\documentclass[reqno]{amsart} \usepackage{graphicx} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 155, pp. 1--10.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/155\hfil Existence of periodic solutions] {Existence of periodic solutions of a delayed \\ predator-prey system on time scales} \author[D. Yang\hfil EJDE-2007/155\hfilneg] {Dandan Yang} \address{Dandan Yang \hfill\break Department of Mathematics\\ Yangzhou University \\ Yangzhou 225002, China} \email{yangdandan2600@sina.com} \thanks{Submitted July 8, 2007. Published November 21, 2007.} \subjclass[2000]{34C25, 92D25} \keywords{Time scales; solution; Fixed-point theorem; predator-prey system; \hfill\break\indent coincidence degree theorem} \begin{abstract} In this paper, we prove the existence of periodic solutions of a delayed periodic predator-prey system based on continuation theorem of coincidence degree. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In recent years, the predator-prey models together with many kinds of functional responses have been of great interest to both applied mathematicians and ecologists \cite{b3,f1,h1,u1,x1,z1}. In 2006, Yu Yang et al. \cite{y1} considered the delayed system with general functional response in Gilpin model $$\begin{gathered} x_1'(t)=x_1(t)\big[r(t)-b(t)x_1^{\theta}(t-\tau_1(t))-\frac {\alpha(t) x_1^{p-1}(t)}{1+mx_1^{p}(t)}x_2(t-\sigma(t))\big], \\ x_2'(t)=x_2(t)\big[-d(t)-a(t)x_2(t-\tau_2(t))+\frac {\beta(t)x_1^p(t-\tau_3(t))}{1+mx_1^p(t-\tau_3(t))}\big], \end{gathered} \label{e1.1}$$ where $x_1(t),x_2(t)$ represent the densities of the prey population and predator population at time t, respectively. They obtained a sufficient condition on the existence of positive periodic solutions of \eqref{e1.1} by using the continuation theorem of coincidence degree theory. In order to unify differential and difference equations, people have done a lot of research about dynamic equations on time scales \cite{a2, a3, a4, e1, m1}, since the theory of time scales is introduced by hilger in \cite{h2}. To the best of our knowledge, only a few results can be found in the literature for predator-prey system by using coincidence degree theorem on time scales. Motivated by \cite{h2,y1}, the aim of this paper is to explore the existence of periodic solutions of the delayed predator-prey system with general functional response, which the prey population growth satisfies Gilpin model on time scales $$\begin{gathered} z_1^{\Delta}(t)=r(t)-b(t)\exp\{{\theta}z_1(t-\tau_1(t))\} -\frac{\alpha(t) \exp\{(p-1)z_1(t)+z_2(t-\sigma(t))\}}{1+m \exp\{pz_1(t)\}},\\ z_2^{\Delta}(t)=-d(t)-a(t)\exp\{z_2(t-\tau_2(t))\} +\frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}}{1+m \exp\{pz_1(t-\tau_3(t))\}}, \end{gathered} \label{e1.2}$$ for $t\in \mathbb{T}$. As we see, if $x_1(t)=\exp{z_1(t)}$, $x_2(t)=\exp{z_2(t)}$, and $\mathbb{T}=\mathbb{R}$, then \eqref{e1.2} reduces