\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 88, pp. 1--11.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/88\hfil Positive periodic solutions] {Positive periodic solutions for a predator-prey model with time delays and impulsive effect} \author[S. Gao, Y. Li\hfil EJDE-2008/88\hfilneg] {Shan Gao, Yongkun Li} % in alphabetical order \address{Shan Gao \newline Department of Mathematics, Yunnan University\\ Kunming, Yunnan, 650091, China} \email{2002711036@163.com} \address{Yongkun Li \newline Department of Mathematics, Yunnan University\\ Kunming, Yunnan, 650091, China} \email{yklie@ynu.edu.cn} \thanks{Submitted April 7, 2008. Published June 21, 2008.} \thanks{Supported by the National Natural Sciences Foundation of China and the Natural Sciences \hfill\break\indent Foundation of Yunnan Province, China} \subjclass[2000]{34K13, 34K45, 92D25} \keywords{Predator-prey model; impulse; positive periodic solution; \hfill\break\indent coincidence degree} \begin{abstract} In this article, a two-species predator-prey model with time delays and impulsive effect is investigated. By using Mawhin's continuation theorem of coincidence degree theory, sufficient conditions are obtained for the existence of positive periodic solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In the past few years, predator-prey models and with many kinds of functional responses have been of great interest to both applied mathematicians and ecologists see references in this article. Recently, by using Floquet theory of linear periodic impulsive equation, Song and Li \cite{s3} considered the following $T$-periodic predator-prey model with modified Leslie-Gower and Holling-type II schemes and impulsive effect \begin{gather*} \left.\begin{gathered} \dot{x}(t)=x(t)\Big(r_{1}(t)-b_{1}(t)x(t)-\frac{a_{1}(t)y(t)}{x(t)+k_{1}(t)} \Big)\\ \dot{y}(t)=y(t)\Big(r_{2}(t)-\frac{a_{2}(t)y(t)}{x(t)+k_{2}(t)}\Big) \end{gathered} \right\} \quad t\neq\tau_{k},\; k\in\mathbb{Z}_{+}, \\ \left.\begin{gathered} x(\tau_{k}^{+})=(1+h_{k})x(\tau_{k})\\ y(\tau_{k}^{+})=(1+g_{k})y(\tau_{k}) \end{gathered} \right\} \quad t=\tau_{k},\; k\in\mathbb{Z}_{+}. \end{gather*} where $b_{1}(t), r_{i}(t), a_{i}(t), k_{i}(t) (i=1,2)$ are continuous $\omega$-periodic functions such that $b_{1}(t)>0, r_{i}(t)>0, a_{i}(t)>0, k_{i}(t)>0 (i=1,2)$ and $\mathbb{Z}_{+}=\{1,2,\dots\}$; $h_{k},g_{k} (k\in \mathbb{Z}_{+})$ are constants and there exists an integer $q>0$ such that $h_{k+q}=h_{k},g_{k+q}=g_{k},\tau_{k+q}=\tau_{k}+\omega$, and $1+h_{k}>0, 1+g_{k}>0$ for all $k\in \mathbb{Z}_{+}$. They obtain some conditions for the linear stability of trivial periodic solution and semitrivial periodic solutions. However, as pointed out in \cite{l2}, naturally, more realistic and interesting models of single or multiple species growth should take into account both the seasonality of the changing environment and the effects of time delays. In this paper, we consider the following $\omega$-periodic predator-prey system with time delays and impulses: \begin{equation} \label{e11} \begin{gathered} \left.