,k}))^T,\quad
k=1,2,\dots,p.
 $$
  Define two projectors $P$ and $Q$ as
$$
P:X\to \ker L,\quad Py=\frac{1}{\omega}\int_0^{\omega}ydt,
$$
$$
Q:Z\to Z,\quad
Q(y,C_1,\dots,Cp)=\Big(\frac{1}{\omega}\int_0^{\omega}ydt+\sum_{k=1}^pC_k,
\Big\{\begin{pmatrix}0\\\vdots\\0\end{pmatrix}\Big\}_{k=1}^{p}\Big),
$$
It is clear that
\begin{gather*}
 \ker L=\{x : x\in X,\;  x=h,\;  h\in R^{n+1}\},\\
 \mathop{\rm Im}L=\{z=(y, C_1,\dots,C_p) \in Z, \;
 {\int_{0}^{\omega}}y(t)dt+\sum_{k=1}^pC_k=0\}\hbox{ is closed in } Z,\\
 \mathop{\rm dim }\ker L=\mathop{\rm codim\, Im}L=n+1.
\end{gather*}
Therefore, $L$ is a Fredholm mapping of index zero. It is easy to
show that $P$ and $Q$ are continuous projectors such that
$$
\mathop{\rm Im}P=\ker L,\quad \ker Q=\mathop{\rm Im}L=\mathop{\rm Im}(I-Q).
$$
Furthermore, the inverse $K_P$ of $L_P$ exists and is given by
$K_P:\mathop{\rm Im}Lto\mathop{\rm Dom}L\cap \ker P$,
$$
K_P(z)=\int_{0}^{t}y(s)ds-
\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}y(s)ds-\sum_{k=1}^pC_k.
$$
Then $QN: X\to Z$ and $K_P(I-Q)N: X\to X$ read
$$
QNx=\Big(\frac{1}{\omega}\int_{0}^{\omega}f(t,x)dt+\frac{1}{\omega}\sum_{k=1}^pC_k,
\Big\{\begin{pmatrix} 0\\\vdots\\0\end{pmatrix}\Big\}_{k=1}^{p}\Big),
$$
\begin{align*}
&K_P(I-Q)Nx\\
&=\frac{1}{\omega}\int_{0}^tf(t,x)dt+\frac{1}{\omega}\sum_{t>\tau_k}C_k
-\Big(\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}f(t,x)ds+\sum_{k=1}^pC_k\Big)\\
&- \Big((\frac{t}{\omega}-\frac{1}{2})\frac{1}{\omega}\int_{0}^{\omega}f(t,x)dt
+\frac{1}{\omega}\sum_{k=1}^pC_k\Big).
\end{align*}
Clearly, $QN$ and $K_P(I-Q)N$ are continuous.

To apply Lemma \ref{lem1}, we need to search for an appropriate
open, bounded subset $\Omega$.
Corresponding to the operator equation $Lx=\lambda Nx$,
$\lambda \in (0,1)$, we obtain
\begin{equation}
\begin{gathered}
\begin{aligned}
\dot y_i(t)=&\lambda \Big[r_i(t)-\sum_{j=1,j\neq
i}^nD_j(t)-a_{ii}(t)e^{y_i(t)}-a_{i,n+1}(t)
e^{y_{n+1}(t)}\Big]\\
&+\sum_{j=1,j\neq i}^nD_j(t)e^{y_j(t)-y_i(t)}],\quad
i=1,\dots, n,
\end{aligned}\\
\begin{aligned}
 \dot y_{n+1}(t)=&\lambda\Big[-r_{n+1}(t)
 +\sum_{j=1}^na_{n+1,j}(t)e^{y_j(t-\sigma_j)}
 -a_{n+1,n+1}(t)e^{y_{n+1}(t-\sigma_{n+1})}\Big],\\
 & t\neq\tau_k,\; k\in Z_+,
\end{aligned}\\
 \Delta (y_i(\tau_k))=\lambda\ln(1-c_{ik}),\quad
  t=\tau_k, \; k\in Z_+, \; i=1,2,\dots,n+1.