to \eqref{e1.1}. The rest of this paper is organized as follows. In section 2, we present some preliminaries, including basic definitions time scales and coincidence degree theorems. We give our main result in section 3 based on the continuation theorem of coincidence degree theorem \cite{g1}. In the last section, we present an example to illustrate our main result. Also the numerical simulations are given to support the theoretical findings. \section{Preliminaries} The study of dynamic equation on time scales goes back to its founder Stefan Hilger \cite{h2} and it is a new area of still fairly theoretical exploration in mathematics. For convenience, we first introduce some definitions and the theory of calculus on timescales, which are needed later. For more details on timescales, please see \cite{a1, b1, b2, h2, l1}. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of real numbers $\mathbb{R}$. The operators $\sigma$ and $\rho$ from $\mathbb{T}$ to $\mathbb{T}$, defined by \cite{h2}, $$\sigma (t)=\inf\{\tau \in \mathbb{T}:\tau >t\}\in \mathbb{T}, \quad \text{and} \quad \rho (t)=\sup \{\tau \in \mathbb{T} : \tau t, respectively. Let  f:\mathbb{T} \to \mathbb{R}  and  t \in \mathbb{T} (assume t is not left-scattered if t = \sup\mathbb{T} ), then the delta derivative of f at the point t is defined to be the number f^\Delta (t) (provided it exists) with the property that for each  \epsilon>0 there is a neighborhood U of t such that$$ |f(\sigma (t))-f(s)-f^ \Delta (t)(\sigma (t)-s)| \le |\sigma(t)-s |, \quad \text{for all} \quad s\in U. $$A function f is said to be delta differentiable on \mathbb{T} if f^{\Delta} exists for all t\in \mathbb{T}. A function F:\mathbb{T}\to \mathbb{R} is called an antiderivative of f:\mathbb{T}\to \mathbb{R} provided F^{\Delta}=f(t) for all t\in\mathbb{T}. Then we define$$ \int _a^b f(t) \Delta t=F(b)-F(a), \quad \text{for } a, b \in \mathbb{T}.$$\subsection*{Notation} Throughout this paper, \mathbb{T} denotes a time scale. Let \omega > 0, the time scale \mathbb{T} is assumed to be \omega-periodic, i.e., t\in\mathbb{T} implies t+\omega \in \mathbb{T}. Let \kappa=\min\{\mathbb{R^{+}}\cap \mathbb{T}\}, and I_{\omega}=[\kappa,\kappa+\omega]\cap \mathbb{T}. A function f : \mathbb{T}\to \mathbb{R} is said to be rd-continuous if it is continuous at right-dense points in \mathbb{T} and it left-sided limits exist (finite)at left-dense points in \mathbb{T}. The set of rd-continuous functions f: \mathbb{T} \to \mathbb{R} will be denoted by C_{rd}(\mathbb{T}). \begin{lemma} \label{lem2.1} If a, b \in \mathbb{T}, \alpha, \beta \in \mathbb{R} and  f, g \in C_{rd}(\mathbb{T}), then \begin{itemize} \item[(a)]$$ \int_a^b[\alpha f(t)+\beta g(t)]\Delta t=\alpha\int_a^b f(t)\Delta t+\beta\int_a^b g(t)\Delta t; $$\item[(b)] if f(t)\ge 0 for all a\le t\le b, then  \int_a^b f(t)\Delta t\ge 0; \item[(c)] if |f(t)|\le g(t)  on [a,b):=\{t\in\mathbb{T}:a\le t< b\}, then$$ \big|\int_a^b f(t)\Delta t\big|\le \int_a^b g(t)\Delta t. $$\end{itemize} \end{lemma} Throughout of this paper, for \eqref{e1.2} we assume that \begin{itemize} \item[(H)] For i=1,2: a(t), b(t), \alpha (t),\beta (t),\sigma(t), \tau_i(t):\mathbb{R} \to [0,+\infty) are rd-continuous positive periodic functions with period \omega and \alpha(t)\ne 0, \beta(t)\ne 0; r(t),d(t):\mathbb{R}\to \mathbb{R} are rd-continuous functions of period \omega and \int_{\kappa}^{\kappa+\omega}d(t)\Delta t>0, \int_{\kappa}^{\kappa+\omega}r(t)\Delta t>0; p is a positive constant and p \ge 1; m and \theta are positive constants. \end{itemize} In view of the actual applications of system \eqref{e1.2}, we consider the initial value problem \begin{gather*} z_i(s)=\Phi_i(s), s\in [\kappa-\tau, \kappa]\cap \mathbb{T}, \Phi_i(\kappa)>0,\\ \Phi_i(s)\in C_{rd}([\kappa-\tau, \kappa]\cap \mathbb{T}, \mathbb{R}^{+}), \quad i=1,2, \end{gather*} where \tau=\max_{t\in [\kappa,\kappa+\omega]} \{\tau_1(t), \tau_2(t), \tau_3(t), \sigma(t)\}. Next we give some fundamental definitions about coincidence degree theorem. These concepts will be used for proving the existence of solutions of \eqref{e1.2}. Let X and Z be two Banach spaces, L: \mathop{\rm Dom}L \subset X \to Z be a continuous mapping. The mapping L will be 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 follows that L |\mathop{\rm Dom}L \cap \ker P : (I-P)X\to ImL is invertible. We denote the inverse of that map 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. The following Lemma is important for the proof of our main results. \begin{lemma}(Continuation Theorem [1]) \label{lem2.2} Let L be a Fredholm mapping of index zero and let N be L-compact on \Omega. Suppose \begin{itemize} \item[(a)] for each \lambda\in (0,1), every solution x of Lx=\lambda Nx is such that x \notin \partial\Omega; \item[(b)] QNx \ne 0 for each x\in \partial \Omega \cap \ker L and$$\mathop{\rm deg} \{JQN, \Omega \cap \ker L, 0\}\ne0.$$\end{itemize} Then the equation Lx =Nx  has at least one solution lying in \mathop{\rm Dom} L\cap \overline \Omega. \end{lemma} The following lemma will be used in the proof of our results. The proof is similar to that of Lemma 3.2 established in \cite{x1}. So we omit it here. \begin{lemma} \label{lem2.3} Let t_1,t_2\in \mathbb{T} and t\in \mathbb{T}. If g: \mathbb{T} \to \mathbb{R} \in C_{rd}(\mathbb{T}) is \omega-periodic, then$$ g(t)\le g(t_1)+ \int_\kappa^{\kappa+\omega}|g^{\Delta}(s)|\Delta s, \quad \text{and} \quad g(t)\ge g(t_2)-\int_\kappa^{\kappa+\omega}|g^{\Delta}(s)|\Delta s. $$\end{lemma} By simple calculation, we get the following two lemmas. \begin{lemma} \label{lem2.4} The following algebraic equation \begin{gather*} \bar b \exp\{\theta z_1\}-\bar r=0,\\ \bar \beta \frac {\exp\{pz_1\}} {1+m\exp\{pz_1\}}-\bar a \exp\{z_2\}-\bar d=0, \end{gather*} has a unique solution. \end{lemma} \begin{lemma} \label{lem2.5} If y(t)>0 for t \in \mathbb{T}, then$$ \frac{y^{p-1}(t)}{1+my^{p}(t)}\le \max \{\frac 1 m, 1 \}. $$\end{lemma} \section{Main result} For convenience, we denote $$z_i(\xi_i)=\min_{t\in I_\omega}z_i(t),\quad z_i(\eta_i)=\max_{t\in I_\omega}z_i(t), \quad i=1,2.