\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 88, pp. 1--7.\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-2007/88\hfil Periodic solutions] {Periodic solutions for a stage-structure ecological model on time scales} \author[K. Zhuang\hfil EJDE-2007/88\hfilneg] {Kejun Zhuang} \address{Kejun Zhuang \newline School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui, 233030, China} \email{zhkj123@163.com} \thanks{Submitted April 9, 2007. Published June 15, 2007.} \subjclass[2000]{92D25, 34C25} \keywords{Time scales; stage-structure; coincidence degree; periodic solutions} \begin{abstract} In this paper, by using the Mawhin's continuation theorem, we prove the existence of periodic solutions for a stage-structure ecological model on time scales. This unifies the results for differential and difference equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Recently, Zheng and Cui constructed the following stage-structure population model, with patches and diffusion, $$\begin{gathered} I'_1(t)=aM_1(t)-bI_1(t)-cI_1(t), \\ M'_1(t)=cI_1(t)-\alpha M^2_1(t)+D_{12}(M_2(t)-M_1(t)), \\ M'_2(t)=-\beta M_2(t)+D_{21}(M_1(t)-M_2(t)), \end{gathered} \label{e1.1}$$ where $a, b, c, \alpha, \beta, D_{12}$ and $D_{21}$ are positive constants. $I_1(t)$ is the density of immature rana chensinensis in patch $1$, $M_i(t)$ denotes the density of mature rana chensinensis in the $i$-patch $(i=1,2)$. $D_{ij}$ is the diffusion coefficient of the species from patch $j$ to patch $i$. The permanence and stability of equilibrium of system \eqref{e1.1} were investigated in \cite{z3}. Taking into account the periodicity of environment, Zhang and Zheng reconstructed the model as follows: $$\begin{gathered} I'_1(t)=a(t)M_1(t)-b(t)I_1(t)-c(t)I_1(t), \\ M'_1(t)=c(t)I_1(t)-\alpha (t)M^2_1(t)+D_{12}(t)(M_2(t)-M_1(t)), \\ M'_2(t)=-\beta (t)M_2(t)+D_{21}(t)(M_1(t)-M_2(t)), \end{gathered} \label{e1.2}$$ where all the coefficients are positive continuous $\omega$-periodic functions. Based on the theory of coincidence degree, the existence of positive periodic solution was established in \cite{z2}. The corresponding discrete system $$\begin{gathered} I_1(k+1)=I_1(k)\exp \big\{ -b(k)-c(k)+a(k)\frac{M_1(k)}{I_1(k)} \big\}, \\ M_1(k+1)=M_1(k)\exp \big\{ -D_{12}(k)-\alpha(k)M_1(k)+\frac{c(k)I_1(k) +D_{12}(k)M_2(k)}{M_1(k)} \big\}, \\ M_2(k+1)=M_2(k)\exp \big\{ -\beta(k)-D_{21}(k) +\frac{D_{21}(k)M_1(k)}{M_2(k)} \big\}, \end{gathered} \label{e1.3}$$ was considered in \cite{z1}. The coefficients are all strictly positive $\omega$-periodic sequences. The existence of periodic solutions for \eqref{e1.3} was done. However, the work of \cite{z1,z2} was repeated to some extent. It is natural to ask whether there is a unified way to explore such kind of problem. To unify the continuous and discrete analysis, Stefan Hilger in his Ph.D. Thesis initiated the theory of calculus on time scales in \cite{h1}. The theme has received much attention in recent years, such as \cite{b1,b2,b3}. It is true that unification and extension are the two main features of the calculus on time scales. In this paper, we consider the following ecological model