\begin{gathered} \dot{x}(t)=x(t)\Big(r_{1}(t)-b_{1}(t)x(t-\tau(t)) -\frac{a_{1}(t)y(t-\sigma_{1}(t))}{x(t-\tau_{1}(t))+k_{1}(t)}\Big)\\ \dot{y}(t)=y(t)\Big(r_{2}(t)-\frac{a_{2}(t)y(t-\sigma_{2}(t))} {x(t-\tau_{2}(t))+k_{2}(t)}\Big) \end{gathered} \right\} \quad t\neq t_{k},\; k\in\mathbb{Z}_{+}, \\ \left.\begin{gathered} x(t_{k}^{+})=I_{k}(x(t_{k}))+x(t_{k}^{-})\\ y(t_{k}^{+})=J_{k}(y(t_{k}))+y(t_{k}^{-}) \end{gathered} \right\} \quad t=t_{k},\; k\in\mathbb{Z}_{+}. \end{gathered} \end{equation} where $x(t_{k}^{+}),x(t_{k}^{-}),y(t_{k}^{+}),y(t_{k}^{-})$ represent the right and the left limit of $x(t_{k})$ and $y(t_{k})$, respectively, in this paper, we assume that $x$, $y$ are left continuous at $t_{k}$; $b_{1}(t)$, $\tau(t)$, $a_{i}(t)$, $r_{i}(t)$, $k_{i}(t)$, $\sigma_{i}(t)$, $\tau_{i}(t)$ ($i=1,2$) are all positive periodic continuous functions with period $\omega>0$ and $\mathbb{Z}_{+}=\{1,2,\dots\}$; $I_{k},J_{k}\in C(\mathbb{R}^{+},\mathbb{R})$ satisfy that $I_{k}(u)>-u, J_{k}(v)>-v$, and there exists a positive integer $p$ such that $t_{k+p}=t_{k}+\omega$, $I_{k+p}=I_{k}$, $J_{k+p}=J_{k}$, $k\in \mathbb{Z_+}$. Without loss of generality, we also assume that $[0,\omega) \cap \{t_{k}: k\in \mathbb{Z_+}\}=\{t_{1},t_{2},\dots,t_{p}\}$. Our purpose of this paper is by using continuation theorem of coincidence degree theory \cite{g1} to establish criteria to guarantee the existence of positive periodic solutions of system \eqref{e11}. \section{Notation and preliminaries} To obtain our main result of this paper, we first need to make the following preparations. For any non-negative integer $q$, let \begin{align*} &C^{(q)}[0,\omega;\,t_{1},t_{2},\dots,t_{p}]\\ & = \Big\{x:[0,\omega]\to\mathbb{R}\text{ such that $x^{(q)}(t)$ exists for $t \neq t_{1}, \dots, t_{p}$; $x^{(q)}(t_{k}^{+}),\; x^{(q)}(t_{k}^{-})$} \\ &\quad \text{exists at $t_{1},\dots,t_{p}$; and $x^{(j)}(t_{k}) = x^{(j)}(t_{k}^{-})$, $k =1,2,\dots,p$, $j = 0,1,2,\dots,q$}\Big\} \end{align*} with the norm \[ \|x\|_{q} =\max\{\sup_{t\in[0,\omega]}|x^{(j)}(t)|\}_{j=0}^{q}. \] It is easy to see that $C^{(q)}[0,\omega; t_{1},t_{2},\dots,t_{p}]$ is a Banach space and the functions in $C[0,\omega;\ t_{1},t_{2},\dots,t_{p}]$ are continuous with respect to $t$ different from $t_{1},t_{2},\dots,t_{p}$. Let \[ PC_{\omega}=\big\{x\in C[0,\omega;t_{1},t_{2},\dots, t_{p}]: x(0)=x(\omega)\big\} \] with the same norm as that of $C[0,\omega;t_{1},t_{2}, \dots,t_{p}]$. Let $X, Y$ be normed