\label{e3.1}$$ \begin{theorem}\label{thm3.1} Assume that condition (H) holds and$$ \bar a \bar r - \max\{\frac 1 m, 1\} \overline \alpha \overline \beta \exp\{(\overline D+\overline d)\omega\}>0, \quad \frac {\overline \beta \exp\{pH_2\}}{1+m\exp\{pH_2\}}-\overline d>0, $$where \begin{gather*} H_2=\frac 1 {\theta } ln\Big(\frac {m \bar a \bar r- \max\{\frac 1 m, 1\}\bar \alpha \bar \beta exp\{(\bar D+\bar d)\omega\}}{m \bar a\bar b}\Big)-(\bar R+\bar r)\omega,\\ \bar a = \frac 1 \omega \int_\kappa^{\kappa+\omega} a(t) \Delta t, \quad \bar r = \frac 1 \omega \int_\kappa^{\kappa+\omega} r(t)\Delta t, \\ \bar R = \frac 1 \omega \int_\kappa^{\kappa+\omega} |r(t)| \Delta t, \quad \bar \alpha = \frac 1 \omega \int_\kappa^{\kappa+\omega} \alpha(t) \Delta t,\\ \bar d = \frac 1 \omega \int_\kappa^{\kappa+\omega} d(t)\Delta t, \quad \bar D = \frac 1 \omega \int_\kappa^{\kappa+\omega} |d(t)| \Delta t, \\ \bar\beta(t) = \frac 1 \omega \int_\kappa^{\kappa+\omega} \beta \Delta t, \end{gather*} then system \eqref{e1.2}has at least one \omega-periodic solution. \end{theorem} \begin{proof} Define \begin{gather*} X=Z=\{(z_1,z_2)^{T}\in C(\mathbb{T},\mathbb{R}^2): z_i(t+\omega)=z_i(t),\; i=1,2,\; t\in \mathbb{T}\}, \\ \| (z_1,z_2)^T\|=\sum_{i=1}^2\max|z_i(t)|,(z_1,z_2)^T \in X(Z). \end{gather*} then X, Z are both Banach spaces endowed with norm \|\cdot\|. Let$$ L:\mathop{\rm Dom}L\to Z, \quad L \begin{pmatrix} z_1\\ z_2 \end{pmatrix} = \begin{pmatrix} z_1^{\Delta}(t)\\ z_2^{\Delta}(t) \end{pmatrix}, where \mathop{\rm Dom}L=X, and N:\mathop{\rm Dom}L \to Z, \begin{align*} &N \begin{pmatrix} z_1\\ z_2 \end{pmatrix}\\ &= \begin{pmatrix} r(t)-b(t)\exp\{{\theta}z_1(t-\tau_1(t))\}-\frac {\alpha(t) \exp\{(p-1)z_1(t)\}}{1+m \exp\{pz_1(t)\}}\exp\{z_2(t-\sigma(t))\} \\[5pt] -d(t)-a(t)\exp\{z_2(t-\tau_2(t))\} +\frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}}{1+m\exp\{p(t-\tau_3(t))\}} \end{pmatrix}, \end{align*} P \begin{pmatrix} z_1\\ z_2 \end{pmatrix} =Q \begin{pmatrix} z_1\\ z_2 \end{pmatrix} = \begin{pmatrix} \frac 1 \omega \int_\kappa^{\kappa+\omega} z_1(t)\Delta t\\ \frac 1 \omega \int_\kappa^{\kappa+\omega }z_2(t)\Delta t \end{pmatrix}, $$where (z_1,z_2)^T \in X. Then \begin{gather*} \ker L=\{(z_1,z_2)^T \in X| (z_1,z_2)^T =(h_1,h_2)^T\in \mathbb{R}^{2}, t\in \mathbb{T}\},\\ \mathop{\rm Im}L= \{(z_1,z_2)^T\in Z|\int_\kappa^{\kappa+\omega}z_1(t)\Delta(t)=0, \quad \int_\kappa^{\kappa+\omega}z_2(t)\Delta(t)=0 \}, \\ \mathop{\rm dim}\ker L= 2 =\mathop{\rm codim}\mathop{\rm Im} L . \end{gather*} Since \mathop{\rm Im}L is closed in Z , then L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that$$ \mathop{\rm Im}P= \ker L, \ker Q= \mathop{\rm Im}L= \mathop{\rm Im} (I-Q). $$Furthermore, the generalized inverse (of L) K_p: \mathop{\rm Im}L \to \ker P \cap \mathop{\rm Dom}L exists and is given by$$ K_p \begin{pmatrix} z_1\\ z_2 \end{pmatrix} =\begin{pmatrix} \int_\kappa^t z_1(s)\Delta s- \frac 1 \omega \int_\kappa^{\kappa+\omega }\int_\kappa^t z_1(s)\Delta s \Delta t\\ \int_\kappa^t z_2(s)\Delta s- \frac 1 \omega \int_\kappa^{\kappa+\omega} \int_\kappa^t z_2(s)\Delta s \Delta t \end{pmatrix}. Thus \begin{align*} & QN \begin{pmatrix} z_1\\ z_2 \end{pmatrix} \\ & = \begin{pmatrix} \frac 1 \omega\int_\kappa^{\kappa+\omega}(r(t)-b(t) \exp\{{\theta}z_1(t-\tau_1(t))\} -\frac {\alpha(t) \exp\{(p-1)z_1(t)\}}{1+m \exp\{pz_1(t)\}}\exp\{z_2(t-\sigma(t))\})\Delta t \\[5pt] \frac 1 \omega \int_\kappa^{\kappa+\omega}( -d(t)-a(t)\exp\{z_2(t-\tau_2(t))\} + \frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}}{1+m\exp\{p(t-\tau_3(t))\}})\Delta t \end{pmatrix}, \end{align*} \begin{align*} & K_p(I-Q)N \begin{pmatrix} z_1\\ z_2 \end{pmatrix} \\ & =\begin{pmatrix} \int_\kappa^t z_1(s)\Delta s- \frac 1 \omega \int_\kappa^{\kappa+\omega }\int_\kappa^t z_1(s)\Delta s \Delta t-(t-\kappa-\frac 1 \omega \int_\kappa^{\kappa+\omega}(t-\kappa)\Delta t)\bar z_1\\[4pt] \int_\kappa^t z_2(s)\Delta s- \frac 1 \omega \int_\kappa^{\kappa+\omega} \int_\kappa^t z_2(s)\Delta s \Delta t-(t-\kappa-\frac 1 \omega \int_\kappa^{\kappa+\omega}(t-\kappa)\Delta t)\bar z_2 \end{pmatrix}. \end{align*} Obviously, QN and K_p(I-Q)N are continuous. According to Arela-Ascoli theorem, it is easy to show that K_p(I-Q)N(\bar \Omega) is compact for any open bounded set \Omega \in X and QN(\bar \Omega) is bounded. Thus, N is L-compact on \Omega. Now,we shall search an appropriate open bounded subset \Omega for the application of the continuation theorem. For the operator equation Lx= \lambda Nx, \lambda \in (0,1), we have \begin{gathered} \begin{aligned} z_1^{\Delta}(t) &=\lambda \Big[r(t)-b(t)\exp\{{\theta}z_1(t-\tau_1(t))\}\\ &\quad -\frac {\alpha(t) \exp\{(p-1)z_1(t)\}}{1+m \exp\{pz_1(t)\}}\exp\{z_2(t-\sigma(t))\}\Big], \end{aligned} \\ z_2^{\Delta}(t)=\lambda \Big[-d(t)-a(t)\exp\{z_2(t-\tau_2(t))\} +\frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}}{1+m\exp\{pz_1(t-\tau_3(t))\}}\Big], \end{gathered} \label{e3.2} where t \in \mathbb{T}. Assume (z_1(t),z_2(t))^{T} is a solution of \eqref{e3.2}. Integrating \eqref{e3.2}, we get \begin{aligned} &\int_\kappa^{\kappa+\omega }b(t)\exp\{\theta z_1(t-\tau_1)\}\Delta t\\ &+ \int_\kappa^{\kappa+\omega} \frac {\alpha(t)\exp\{(p-1)z_1(t)\}\exp\{z_2(t-\sigma (t))\}}{1+m\exp\{pz_1(t)\}}\Delta t=\bar r\omega, \end{aligned}\label{e3.3} $$\int_\kappa^{\kappa+\omega} \frac {\beta(t)\exp\{pz_1(t-\tau_3)\}} {1+m\exp\{pz_1(t-\tau_3)\}}\Delta t- \int_\kappa^{\kappa+\omega} a(t)\exp\{z_2(t-\tau_2)\}\Delta t=\bar d \omega,\label{e3.4}$$ By