\end{gathered} \label{e2.3}
\end{equation}
 Suppose that $ (y_1(t), \dots, y_{n+1}(t))^T \in X $ is a solution of
 \eqref{e2.3} for  some $\lambda \in (0,1)$. Integrating system \eqref{e2.3}
 over $[0,\omega]$, for $i=1,2,\dots,n$, we have
\begin{equation}
 \begin{aligned}
&{\int_{0}^{\omega}}a_{ii}(t)e^{y_i(t)}dt+
{\int_{0}^{\omega}}a_{i,n+1}(t)e^{y_{n+1}(t)}dt+\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\\
&={\int_{0}^{\omega}}(r_i(t)-\sum_{j=1,j\neq
i}^nD_j(t))dt+\sum_{j=1,j\neq i}^n{\int_0^\omega}
D_j(t)e^{y_j(t)-y_i(t)}dt,
\end{aligned}  \label{e2.4}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\int_{0}^{\omega}r_{n+1}(t)dt+  \int_{0}^{\omega}a_{n+1,n+1}(t)
e^{y_{n+1}(t-\sigma_{n+1})} dt+\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\\
&=\sum_{j=1}^n\int_{0}^{\omega}a_{n+1,j}(t)e^{y_j(t-\sigma_j)}dt.
\end{aligned} \label{e2.5}
\end{equation}
 Multiplying the $i$th  equation \eqref{e2.3} by $e^{y_i(t)}$ and
 integrating over $[0,\omega]$  gives
\begin{align*}
&\int_{0}^{\omega}a_{ii}(t)e^{2y_i(t)}dt\\
&\leq \int_{0}^{\omega}(r_i(t)-\sum_{j=1,j\neq
i}^nD_j(t))e^{y_i(t)}dt+\sum_{j=1,j\neq i}^n\int_0^\omega
D_j(t)e^{y_j(t)}dt+\sum_{k=1}^p\Delta(e^{y_i(\tau_k)}),
\end{align*}
in which
$\Delta(e^{y_i(\tau_k)})=[(1-c_{ik})^\lambda-1]e^{y_i(\tau_k)}\leq 0$,
then we have
\begin{equation}
a_{ii}^L\int_{0}^{\omega}e^{2y_i(t)}dt\leq
[(r_i-\sum_{j=1,j\neq
i}^nD_j)^M]\int_{0}^{\omega}e^{y_i(t)}dt+\sum_{j=1,j\neq
i}^nD_j^M\int_0^\omega e^{y_j(t)}dt\,. \label{e2.6}
\end{equation}
Using the inequality
$$
\Big(\int_{0}^{\omega}e^{y_i(t)}dt\Big)^2\leq \omega
\int_{0}^{\omega}e^{2y_i(t)}dt,
$$
it follows from \eqref{e2.6} that
\begin{equation}
\frac{1}{\omega}a_{ii}^L(\int_{0}^{\omega}e^{y_i(t)}dt)^2 \leq
\big[(r_i-\sum_{j=1,j\neq
i}^nD_j)^M\big]\int_{0}^{\omega}e^{y_i(t)}dt+\sum_{j=1,j\neq
i}^nD_j^M\int_0^\omega e^{y_j(t)}dt.\label{e2.7}
\end{equation}
Using the fact that if  $i=k$,
$$
\int_{0}^{\omega}e^{y_k(t)}dt\geq
\max\big\{\int_{0}^{\omega}e^{y_i(t)}dt, i=1,\dots,n\big\}
$$
this , together with \eqref{e2.7},  leads to
\begin{align*}
&\frac{1}{\omega}a_{kk}^L(\int_{0}^{\omega}e^{y_k(t)}dt)^2\\
&\leq \big[(r_k-\sum_{j=1,j\neq
k}^nD_j)^M\big]\int_{0}^{\omega}e^{y_k(t)}dt+(\sum_{j=1,j\neq
k}^nD_j^M)\int_0^\omega e^{y_k(t)}dt,
\end{align*}
which implies
\begin{equation}
\begin{aligned}
\max\{\int_{0}^{\omega}e^{y_i(t)}dt, i=1,\dots,n\}
\leq& \int_{0}^{\omega}e^{y_k(t)}dt \\
\leq& \frac{[(r_k-\sum_{j=1,j\neq
k}^nD_j)^M+\sum_{j=1,j\neq k}^nD_j^M]\omega}{a_{kk}^L}.