with stage structure and diffusion on time scales: $$\begin{gathered} u_1^\Delta (t)=-b(t)-c(t)+a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}}, \\ u_2^\Delta (t)=-D_{12}(t)-\alpha(t)e^{u_2(t)}+\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}, \\ u_3^\Delta (t)=-\beta(t)-D_{21}(t)+\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}}, \end{gathered} \label{e1.4}$$ where $a(t), b(t), c(t), \alpha(t), D_{12}(t), \beta(t)$ and $D_{21}(t)$ are rd-continuous positive $\omega$-pe\-riodic functions on time scales $\mathbb{T}$. Set $I_1(t)=e^{u_1(t)}$, $M_1(t)=e^{u_2(t)}$ and $M_2(t)=e^{u_3(t)}$, if $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$, then \eqref{e1.4} can be reduced to \eqref{e1.2} and \eqref{e1.3}, respectively. As a result, it is unnecessary to investigate the periodic solutions of \eqref{e1.2} and \eqref{e1.3} separately. Thus, we shall prove the periodicity of system \eqref{e1.4} by Mawhin's continuation theorem in coincidence degree theory to unify the results in \cite{z1,z2}. This approach has been widely applied to deal with the existence of periodic solutions of differential equations and difference equations but rarely applied to the dynamic equations on time scales \cite{b2,b3}. \section{Preliminary results} For the convenience of the reader, we first present some basic definitions and lemmas about time scales and the continuation theorem of the coincidence degree theory; more details can be found in \cite{b1,g1}. A time scale $\mathbb{T}$ is an arbitrary nonempty closed subset of real numbers $\mathbb{R}$. Throughout this paper, we assume that the time scale $\mathbb{T}$ is unbounded above and below, such as $\mathbb{R}$, $\mathbb{Z}$ and $\bigcup_{k\in\mathbb{Z}}[2k,2k+1]$. The following definitions and lemmas about time scales are from \cite{b1}. \begin{definition} \label{def2.1} \rm The forward jump operator $\sigma:\mathbb{T}\to \mathbb{T}$, the backward jump operator $\rho:\mathbb{T}\to\mathbb{T}$, and the graininess $\mu:\mathbb{T}\to\mathbb{R}^+=[0,+\infty)$ are defined, respectively, by $$\sigma(t):=\inf\{s\in\mathbb{T}:s>t\}, \quad \rho(t):=\sup\{s\in\mathbb{T}:s0, there is a neighborhood U of t such that$$ | f(\sigma(t))-f(s)-f^\Delta(t)(\sigma(t)-s) |\leq\varepsilon|\sigma(t)-s|\quad \mbox{for all } s\in U. $$In this case, f^\Delta(t) is called the delta (or Hilger) derivative of f at t. Moreover, f is said to be delta or Hilger differentiable on \mathbb{T} if f^\Delta(t) exists for all t\in\mathbb{T}. A function F:\mathbb{T}\to\mathbb{R} is called an antiderivative of f:\mathbb{T}\to\mathbb{R} provided F^\Delta(t)=f(t) for all t\in \mathbb{T}. Then we define$$ \int_r^sf(t)\Delta t=F(s)-F(r)\quad \mbox{for } r,s\in\mathbb{T}. $$\end{definition} \begin{definition} \label{def2.3} \rm A function f:\mathbb{T}\to\mathbb{R} is said to be rd-continuous if it is continuous at right-dense points in \mathbb{T} and its left-sided limits exist(finite) at left-dense points in \mathbb{T}. The set of rd-continuous functions f:\mathbb{T}\to\mathbb{R} will be denoted by C_{rd}(\mathbb{T}). \end{definition} \begin{lemma} \label{lem2.1} Every rd-continuous function