vector spaces, $L :$ Dom\,$L \subset X \to Y$ be a linear mapping, and $N : X \to Y$ be a continuous mapping. The mapping $L$ will be called a Fredholm mapping of index zero if dim $\ker L = \mathop{\rm codim}\mathop{\rm Im}L < +\infty$ and $\mathop{\rm Im}L$ is closed in $Y$. If $L$ is a Fredholm mapping of index zero, and there exist continuous projectors: $P : X \to X$ and $Q : Y \to Y$ such that $\mathop{\rm Im}P =\ker L$, $\ker Q =\mathop{\rm Im}L =\mathop{\rm Im}(I-Q)$. It follows that mapping $L |_{\mathop{\rm Dom}L \cap \ker P}: (I-P)X \to \mathop{\rm Im}L$ is invertible. We denote the inverse of that mapping by $K_{P}$. If $\Omega$ is an open bounded subset of $X$, the mapping $N$ will be called $L$-compact on $\overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and $K_{P}(I-Q)N : \overline{\Omega} \to X$ is compact. Since $\mathop{\rm Im}Q$ is isomorphic to $\ker L$, there exists an isomorphism $J :\mathop{\rm Im}Q \to \ker L$. \begin{definition} \label{def21} \rm The set $F$ is said to be quasi-equicontinuous in $[0,\omega]$ if for any $\epsilon>0$ there exists $\delta>0$ such that if $x\in F$, $k\in\mathbb{Z}_{+}$, $t_{1},t_{2}\in(t_{k-1},t_{k})\cap[0,\omega]$, $|t_{1}-t_{2}|<\delta$, then $|x(t_{1})-x(t_{2})|<\epsilon$. \end{definition} \begin{lemma}[\cite{b1}] \label{lem21} The set $F\subset PC_{\omega}$ is relatively compact if and only if \begin{itemize} \item[(1)] $F$ is bounded, that is, $\|f\|_{PC_{\omega}}=\|f\|_{0}=\sup_{t\in[0,\omega]}|f(t)|\leq M$ for each $f\in F$ and some $M>0$; \item[(2)] $F$ is quasi-equicontinuous in $\mathop{\rm Dom}f$. \end{itemize} \end{lemma} Now, we introduce Mawhin's continuation theorem. \begin{lemma}[\cite{g1}] \label{lem22} Let $\Omega\subset X$ be an open bounded set and let $N :X\to Y$ be a continuous operator which is $L$-compact on $\overline{\Omega}$. 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 $x\in\partial\Omega\cap\ker L$, $QNx\neq0$, \item[(c)] $\deg(JQN,\Omega\cap\ker L,0)\neq0$. \end{itemize} Then $Lx=Nx$ has at least one solution in $\overline{\Omega}\cap\mathop{\rm Dom}L$. \end{lemma} Throughout this paper, we assume that there exist $p_{1k},p_{2k},q_{1k},q_{2k}\in\mathbb{R}$, $k\in Z_+$ such that \begin{gather*} \inf_{u>0}\frac{I_{k}(u)}{u}\geq q_{1k}>-1,\quad \sup_{u>0}\frac{I_{k}(u)}{u}\leq p_{1k}<+\infty, \\ \inf_{v>0}\frac{J_{k}(v)}{v}\geq q_{2k}>-1,\quad \sup_{v>0}\frac{J_{k}(v)}{v}\leq p_{2k}<+\infty. \end{gather*} For convenience, we introduce the notation \begin{gather*} \overline{f}=\frac{1}{\omega}\int_{0}^{\omega}f(t)\,\mathrm{d}t,\quad f^{M}=\max_{t\in[0,\,\omega]}\{f(t)\},\\ l_{1k}=\max\big\{|\ln(1+p_{1k})|,|\ln(1+q_{1k})|\big\},\quad l_{2k}=\max\big\{|\ln(1+p_{2k})|,|\ln(1+q_{2k})|\big\} \end{gather*} where $f$ is a continuous $\omega$-periodic function and $k\in Z_+$. \section{Main result} Let \begin{gather*} H_{1}=\ln\Big(\frac{\overline{r_{1}}+\frac{1}{\omega} \ln(\prod_{k=1}^{p}(1+p_{1k}))}{\overline{b_{1}}}\Big),\\ M_{1}=H_{1}+2\omega\overline{r_{1}} +\ln\Big(\prod_{k=1}^{p}(1+p_{1k})\Big)+\sum_{k=1}^{p}l_{1k}; \\ H_{2}=\ln\Big(\frac{\overline{r_{2}}+\frac{1}{\omega} \ln(\prod_{k=1}^{p}(1+q_{2k}))} {\overline{(\frac{a_{2}}{k_{2}})}}\Big), \\ M_{2}=H_{2}-2\omega\overline{r_{2}}-\ln\Big(\prod_{k=1}^{p}(1+p_{2k})\Big) -\sum_{k=1}^{p}l_{2k}; \\ H_{3}=\ln\Big(\frac{(k_{2}^{M}+\mathrm{e}^{M_{1}})(\overline{r_{2}} +\frac{1}{\omega}\ln(\prod_{k=1}^{p} (1+p_{2k})))}{\overline{a_{2}}}\Big),\\ M_{3}=H_{3}+2\omega\overline{r_{2}}+\ln\Big(\prod_{k=1}^{p}(1+p_{2k})\Big) +\sum_{k=1}^{p}l_{2k}; \\ H_{4}=\ln\Big(\frac{\overline{r_{1}}+\frac{1}{\omega} \ln(\prod_{k=1}^{p}(1+q_{1k}))-\mathrm{e}^{M_{3}} \overline{(\frac{a_{1}}{k_{1}})}}{\overline{b_{1}}}\Big), \\ M_{4}=H_{4}-2\omega\overline{r_{1}}-\ln\Big(\prod_{k=1}^{p}(1+p_{1k})\Big) -\sum_{k=1}^{p}l_{1k}; \end{gather*} Our main result of this paper is as follows: \begin{theorem} \label{thm31} If \begin{gather*} \overline{r_{1}}+\frac{1}{\omega}\ln\Big(\prod_{k=1}^{p}(1+p_{1k})\Big)>0,\\ \overline{r_{2}}+\frac{1}{\omega}\ln\Big(\prod_{k=1}^{p}(1+q_{2k})\Big)>0,\\ \overline{r_{1}}+\frac{1}{\omega}\ln\Big(\prod_{k=1}^{p}(1+q_{1k})\Big)- \mathrm{e}^{M_{3}}\overline{(\frac{a_{1}}{k_{1}})}>0, \end{gather*} then \eqref{e11} has at least one $\omega$-periodic positive solution. \end{theorem} \begin{proof} Let $x(t)=\mathrm{e}^{u(t)},\ y(t)=\mathrm{e}^{v(t)}$ then \eqref{e11} is reformulated as \begin{equation} \label{e31} \begin{gathered} \left.\begin{gathered} \dot{u}(t)=r_{1}(t)-b_{1}(t)\exp\{u(t-\tau(t))\} -\frac{a_{1}(t)\exp\{v(t-\sigma_{1}(t))\}} {\exp\{u(t-\tau_{1}(t))\}+k_{1}(t)}\\ \dot{v}(t)=r_{2}(t)-\frac{a_{2}(t) \exp\{v(t-\sigma_{2}(t))\}}{\exp\{u(t-\tau_{2}(t))\}+k_{2}(t)} \end{gathered} \right\} \\ \text{ for $t\neq t_{k},\; k\in\mathbb{Z}_{+}$, and} \\ \left.\begin{gathered} u(t_{k}^{+})=e_{k}(u(t_{k}))+u(t_{k}^{-})\\ v(t_{k}^{+})=f_{k}(v(t_{k}))+v(t_{k}^{-}) \end{gathered} \right\} \quad t=t_{k},k\in\mathbb{Z}_{+}, \end{gathered} \end{equation} where \begin{gather*} e_{k}(u(t_{k}))=\ln\Big(\frac{I_{k}(\exp\{u(t_{k})\}) +\exp\{u(t_{k})\}}{\exp\{u(t_{k})\}}\Big), \\ f_{k}(v(t_{k}))=\ln\Big(\frac{J_{k}(\exp\{v(t_{k})\}) +\exp\{v(t_{k})\}}{\exp\{v(t_{k})\}}\Big). \end{gather*} It is easy to see that \begin{gather*} \ln(1+q_{1k})\leq e_{k}(u(t_{k})) \leq \ln(1+p_{1k}), \\ \ln(1+q_{2k})\leq f_{k}(v(t_{k})) \leq \ln(1+p_{2k}). \end{gather*} If system \eqref{e31} has an $\omega$-periodic solution $(u(t),v(t))$, then \[ (\mathrm{e}^{u(t)},\ \mathrm{e}^{v(t)})= (x^{\ast}(t),\ y^{\ast}(t)) \] is a positive $\omega$-periodic