\end{aligned} \label{e2.8}
\end{equation}
Set
$$
A=\max_{1\leq i\leq n}\Big\{\frac{(r_i-\sum_{j=1,j\neq
i}^nD_j)^M+\sum_{j=1,j\neq i}^nD_j^M}{a_{ii}^L}\Big\},
$$
we have from \eqref{e2.8} that
\begin{equation}
\int_{0}^{\omega}e^{y_i(t)}dt\leq A\omega,\quad i=1,\dots,n.
\label{e2.9}
\end{equation}
Since $y_i\in PC_\omega$, there exists $t_i,T_i\in [0,T]\cup
\{\tau_1^+,\tau_2^+,\dots,\tau_p^+\}$ such that
 $$
 y_i(t_i)=\min_{t\in [0,\omega]}y_i(t), \quad
 y_i(T_i)=\max_{t\in [0,\omega]}y_i(t),\quad
 i=1,2,\dots,n.
 $$
 It follows from \eqref{e2.9} that
\begin{equation}
 y(t_i)\leq \ln A,\quad i=1,2,\dots,n. \label{e2.11}
\end{equation}
We derive from \eqref{e2.5} that
\begin{equation}
\begin{aligned}
&{\int_0^\omega}
a_{n+1,n+1}(t)e^{y_{n+1}(t-\sigma_{n+1})}dt\\
&\leq \sum_{j=1}^na_{n+1,j}^M{\int_0^\omega}e^{y_j(t-\sigma_j)}dt
-\overline{r_{n+1}}\omega-\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\\
&=\sum_{j=1}^na_{n+1,j}^M{\int_0^\omega} e^{y_j(t)}dt-\overline{r_{n+1}}\omega
-\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\\
&\leq (A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}})\omega
-\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}},
\end{aligned}\label{e2.12}
\end{equation}
which yields
$$
y_{n+1}(t_{n+1})\leq \ln\frac{A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}{\overline{a_{n+1,n+1}}},
$$
and
\begin{equation}
\begin{aligned}
{\int_0^\omega} e^{y_{n+1}(t)}dt
&=\int_0^\omega e^{y_{n+1}(t-\sigma_{n+1})}dt\\
&\leq \frac{A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}{a_{n+1,n+1}^L}\omega\,.