the first equation of \eqref{e3.2} and \eqref{e3.3}, we get \begin{align*} \int_\kappa^{\kappa+\omega}|z_1^\Delta (t)| \Delta t & \le \int_\kappa^{\kappa+\omega } |r(t)|\Delta t + \int_\kappa^{\kappa+\omega} \Big[b(t)\exp\{\theta z_1(t-\tau_1(t))\} \\ & \quad +\frac {\alpha(t)\exp\{(p-1)z_1(t)\}\exp\{z_2(t-\sigma (t))\}}{1+m\exp\{pz_1(t)\}} \Big] \Delta t \\ & \le (\bar R + \bar r )\omega. \end{align*} By the second equation of \eqref{e3.2} and \eqref{e3.4}, we have \begin{align*} &\int_\kappa^{\kappa+\omega }|z_2^\Delta (t)| \Delta t \\ &\le \int_\kappa^{\kappa+\omega } |d(t)| \Delta t + \int_\kappa^{\kappa+\omega }\Big[\frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}} {1+m\exp\{pz_1(t-\tau_3(t))\}}+ a(t)\exp\{z_2(t-\tau_2(t))\Big] \Delta t \\ & \le (\bar D + \bar d )\omega. \end{align*} By \eqref{e3.1} and \eqref{e3.4}, we obtain \begin{align*} \bar a \omega \exp\{z_2\{\xi_2\}\}&\le \int_\kappa^{\kappa+\omega} a(t)\exp\{z_2(t-\tau_2(t))\}\Delta t\\ & = \int_\kappa^{\kappa+\omega }\frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}}{1+m\exp\{pz_1(t-\tau_3(t))\}}\Delta t-\bar d \omega \le \frac {\bar \beta \omega }{m}; \end{align*} that is, z_2(\xi_2)\le \ln \{\frac {\bar \beta}{m \bar a}\}:=L_2, $$hence $$z_2(t)\le z_2(\xi_2)+\int_\kappa^{\kappa+\omega} |z_2^{\Delta}(t)|\Delta t \le \ln \{\frac {\bar \beta}{m \bar a}\}+(\bar D + \bar d )\omega:=H_3.\label{e3.5}$$ From \eqref{e3.1} and \eqref{e3.3}, we have$$ \bar r \omega \ge \int_\kappa^{\kappa+\omega} b(t)\exp\{\theta z_1(t-\tau_1(t))\}\Delta t \ge \bar b \omega \exp\{\theta z_1(\xi_1)\}; $$that is$$ z_1(\xi_1)\le \frac 1 \theta \ln \{\frac {\bar r} {\bar b}\}:=L_1, then $$z_1(t)\le z_1(\xi_1)+\int_\kappa^{\kappa+\omega} |z_1^\Delta(t)|\Delta t \le \frac 1 \theta \ln \{\frac {\bar r} {\bar b}\}+(\bar R + \bar r )\omega :=H_1.\label{e3.6}$$ By \eqref{e3.1}, \eqref{e3.3}, \eqref{e3.6}, lemma \ref{lem2.5} and under the assumptions of theorem \ref{thm3.1}, we have \begin{align*} \bar b \omega \exp\{\theta z_1(\eta_1)\}&\ge \int_\kappa^{\kappa+\omega} b(t)\exp\{\theta z_1(\eta_1)\} \Delta t \\ & = \bar r \omega- \int_\kappa^{\kappa+\omega} \frac {\alpha(t) \exp\{(p-1)z_1(t)\}\exp\{z_2(t-\sigma(t))\}}{1+m \exp\{pz_1(t)\}} \\ & \ge \bar r \omega - \frac{ \bar \alpha \bar \beta}{m \bar a}\exp\{(\bar D +\bar d)\omega\}, \end{align*} thus z_1(\eta_1)\ge \frac 1 {\theta } \ln(\frac {m \bar a \bar r- \max\{\frac 1 m, 1\}\bar \alpha \bar \beta \exp\{(\bar D+\bar d)\omega\}}{m \bar a\bar b}):=l_1. We also can get that \begin{aligned} z_1(t) & \ge z_1(\eta_1)-\int_\kappa^{\kappa+\omega}|z_1^{\Delta}(t)|\Delta t\\ & \ge \frac 1 {\theta } \ln(\frac {m \bar a \bar r- \max \{\frac 1 m, 1\}\bar \alpha \bar \beta \exp\{(\bar D+\bar d)\omega\}}{m \bar a\bar b})-(\bar R+\bar r)\omega :=H_2.\label{e3.7} \end{aligned} By \eqref{e3.6} and \eqref{e3.7}, we have $$\max _{t\in [0,\omega]}|z_1(t)|\le \max\{|H_1|,|H_2|\}:=H_5.