solution to system \eqref{e11}. So, in the following, we discuss the existence of $\omega$-periodic solution to system $\eqref{e31}$. Here, we denote \begin{gather*} A(t)= r_{1}(t)-b_{1}(t)\exp\{u(t-\tau(t))\}-\frac{a_{1}(t)\exp\{v(t-\sigma_{1}(t))\}} {\exp\{u(t-\tau_{1}(t))\}+k_{1}(t)},\\ B(t)= r_{2}(t)-\frac{a_{2}(t)\exp\{v(t-\sigma_{2}(t))\}}{\exp\{u(t-\tau_{2}(t))\}+k_{2}(t)}. \end{gather*} To use the continuation theorem of coincidence degree theory to establish the existence of an $\omega$-periodic solution of \eqref{e31}, we take \[ X= PC_{\omega} \times PC_{\omega},\quad Y = X \times \mathbb{R}^{2p}. \] Then $X$ is a Banach space with the norm \[ \|x\|_{X} = \|(u,v)\|_{X} = \|u\|_{0} + \|v\|_{0} = \sup_{t\in[0, \omega]}|u(t)| + \sup_{t\in[0, \omega]}|v(t)|, \] and $Y$ is also a Banach space with the norm \[ \|z\|_{Y} = \|x\|_{X}+\|y\|_{2},\quad x\in X,\; y\in\mathbb{R}^{2p}, \] where $\|\cdot\|_{2}$ in $\mathbb{R}^{n}$ is defined as \[ \|\xi\|_{2}=\|(\xi_{1},\xi_{2},\dots,\xi_{n})\|_{2}=\sum_{i=1}^{n}|\xi_{i}|. \] So if $x=(u,v)\in X\cap\mathbb{R}^{2}$, then $\|x\|_{X}=\|x\|_{2}$. Let \begin{gather*} \mathop{\rm Dom}L = \Big\{(u,v): u,v \in C^{(1)}[0,\omega; t_{1}, t_{2}, \dots, t_{p}]; u(0)=u(\omega), v(0)=v(\omega)\Big\}, \\ L : \mathop{\rm Dom}L \cap X \to Y \\ (u,v) \to (\dot{u},\dot{v},\Delta u(t_{1}),\dots,\Delta u(t_{p}),\Delta v(t_{1}), \dots,\Delta v(t_{p})), \end{gather*} and let $ N : X \to Y$ with \[ N(x)=N(u,v)=\big(A(t),B(t),\Delta u(t_{1}),\dots,\Delta u(t_{p}),\Delta v(t_{1}),\dots,\Delta v(t_{p})\big). \] Obviously, $\ker L = \Big\{(u,v): u,v \in \mathbb{R}, t\in[0, \omega] \Big\} = \mathbb{R}^{2}$, \begin{align*} \mathop{\rm Im}L &= \Big\{z=(f,g,c_{1},\dots,c_{p},d_{1},\dots,d_{p}) \in Y : \int_{0}^{\omega}f(s)\,\mathrm{d}s+\sum_{k=1}^{p}c_{k}=0,\\ &\quad \int_{0}^{\omega}g(s)\,\mathrm{d}s + \sum_{k=1}^{p}d_{k}=0\,\Big\} \end{align*} and $\dim \ker L = \mathop{\rm codim}\mathop{\rm Im}L = 2$. So that, $\mathop{\rm Im}L$ is closed in $Y$, $L$ is a Fredholm mapping of index zero. Define the two projectors \begin{gather*} Px=\frac{1}{\omega},\int_{0}^{\omega}x(t)\,\mathrm{d}t,\\ Qz=\Big(\frac{1}{\omega}\big[\int_{0}^{\omega}f(s)\,\mathrm{d}s +\sum_{k=1}^{p}c_{k}\big], \frac{1}{\omega}\,\big[\int_{0}^{\omega}g(s)\,\mathrm{d}s +\sum_{k=1}^{p}d_{k}\big], \underbrace{0,\dots,0}_{2p}\Big). \end{gather*} It is easy to show that $P$ and $Q$ are continuous and satisfy \[ \mathop{\rm Im}P=\ker L=\mathbb{R}^{2},\quad \mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q). \] Further, let $L_{P}=L\big|_{\mathop{\rm Dom}L\cap\ker P}$ and the generalized inverse $K_{P}=L_{P}^{-1}$ is given by \begin{align*} K_{P}z&=\Big(\int_{0}^{t}f(s)\,\mathrm{d}s+\sum_{t>t_{k}}c_{k} -\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}f(s)\,\mathrm{d}s\,\mathrm{d}t -\sum_{k=1}^{p}c_{k}+\frac{1}{\omega}\sum_{k=1}^{p}t_{k}c_{k},\\ & \quad\int_{0}^{t}g(s)\,\mathrm{d}s+\sum_{t>t_{k}}d_{k} -\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}g(s)\,\mathrm{d}s\,\mathrm{d}t -\sum_{k=1}^{p}d_{k}+\frac{1}{\omega}\sum_{k=1}^{p}t_{k}d_{k}\,\Big). \end{align*} Thus, the expression of $QNx$ is \[ \Big(\frac{1}{\omega}\big[\int_{0}^{\omega}A(s)\, \mathrm{d}s\,+\,\sum_{k=1}^{p}e_{k}(u(t_{k}))\big] ,\ \frac{1}{\omega}\,\big[\int_{0}^{\omega}B(s)\,\mathrm{d}s\ +\,\sum_{k=1}^{p}f_{k}(v(t_{k}))\big],\ \underbrace{0, \dots, 0}_{2p}\Big), \] and then \begin{align*} &K_{P}(I-Q)Nx\\ &=\Big(\int_{0}^{t}A(s)\,\mathrm{d}s+\sum_{t>t_{k}}e_{k}(u(t_{k})) -\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}A(s)\,\mathrm{d}s\,\mathrm{d}t +\frac{1}{\omega}\sum_{k=1}^{p}t_{k}e_{k}(u(t_{k}))\\ &\quad +(\frac{1}{2}-\frac{t}{\omega})\,\int_{0}^{\omega}A(s)\,\mathrm{d}s -(\frac{1}{2}+\frac{t}{\omega})\,\sum_{k=1}^{p}e_{k}(u(t_{k}))\ ,\\ & \quad \int_{0}^{t}B(s)\,\mathrm{d}s+\sum_{t>t_{k}}f_{k}(v(t_{k})) -\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}B(s)\,\mathrm{d}s\,\mathrm{d}t +\frac{1}{\omega}\sum_{k=1}^{p}t_{k}f_{k}(v(t_{k}))\\ &\quad +(\frac{1}{2}-\frac{t}{\omega})\,\int_{0}^{\omega}B(s)\,\mathrm{d}s -(\frac{1}{2}+\frac{t}{\omega})\,\sum_{k=1}^{p}f_{k}(v(t_{k}))\Big). \end{align*} Hence, $QN$ and $K_{P}(I-Q)N$ are both continuous. Using Lemma \ref{lem21}, it is easy to show that $K_{P}(I-Q)N(\overline{\Omega})$ is compact for any open bounded set $\Omega\subset X$. Moreover, $QN(\overline{\Omega})$ is bounded. Therefore, $N$ is $L$-compact on $\overline{\Omega}$ for any open bounded set $\Omega\subset X$. Now, it needs to show that there exists a domain $\Omega$ that satisfies all the requirements given in Lemma \ref{lem22}. Corresponding to operator equation $Lx=\lambda Nx$, $\lambda\in(0,1)$, $x=(u,v)$, we get \begin{equation} \label{e32} \begin{gathered} \left.\begin{gathered} \dot{u}(t)=\lambda\Big[r_{1}(t)-b_{1}(t)\exp\{u(t-\tau(t))\} -\frac{a_{1}(t)\exp\{v(t-\sigma_{1}(t))\}}{\exp\{u(t-\tau_{1}(t))\} +k_{1}(t)}\Big]\\ \dot{v}(t)=\lambda\Big[r_{2}(t)-\frac{a_{2}(t)\exp\{v(t-\sigma_{2}(t))\}} {\exp\{u(t-\tau_{2}(t))\}+k_{2}(t)}\Big] \end{gathered} \right\}\\ \text{for $t\neq t_{k}$, $k\in\mathbb{Z}_{+}$, and} \\ \left.\begin{gathered} u(t_{k}^{+})=\lambda e_{k}(u(t_{k}))+u(t_{k}^{-})\\ v(t_{k}^{+})=\lambda f_{k}(v(t_{k}))+v(t_{k}^{-}) \end{gathered} \right\} \quad t=t_{k},\; k\in\mathbb{Z}_{+}. \end{gathered} \end{equation} Suppose $x=(u,v)$ is an $\omega$-periodic solution to system \eqref{e32}. By integrating \eqref{e32} over $[0,\omega]$ we obtain \begin{equation} \label{e33} \begin{gathered} \begin{aligned} &\int_{0}^{\omega}\,r_{1}(t)\,\mathrm{d}t+\sum_{k=1}^{p}\,e_{k}(u(t_{k}))\\ &=\int_{0}^{\omega}\Big[b_{1}(t)\exp\{u(t-\tau(t))\} +\frac{a_{1}(t)\exp\{v(t-\sigma_{1}(t))\}}{\exp\{u(t-\tau_{1}(t))\} +k_{1}(t)}\Big]\,\mathrm{d}t, \end{aligned}\\ \int_{0}^{\omega}\,r_{2}(t)\,\mathrm{d}t+\sum_{k=1}^{p}\,f_{k}(v(t_{k})) =\int_{0}^{\omega}\frac{a_{2}(t)\exp\{v(t-\sigma_{2}(t))\}}{\exp\{u(t-\tau_{2}(t))\}+k_{2}(t)}\,\mathrm{d}t. \end{gathered} \end{equation} From \eqref{e32} and \eqref{e33}, we have \begin{gather} \int_{0}^{\omega}\,|\dot{u}(t)|\,\mathrm{d}t <2\int_{0}^{\omega}r_{1}(t)\,\mathrm{d}t +\sum_{k=1}^{p}\,e_{k}(u(t_{k})), \label{e34}\\ \int_{0}^{\omega}\,|\dot{v}(t)|\,\mathrm{d}t <2\int_{0}^{\omega}r_{2}(t)\,\mathrm{d}t +\sum_{k=1}^{p}\,f_{k}(v(t_{k})). \label{e35} \end{gather} Since $u(t),v(t)\in PC_{\omega}$, there exist $\xi_{1},\xi_{2},\eta_{1},\eta_{2}\in[0,\omega]$ such that \begin{equation} \label{e36} \begin{gathered} u(\xi_{1})=\min_{t\in[0,\omega]}u(t),\quad u(\eta_{1})=\max_{t\in[0,\omega]}u(t),\\ v(\xi_{2})=\min_{t\in[0,\omega]}v(t),\quad v(\eta_{2})=\max_{t\in[0,\omega]}v(t). \end{gathered} \end{equation} Then by \eqref{e33} and \eqref{e36}, we obtain \begin{gather*} \overline{r_{1}}+\frac{1}{\omega}\sum_{k=1}^{p}\,e_{k}(u(t_{k})) \geq\frac{1}{\omega}\int_{0}^{\omega}\,b_{1}(t)\mathrm{e}^{u(\xi_{1})}\, \mathrm{d}t\,,\\ \overline{r_{2}}+\frac{1}{\omega}\sum_{k=1}^{p}\,f_{k}(v(t_{k})) \leq\frac{1}{\omega}\int_{0}^{\omega}\,\frac{a_{2}(t) \mathrm{e}^{v(\eta_{2})}}{k_{2}(t)}\,\mathrm{d}t\,; \end{gather*} that is, \[ u(\xi_{1})\leq\ln\big[\frac{\overline{r_{1}}+ \frac{1}{\omega}\sum_{k=1}^{p}\,e_{k}(u(t_{k}))}{\overline{b_{1}}}\Big] \leq \ln\big[\frac{\overline{r_{1}}+\frac{1}{\omega} \ln\big(\prod_{k=1}^{p}(1+p_{1k})\big)}{\overline{b_{1}}}\big]=: H_{1} \] and \[ v(\eta_{2})\geq \ln\big[\frac{\overline{r_{2}}+\frac{1}{\omega} \sum_{k=1}^{p}\,f_{k}(v(t_{k}))} {\overline{\big(\frac{a_{2}}{k_{2}}\big)}}\big] \geq\ln\big[\frac{\overline{r_{2}}+\frac{1}{\omega} \ln\big(\prod_{k=1}^{p}(1+q_{2k})\big)} {\overline{\big(\frac{a_{2}}{k_{2}}\big)}}\big]=: H_{2}. \] Hence \begin{align*} u(t)&\leq u(\xi_{1})\,+\,\int_{0}^{\omega}\,|\dot{u}(t)|\,\mathrm{d}t +\sum_{k=1}^{p}\,|e_{k}(u(t_{k}))|\\ &\leq H_{1}+2\omega\overline{r_{1}}+\ln\big(\prod_{k=1}^{p}(1+p_{1k})\big) +\sum_{k=1}^{p}l_{1k}=: M_{1} \end{align*} and \begin{align*} v(t)&\geq v(\eta_{2})-\int_{0}^{\omega}\,|\dot{v}(t)|\,\mathrm{d}t-\sum_{k=1}^{p} \,|f_{k}(v(t_{k}))|\\ &\geq H_{2}-2\omega\overline{r_{2}}-\ln\big(\prod_{k=1}^{p}(1 +p_{2k})\big)-\sum_{k=1}^{p}l_{2k}=: M_{2}. \end{align*} So we have \[ \overline{r_{2}}+\frac{1}{\omega}\sum_{k=1}^{p}\,f_{k}(v(t_{k})) \geq\frac{1}{\omega}\int_{0}^{\omega}\, \frac{a_{2}(t)\mathrm{e}^{v(\xi_{2})}}{k_{2}(t)+\mathrm{e}^{M_{1}}}\, \mathrm{d}t \geq\frac{1}{\omega}\int_{0}^{\omega}\,\frac{a_{2}(t)\mathrm{e}^{v(\xi_{2})}} {k_{2}^{M}+\mathrm{e}^{M_{1}}}\,\mathrm{d}t\,; \] that is, \begin{align*} v(\xi_{2})&\leq\ln\Big(\frac{\big(k_{2}^{M} +\mathrm{e}^{M_{1}}\big)\big(\overline{r_{2}}+\frac{1}{\omega}\sum_{k=1}^{p}\,f_{k}(v(t_{k}))\big)} {\overline{a_{2}}}\Big)\\ &\leq\ln\Big(\frac{\big(k_{2}^{M} +\mathrm{e}^{M_{1}}\big)\big(\overline{r_{2}}+\frac{1}{\omega}\ln(\prod_{k=1}^{p} (1+p_{2k}))\big)}{\overline{a_{2}}}\Big)=: H_{3}. \end{align*} Thus \begin{align*} v(t)&\leq v(\xi_{2})+\int_{0}^{\omega}\,|\dot{v}(t)|\,\mathrm{d}t +\sum_{k=1}^{p}\,|f_{k}(v(t_{k}))|\\ &\leq H_{3} +2\omega\overline{r_{2}}+\ln\big(\prod_{k=1}^{p}(1+p_{2k})\big) +\sum_{k=1}^{p}l_{2k}=: M_{3}. \end{align*} Similarly, we have \[ \overline{r_{1}}+\frac{1}{\omega}\sum_{k=1}^{p}\,e_{k}(u(t_{k})) \leq\frac{1}{\omega}\int_{0}^{\omega}\,b_{1}(t)\mathrm{e}^{u(\eta_{1})}\,\mathrm{d}t +\frac{1}{\omega}\int_{0}^{\omega}\,\frac{a_{1}(t) \mathrm{e}^{M_{3}}}{k_{1}(t)}\,\mathrm{d}t; \] that is, \begin{align*} u(\eta_{1})&\geq \ln\Big[\frac{\overline{r_{1}} +\frac{1}{\omega}\sum_{k=1}^{p}\,e_{k}(v(t_{k})) -\mathrm{e}^{M_{3}}\overline{\big(\frac{a_{1}}{k_{1}}\big)}} {\overline{b_{1}}}\Big]\\ &\geq \ln\Big[\frac{\overline{r_{1}}+ \frac{1}{\omega}\ln\big(\prod_{k=1}^{p}(1+q_{1k})\big) -\mathrm{e}^{M_{3}}\overline{\big(\frac{a_{1}}{k_{1}}\big)}} {\overline{b_{1}}}\Big]=: H_{4}. \end{align*} Then \begin{align*} u(t)&\geq u(\eta_{1})-\int_{0}^{\omega}\,|\dot{u}(t)|\,\mathrm{d}t -\sum_{k=1}^{p}\,|e_{k}(u(t_{k}))|\\ &\geq H_{4}-2\omega\overline{r_{1}}-\ln\big(\prod_{k=1}^{p}(1+p_{1k})\big) -\sum_{k=1}^{p}l_{1k}=: M_{4}. \end{align*} Now, we have \[ M_{4}\leq u(t)\leq M_{1},\quad M_{2}\leq v(t)\leq M_{3}. \] Let $D=|M_{1}|+|M_{2}|+|M_{3}|+|M_{4}|$. We have \[ \|x\|_{X}=\|u\|_{0}+\|v\|_{0}\leq D. \] Clearly, $D$ is independent of $\lambda$. Denote $M=D+D_{0}$, where $D_{0}$ is taken sufficiently large such that each solution $(u^{*},v^{*})$ of \begin{equation} \label{e37} \begin{gathered} \int_{0}^{\omega}\,r_{1}(t)\,\mathrm{d}t+\sum_{k=1}^{p}\,e_{k}(u(t_{k})) =\int_{0}^{\omega}\Big[b_{1}(t)\mathrm{e}^{u} +\frac{a_{1}(t)\mathrm{e}^{v}}{\mathrm{e}^{u}+k_{1}(t)}\Big]\,\mathrm{d}t, \\ \int_{0}^{\omega}\,r_{2}(t)\,\mathrm{d}t+\sum_{k=1}^{p}\,f_{k}(v(t_{k})) =\int_{0}^{\omega}\frac{a_{2}(t)\mathrm{e}^{v}}{\mathrm{e}^{u}+k_{2}(t)}\,\mathrm{d}t \end{gathered} \end{equation} satisfies $\|(u^{*},v^{*})\|_{X}0.40-0.32=0.08>0. \end{gather*} So that, all conditions of Theorem \ref{thm31} are satisfied. Therefore, \eqref{e41} has at least one $\omega$-periodic positive solution. \begin{thebibliography}{00} \bibitem{a1}Z. Amine, R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl. 185 (1994) 477-489. \bibitem{b1} D. D. Bainov, P. 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