\end{aligned}\label{e2.13}
\end{equation}
It follows from \eqref{e2.4}, \eqref{e2.5}, \eqref{e2.9} and \eqref{e2.13} that
\begin{equation}
\begin{aligned}
 &{\int_{0}^{\omega}}|\dot y_{n+1}(t)|dt\\
 &={\int_{0}^{\omega}}\lambda|-r_{n+1}(t)+\sum_{j=1}^na_{n+1,j}(t)
 e^{y_j(t-\sigma_j)}-a_{n+1,n+1}(t)e^{y_{n+1}(t-\sigma_{n+1})}|dt\\
&\leq{\int_{0}^{\omega}}[r_{n+1}(t)+\sum_{j=1}^na_{n+1,j}(t)
e^{y_{n+1}(t-\sigma_j)}+a_{n+1,n+1}(t)e^{y_{n+1}(t-\sigma_{n+1})}]dt\\
&\leq 2{\int_{0}^{\omega}}\sum_{j=1}^na_{n+1,j}(t)e^{y_j(t-\sigma_j)}dt
-\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\\
&< 2A\omega\sum_{j=1}^na_{n+1,j}^M
 :=d_{n+1},
 \end{aligned}  \label{e2.14}
\end{equation}
and
\begin{equation}
 \begin{aligned}
 {\int_{0}^{\omega}}|\dot y_i(t)|dt
 &={\int_{0}^{\omega}}\lambda|r_i(t)-\sum_{j=1,j\neq i}^nD_j(t)-a_{ii}(t)
 e^{y_i(t)}-a_{i,n+1}(t)e^{y_{n+1}(t)}\\
&\quad +\sum_{j=1,j\neq i}^nD_j(t)e^{y_j(t)-y_i(t)}|dt\\
&\leq{\int_{0}^{\omega}}[r_i(t)+\sum_{j=1,j\neq
i}^nD_j(t)+a_{ii}(t)e^{y_i(t)}+a_{i,n+1}(t)e^{y_{n+1}(t)}\\
&\quad +\sum_{j=1,j\neq i}^nD_j(t)e^{y_j(t)-y_i(t)}]dt\\
&\leq 2{\int_0^\omega}
[a_{ii}(t)e^{y_i(t)}+a_{i,n+1}(t)e^{y_{n+1}(t)}]dt+\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\\
&\leq 2\big[a_{ii}^M\int_0^\omega
e^{y_i(t)}dt+a_{i,n+1}^M{\int_0^\omega} e^{y_{n+1}(t)}dt\big]
+\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\\
&<2A\big[a_{ii}^M
+a_{i,n+1}^M(\sum_{j=1}^na_{n+1,j}^M)/a_{n+1,n+1}^L\big]\omega
+\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}
:=d_i,
 \end{aligned}  \label{e2.15}
\end{equation}
$i=1,\dots,n$. From \eqref{e2.11}, \eqref{e2.15} and \eqref{e2.13}, \eqref{e2.14}, we have
\begin{equation}
 \begin{aligned}
 &y_{n+1}(t)\\
 & \leq y_{n+1}(t_{n+1})+{\int_{0}^\omega}|\dot
 y_{n+1}(t)|dt+\big|\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\big|\\
 &\leq \ln\frac{A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}
 -\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}{\overline{a_{n+1,n+1}}}
 +d_{n+1}+\big|\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\big|,
 \end{aligned}  \label{e2.16}
\end{equation}
and
\begin{equation}
 \begin{aligned}
 y_i(t)\leq&
 y_i(t_i)+{\int_{0}^\omega}|\dot y_i(t)|dt+\big|\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\big|  \\
\leq&\ln
A+d_i+\big|\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\big|, \quad i=1,2,\dots, n.
\end{aligned} \label{e2.17}
\end{equation}
On the other hand, it follows from \eqref{e2.4} and \eqref{e2.9} that
\begin{align*}
a_{ii}^Me^{y_i(T_i)}\omega
&\geq(\overline{r_i}-\sum_{j=1,j\neq i}^n\overline{D_j})\omega-\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}
-a_{i,n+1}^M{\int_0^\omega} e^{y_{n+1}(t)}dt\\
&\geq (\overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j})\omega-\ln\prod_{k=1}^p\frac{1}{1-c_{ik}} \\
&\quad-a_{i,n+1}^M\Big(A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\Big)\omega/a_{n+1,n+1}^L,
\end{align*}
which implies
\begin{equation}
\begin{aligned}
y_i(T_i)\geq&
\ln\Big[ \frac 1{a_{ii}^M}
\Big( \overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\\
&\quad -a_{i,n+1}^M(A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}})/a_{n+1,n+1}^L
\Big)\Big]
:=\rho_i.