\label{e3.8}$$ Now we are in a position to estimate z_2(\eta_2). From \eqref{e3.1}, \eqref{e3.4} and \eqref{e3.7}, we get \begin{align*} \bar a \omega \exp\{z_2(\eta_2)\} &\ge \int_\kappa^{\kappa+\omega} a(t)\exp\{z_2(t-\tau_2)\}\Delta t\\ &=\int_\kappa^{\kappa+\omega }\frac {\beta(t)\exp\{pz_1(t-\tau_3(t))\}}{1+m\exp\{pz_1(t-\tau_3(t))\}}-\bar d \omega\\& \ge\frac {\bar \beta \omega exp\{pH_2\}}{1+m\exp\{pH_2\}}-\bar d \omega, \end{align*} thus z_2(\eta_2)\ge \ln\{\frac {\frac{\bar \beta \exp\{pH_2\}}{1+m\exp\{pH_2\}}-\bar d}{\bar a}\}:=l_2, $$we have also $$z_2(t)\ge z_(\eta_2)-\int_\kappa^{\kappa+\omega } |z_2^{\Delta}|\Delta t \ge \ln\{\frac {\frac{\bar \beta \exp\{pH_2\}}{1+m\exp\{pH_2\}}-\bar d}{\bar a}\}-(\bar D+ \bar d)\omega :=H_4.\label{e3.9}$$ By \eqref{e3.5} and \eqref{e3.9}, we get$$ \max_{t\in[0,\omega]}|z_2(t)|\le \max\{|H_3|,|H_4|\}:=H_6, $$clearly, H_5, H_6 are dependent on \lambda. Let H_8=H_5+H_6+H_7, where H_7 is large enough, such that H_8\ge|l_1|+|L_1|+|l_2|+|L_2|. Next, for (z_1, z_2)^T\in \mathbb{R}^2, \mu \in [0,1], we shall consider the following algebraic equations: $$\begin{gathered} \bar b \exp\{\theta z_1\}+ \mu \frac {\bar \alpha \exp\{(p-1)z_1\}\exp\{z_2\}}{1+m\exp\{pz_1\}}-\bar r=0,\\ \frac {\bar \beta \exp\{pz_1\}} {1+m\exp\{pz_1\}}- \bar a \exp\{z_2\}-\bar d =0. \end{gathered} \label{e3.10}$$ Similar to the above discussion, we can easily check that, every solution (z_1^*, z_2^*)^T of \eqref{e3.10} satisfies$$ l_1\le z_1^* \le L_1,l_2\le z_2^* \le L_2. $$Take \Omega =\{(z_1(t),z_2(t))^T\in z:\|(z_1,z_2)^T\| 0,$$ and $$\frac{\bar \beta \exp\{pH_2\}}{1+m\exp\{pH_2\}}-\bar d=0.05 > 0.$$ According to theorem \ref{thm3.1}, it is easy to see that \eqref{e4.2} has at least one $2\pi$-periodic solution. Numerical simulations of solution for \eqref{e4.2} and the solution tends to the $2\pi$-periodic solution see Figure 1a and Figure 1b, respectively. The simulation is performed using MATLAB software. \begin{figure}[ht] \begin{center} \includegraphics[width=0.45\textwidth]{fig1a} \includegraphics[width=0.45\textwidth]{fig1b} \end{center} \caption{(a) Numerical solution $x_1(t)$, $x_2(t)$ of system \eqref{e4.2}, where $x_1(s) = x_2(s) = 0$ for $s \in [- 0.8, 0]$. (b) Phase trajectories of system \eqref{e4.2}, where $x_1(s) = x_2(s) = 0$ for $s \in [- 0.8, 0].$} \end{figure} Numerical simulations of solution for \eqref{e4.2} and the solution tends to the $2\pi$-periodic solution; see Fig. 1. \subsection*{Acknowledgements} The author is deeply indebted to the the anonymous referee for his/her excellent suggestions, which greatly improve the presentation of this paper. \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal, M .Bohner; \emph{Basic calculus on time scales and some of its applications}, Results Math. 35 (1999) 3-22. \bibitem{a2} R. P. Agarwal, D. 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