\end{aligned} \label{e2.18}
\end{equation}
 From \eqref{e2.14} and \eqref{e2.18} it follows that for $i=1,2\dots,n$,
\begin{equation}
y_i(t)\geq y_i(T_i)-\int_0^\omega |\dot
y_i(t)|dt-|\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}|\geq
\rho_i-d_i-|\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}|,
\label{e2.19}
\end{equation}
Note that
$$
\int_0^\omega a_{ii}(t)e^{y_i(t)}dt+\int_0^\omega a_{i,n+1}(t)e^{y_{n+1}(t)}dt
\geq (\overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j})\omega-\ln\prod_{k=1}^p\frac{1}{1-c_{ik}},
$$
which, together with \eqref{e2.5}, leads to
\begin{align*}
&a_{n+1,n+1}^My_{n+1}(T_{n+1})\omega\\
&\geq\sum_{j=1}^na_{n+1,j}^L{\int_0^\omega}
e^{y_j(t)}dt -\overline{r_{n+1}}\omega\\
&\geq \sum_{j=1}^na_{n+1,j}^L\frac{(\overline{r_j}-\sum_{k=1,k\neq
j}^n\overline{D_k}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{jk}}
-\overline{a_{j,n+1}}e^{y_n+1(T_{n+1})})\omega}{a_{jj}^M}\\
&\quad -\overline{r_{n+1}}\omega-\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
\end{align*}
which yields
\begin{equation}
\begin{aligned}
y_{n+1}(T_{n+1})\geq&
\frac 1{a_{n+1,n+1}^M+(\sum_{j=1}^na_{n+1,j}^M\overline{a_{j,n+1}})/a_{jj}^M}\\
&\times\Big(
\sum_{j=1}^na_{n+1,j}^M\big(\overline{r_j}-\sum_{k=1,k\neq
j}^n\overline{D_k}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{jk}}\big)/a_{jj}^M\\
&-\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}} \Big)
:=\rho_{n+1}.
\end{aligned} \label{e2.20}
\end{equation}
 From what has been discussed above, we finally derive that for
$i=1,\dots,n$,
\begin{align*}
&\max_{t\in [0, \omega]}|y_i(t)|\\
&\leq \max\big\{|\ln A|+d_i+|\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}|,
  |\rho_i|+d_i+|\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}|\big\}
 :=B_i,
\end{align*}
and
\begin{equation}
\begin{aligned}
& \max_{t\in [0, \omega]}|y_{n+1}(t) \\
&\leq| \max \{|\ln\frac{A\sum_{j=1}^na_{n+1,j}^M-\overline{r_{n+1}}
\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}{\overline{a_{n+1,n+1}}}|\\
 &\quad +d_{n+1}+|\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}|
 ,|\rho_{n+1}|+d_{n+1}+|\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}|\big\}
 :=B_{n+1}.
 \end{aligned} \label{e2.21}
\end{equation}
Clearly, $B_i$ $(i=1,2,\dots,n+1)$ are independent of $\lambda$.
 Denote $B=\sum_{i=1}^{n+1}B_i+B_0$,
 here $B_0$ is taken sufficiently large such that each solution
$(v_1^\ast,v_2^\ast,...,v_n^\ast,v_{n+1}^\ast)^T$
 of the system of algebraic equations
\begin{equation}
 \begin{gathered}
\overline{r_i}-\sum_{j=2}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}
-\overline{a_{ii}}e^{v_i}-\overline{a_{i,n+1}}e^{v_{n+1}}
+\sum_{j=1,j\neq i}^n\overline{D_j}e^{v_j-v_i}=0,\\
i=1,2,\dots,n,\\
-\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
+\sum_{j=1}^n\overline{a_{n+1,j}}e^{v_j}-\overline{a_{n+1,n+1}}e^{v_{n+1}}=0,
 \end{gathered}  \label{e2.22}
\end{equation}
 satisfies $\| (v_1^\ast,v_2^\ast,\dots ,v_{n+1}^\ast)^T\|
 =\sum_{i=1}^{n+1}|v_i^\ast|<B$ (if it exists) and
 $\sum_{i=1}^{n+1}C_i<B$,
 where for $i=1,2,\dots,n$,
\begin{align*}
C_i=&\max\big\{|\ln A_0|, \Big|\frac{1}{\overline{a_{ii}}}
\Big(\overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\\
&-\overline{a_{i,n+1}}(A_0\sum_{j=1}^n\overline{a_{n+1,j}}-\overline{r_{n+1}})/
\overline{a_{n+1,n+1}}
\Big|\big\}
\end{align*}
and
\begin{equation}
\begin{aligned}
&C_{n+1}\\
&=\max\big\{|\ln\frac{A_0\sum_{j=1}^n\overline{a_{n+1,j}}-\overline{r_{n+1}}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}{\overline{a_{n+1,n+1}}}|,\\
&\big|\frac{\sum_{i=1}^n\overline{a_{n+1,i}}(\overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}})/\overline{a_{ii}}-\overline{r_{n+1}}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}}
{\overline{a_{n+1,n+1}}+(\sum_{j=1}^n\overline{a_{n+1,j}}\,\overline{a_{j,n+1}})/\overline{a_{jj}}}\big|
\big\}
\end{aligned} \label{e2.23}
\end{equation}
in which
 $$
 A_0=\max_{1\leq i\leq n}\{\overline{r_i}/\overline{a_{ii}}\}.
$$

 Now, we take
$\Omega=\{y\in X: \|y\|<B\}$. Thus, the condition (a) of
 Lemma \ref{lem1} is satisfied.  When $y \in\partial\Omega\cap
\ker L=\partial\Omega\cap R^{n+1}, y=(y_1,y_2,...,y_{n+1})^T$ is a
constant vector in $R^{n+1}$ with $\|y\|=B$. If system \eqref{e2.22} has
solutions, then
  \begin{align*}
&QN(y_1,\dots,y_n,y_{n+1})^T \\
&=\Big( \begin{pmatrix}
\overline{r_1}-\sum_{j=2}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{1k}}-\overline{a_{11}}e^{y_1}-\overline{a_{1,n+1}}e^{y_{n+1}}
+\sum_{j=2}^n\overline{D_j}e^{y_j-y_1}\\
 \vdots\\
\overline{r_n}-\sum_{j=1}^{n-1}\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{nk}}-\overline{a_{nn}}e^{y_n}-\overline{a_{n,n+1}}e^{y_{n+1}}
+\sum_{j=1}^{n-1}\overline{D_j}e^{y_j-y_n}\\
 -\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
 +\sum_{j=1}^n\overline{a_{n+1,j}}e^{y_j}-\overline{a_{n+1,n+1}}e^{y_{n+1}}
 \end{pmatrix},\\
 &\quad \Big\{\begin{pmatrix}0\\ \vdots\\0\end{pmatrix}\Big\}_{k=1}^{p}\Big)
 \neq (0,\dots,0,0)^T.
 \end{align*}
 If system \eqref{e2.22} does not have a solution, then we can directly derive
 $$
 QN \begin{pmatrix}
 y_1\\\vdots\\y_{n+1}
 \end{pmatrix}
 \neq\Big( \Big\{\begin{pmatrix}0\\\vdots\\0\end{pmatrix}\Big\}_{i=1}^{n+1},
 \Big\{\begin{pmatrix}0\\\vdots\\0\end{pmatrix}\Big\}_{k=1}^{p}\Big).
 $$
  Thus, the condition (b) in Lemma \ref{lem1} is satisfied.

  Finally, we will prove that the condition (c) in Lemma \ref{lem1} is
satisfied. To this end, we define $\phi: DomL \times [0,1]\rightarrow X$ by
\begin{align*}
&\phi(y_1, \dots, y_{n+1},\mu)\\
&= \begin{pmatrix}
\overline{r_1}-\sum_{j=2}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{1k}}
-\overline{a_{11}}e^{y_1} \\
 \vdots\\
\overline{r_n}-\sum_{j=1}^{n-1}\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{nk}}
-\overline{a_{nn}}e^{y_n}\\
 -\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
 +\sum_{j=1}^n\overline{a_{n+1,j}}e^{y_j}-\overline{a_{n+1,n+1}}e^{y_{n+1}}
 \end{pmatrix}\\
 &\quad + \mu\begin{pmatrix}
 -\overline{a_{1,n+1}}e^{y_{n+1}}+\sum_{j=2}^n\overline{D_j}e^{y_j-y_1}\\
 \vdots\\
-\overline{a_{nn}}e^{y_n}-\overline{a_{n,n+1}}e^{y_{n+1}}
+\sum_{j=1}^{n-1}\overline{D_j}e^{y_j-y_n}\\ 0
 \end{pmatrix},
 \end{align*}
 where $\mu$ is a parameter. When
 $(y_1,y_2,\dots,y_{n+1})^T \in \partial\Omega\cap
R^{n+1}$, $(y_1,y_2,\dots,y_{n+1})^T$ is a constant vector in
$R^{n+1}$ with $\|y\|= B$.

 We will show that when $(y_1,y_2,\dots,y_{n+1})^T\in \partial\Omega\cap
 \ker L$, $\phi(y_1,y_2,\dots,y_{n+1},\mu)\neq 0$. Otherwise,
there is a constant vector $(y_1, \dots, y_{n+1})^T\in R^{n+1}$
with $\|y\|=B$ satisfying $\phi(y_1,y_2,\dots,y_{n+1},\mu)= 0$,
that is
\begin{gather*}
\overline{r_i}-\sum_{j=1,j\neq i
}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}-\overline{a_{ii}}e^{y_i}
-\mu\overline{a_{i,n+1}}e^{y_{n+1}} +\mu\sum_{j=1,j\neq i}^n\overline{D_j}e^{y_j-y_i}=0,\\
i=1,2,\dots,n,\\
  -\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
+\sum_{j=1}^n\overline{a_{n+1,j}}e^{y_j}-\overline{a_{n+1,n+1}}e^{y_{n+1}}=0.
 \end{gather*}
 By similar argument in \eqref{e2.11}, \eqref{e2.13}, \eqref{e2.18} and \eqref{e2.20}, we have
$$
|y_i|\leq C_i, i=1,\dots,n+1,
$$
where $C_i$ is defined by \eqref{e2.23}. Thus, we have
 $ \sum_{i=1}^{n+1}|y_i|\leq
\sum_{i=1}^{n+1}C_i<B,$
 which is leads to a contradiction. Using the property of topological
degree and taking
$$
J:\mathop{\rm Im}Q\to X,\quad
\Big(\begin{pmatrix}y_1\\\vdots\\y_{n+1}
\end{pmatrix},\Big\{\begin{pmatrix}0\\\vdots
\\0\end{pmatrix}\Big\}_{k=1}^p\Big)\rightarrow
 \begin{pmatrix}y_1\\\vdots\\y_{n+1}
\end{pmatrix},
 $$
  we have
\begin{align*}
&\deg (JQN(y_1, \dots, y_{n+1})^T, \Omega\cap\ker L,(0,\dots,0)^T)\\
=&\deg (\phi(y_1, \dots, y_{n+1}, 1), \Omega\cap\ker L, (0,\dots,0)^T)\\
=&\deg (\phi(y_1, \dots,y_{n+1}, 0), \Omega\cap\ker L, (0,\dots,0)^T)\\
=&\deg \Big(\Big(\overline{r_1}-\sum_{j=2}^n\overline{D_j}
-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{1k}}
-\overline{a_{11}}e^{y_1},\dots,
 \overline{r_n}-\sum_{j=2}^n\overline{D_j} \\
&\quad  -\frac{1}{\omega}\ln\prod_{k=1}^p
 \frac{1}{1-c_{nk}}
 -\overline{a_{nn}}e^{y_n},
 -\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\\
 &\quad +\sum_{j=1}^{n-1}\overline{a_{n+1,j}}e^{y_j}
 -\overline{a_{n+1,n+1}}e^{y_{n+1}}\Big)^T,
  \Omega\cap\ker L, (0,\dots,0)^T\Big).
\end{align*}
Under assumption (H1), one can easily show that the
system of algebraic equations
\begin{equation}
\begin{gathered}
\overline{r_i}-\sum_{j=1,j\neq
i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}-\overline{a_{ii}}u_i=0,\quad
i=1,2,\dots,n,\\
-\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}
+\sum_{j=1}^n\overline{a_{n+1,j}}u_j-\overline{a_{n+1,n+1}}u_{n+1}=0,
 \end{gathered}  \label{e2.24}
\end{equation}
 has a unique solution $(u_1^\ast, \dots, u_{n+1}^\ast)^T$ which
satisfies
$$
 u_i^\ast= \frac{ \overline{r_i}-\sum_{j=1,j\neq
 i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}
 }{\overline{a_{ii}}}>0
$$
for $i=1, \dots, n$, and
\begin{align*}
 u_{n+1}^\ast&=\frac{1}{\overline{a_{n+1,n+1}}}
 \Big( \sum_{j=1}^n\overline{a_{n+1,j}}\big(\overline{r_j}-\sum_{j=1,j\neq
 i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\big)/(\overline{a_{jj}})\\
 &\quad -\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\Big)\\
&\geq \frac{1}{\overline{a_{n+1,n+1}}}
 \Big( \sum_{j=1}^na_{n+1,j}^M\big(\overline{r_j}-\sum_{j=1,j\neq
 i}^n\overline{D_j}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{ik}}\big)/(\overline{a_{jj}})\\
 &\quad -\overline{r_{n+1}}-\frac{1}{\omega}\ln\prod_{k=1}^p\frac{1}{1-c_{n+1,k}}\Big)
  >0.
\end{align*}
A direct calculation shows that
\begin{align*}
 &\deg (JQN(y_1,y_2,\dots,y_{n+1})^T,\Omega\cap KerL,(0,0,\dots,0)^T)\\
&=\mathop{\rm sgn}\big(\prod_{i=1}^{n+1}(-\overline{a_{ii}})\big)=(-1)^{n+1}\neq
0.
\end{align*}
Finally, easily we show that the set
$\{K_P(I-Q)Nu|u\in\bar{\Omega}\}$ is equicontinuous and uniformly bounded.
Using Lemma \ref{lem2} and the Arzela-Ascoli Theorem, we see that
$K_P(I-Q)N:\overline{\Omega}\to X$ is compact. Moreover,
$QN(\bar{\Omega})$ is bounded. Consequently, $N$ is $L-$compact.

 We have proved that $\Omega$ satisfies
all the requirements in Lemma \ref{lem1}. Hence, system \eqref{e2.2} has at least
one $\omega$-periodic solution. Accordingly, system \eqref{e1.1} has at
least one positive $\omega$-periodic solution. This completes the
proof.
\end{proof}

\subsection*{Remark}
In this paper, by borrowing Gaines and
Mawhin's continuation theorem of coincidence degree theory, we
have established sufficient conditions for the existence of
positive periodic solutions to system \eqref{e1.1} with initial
conditions \eqref{e1.2}. We would like to mention here that it is
interesting but challenging to discuss the existence of positive
periodic solutions of \eqref{e1.1} when we incorporate time delays
to the self-regulated terms of the prey in n-patch environments.
We leave this for our future work.

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