has an antiderivative. \end{lemma} \begin{lemma} \label{lem2.2} If a,b\in\mathbb{T}, \alpha, \beta\in\mathbb{R} and f, g\in C_{rd}(\mathbb{T}),then \begin{itemize} \item[(a)] \int_a^b[\alpha f(t)+\beta g(t)]\Delta t=\alpha\int_a^bf(t)\Delta t+\beta\int_a^b g(t) \Delta t; \item[(b)] if f(t)\geq 0 for all a\leq t0,$$ then \eqref{e1.4} has at least one $\omega$-periodic solution. \end{theorem} \begin{proof} Let \begin{gather*} X=Z=\big\{ (u_1,u_2,u_3)^T\in C(\mathbb{T},\mathbb{R}^3): u_i(t+\omega)=u_i(t),\;i=1,2,3, \forall t\in \mathbb{T} \big\}, \\ \| (u_1,u_2,u_3)^T\| =\sum_{i=1}^3 \max_{t\in I_\omega}|u_i(t)|,\quad (u_1,u_2,u_3)^T\in X \quad(\mbox{or in} Z). \end{gather*} Then $X$ and $Z$ are both Banach spaces when they are endowed with the above norm $\| \cdot \|$. Let \begin{gather*} N \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = \begin{bmatrix} N_1 \\ N_2 \\ N_3 \end{bmatrix} = \begin{bmatrix} -b(t)-c(t)+a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}} \\ -D_{12}(t)-\alpha(t)e^{u_2(t)}+\frac{c(t)e^{u_1(t)} +D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}} \\ -\beta(t)-D_{21}(t)+\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}} \end{bmatrix}, \\ L \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} =\begin{bmatrix} u_1^\Delta \\ u_2^\Delta \\ u_3^\Delta \end{bmatrix},\quad P \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} =Q \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = \begin{bmatrix} \frac{1}{\omega}\int_k^{k+\omega}u_1(t)\Delta t \\ \frac{1}{\omega}\int_k^{k+\omega}u_2(t)\Delta t \\ \frac{1}{\omega}\int_k^{k+\omega}u_3(t)\Delta t \end{bmatrix}. \end{gather*} Then \begin{gather*} \ker L= \big\{ (u_1,u_2,u_3)^T\in X : (u_1(t),u_2(t),u_3(t))^T=(h_1,h_2,h_3)^T\in\mathbb{R}^3 ,t\in\mathbb{T} \big\}, \\ \mathop{\rm Im}L=\big\{ (u_1,u_2,u_3)^T\in Z : \bar{u}_1=\bar{u}_2=\bar{u}_3=0,t\in\mathbb{T} \big\}, \\ \dim \ker L=3=\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 not difficult to prove that $P$ and $Q$ are continuous projections such that $\mathop{\rm Im}P=\ker L$ and $\mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q)$. 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 $$K_P \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = \begin{bmatrix} \int_k^tu_1(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega} \int_k^t u_1(s)\Delta s\Delta t \\ \int_k^tu_2(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega} \int_k^t u_2(s)\Delta s\Delta t \\ \int_k^tu_3(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega} \int_k^t u_3(s)\Delta s\Delta t \end{bmatrix} .$$ Thus $$QN\begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} = \begin{bmatrix} \frac{1}{\omega}\int_k^{k+\omega}\big( -b(t)-c(t) +a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}}\big)\Delta t \\ \frac{1}{\omega}\int_k^{k+\omega}\big( -D_{12}(t)-\alpha(t)e^{u_2(t)} +\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}\big)\Delta t \\ \frac{1}{\omega}\int_k^{k+\omega}\big( -\beta(t)-D_{21}(t) +\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}}\big)\Delta t \end{bmatrix},$$ \begin{align*} & K_P(I-Q)N \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \\ &= \begin{bmatrix} \int_k^tu_1(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega} \int_k^tu_1(s)\Delta s\Delta t-\left( t-k-\frac{1}{\omega} \int_k^{k+\omega}(t-k)\Delta t \right)\bar{u}_1 \\ \int_k^tu_2(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega} \int_k^tu_2(s)\Delta s\Delta t-\left( t-k-\frac{1}{\omega} \int_k^{k+\omega}(t-k)\Delta t \right)\bar{u}_2 \\ \int_k^tu_3(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega} \int_k^tu_3(s)\Delta s\Delta t-\left( t-k-\frac{1}{\omega} \int_k^{k+\omega}(t-k)\Delta t \right)\bar{u}_3 \end{bmatrix}. \end{align*} Obviously, $QN$ and $K_P(I-Q)N$ are continuous. According to Arzela-Ascoli theorem, it is easy to show that $K_P(I-Q)N(\bar{\Omega})$ is compact for any open bounded set $\Omega\subset X$ and $QN(\bar{\Omega})$ is bounded. Thus, $N$ is $L$-compact on $\bar{\Omega}$. Now, we shall search an appropriate open bounded subset $\Omega$ for the application of the continuation theorem, Lemma \ref{lem2.3}. For the operator equation $Lu=\lambda Nu$, where $\lambda\in (0,1)$, we have $$\begin{gathered} u_1^\Delta (t)=\lambda\left(-b(t)-c(t) +a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}}\right), \\ u_2^\Delta (t)=\lambda\left(-D_{12}(t) -\alpha(t)e^{u_2(t)}+\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}\right), \\ u_3^\Delta (t)=\lambda\left(-\beta(t)-D_{21}(t) +\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}}\right). \end{gathered} \label{e3.1}$$ Assume that $(u_1,u_2,u_3)^T\in X$ is a solution of system \eqref{e3.1} for a certain $\lambda\in (0,1)$. Integrating \eqref{e3.1} on both sides from $k$ to $k+\omega$, we obtain $$\begin{gathered} \bar{b}\omega+\bar{c}\omega=\int_k^{k+\omega}a(t) \frac{e^{u_2(t)}}{e^{u_1(t)}}\Delta t, \\ \bar{D}_{12}\omega+\int_k^{k+\omega}\alpha(t)e^{u_2(t)}\Delta t =\int_k^{k+\omega}\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}\Delta t, \\ \bar{\beta}\omega+\bar{D}_{21}\omega =\int_k^{k+\omega}\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}} \Delta t. \end{gathered}\label{e3.2}$$ Since $(u_1,u_2,u_3)^T\in X$, there exist $\xi_i,\eta_i\in[k,k+\omega]$, $i=1,2,3$, such that $$u_i(\xi_i)=\min_{t\in[k,k+\omega]}\{u_i(t)\},\quad u_i(\eta_i)=\max_{t\in[k,k+\omega]}\{u_i(t)\},\quad i=1,2,3.\label{e3.3}$$ From \eqref{e3.1} and \eqref{e3.3}, we have \begin{gather} a(\eta_1)e^{u_2(\eta_1)}=(b(\eta_1)+c(\eta_1))e^{u_1(\eta_1)}, \label{e3.4} \\ c(\eta_2)e^{u_1(\eta_2)}+D_{12}(\eta_2)e^{u_3(\eta_2)} =D_{12}(\eta_2)e^{u_2(\eta_2)}+\alpha(\eta_2)e^{2u_2(\eta_2)}, \label{e3.5} \\ D_{21}(\eta_3)e^{u_2(\eta_3)}=\beta(\eta_3)e^{u_3(\eta_3)} +D_{21}(\eta_3)e^{u_3(\eta_3)}. \label{e3.6} \end{gather} Thus, \begin{gather} (b+c)^Le^{u_1(\eta_1)}\leq a^Me^{u_2(\eta_2)},\label{e3.7} \\ (\beta+D_{21})^Le^{u_3(\eta_3)}\leq D_{21}^Me^{u_2(\eta_2)}.\label{e3.8} \end{gather} From \eqref{e3.5}, \eqref{e3.7} and \eqref{e3.8}, we have \begin{align*} \alpha^Le^{2u_2(\eta_2)}+D_{12}^Le^{u_2(\eta_2)} &\leq c^Me^{u_1(\eta_1)}+D_{12}^Me^{u_3(\eta_3)}\\ &\leq \frac{c^Ma^M}{(b+c)^L}e^{u_2(\eta_2)} +\frac{D_{12}^MD_{21}^M}{(\beta+D_{21})^L}e^{u_2(\eta_2)}. \end{align*} From the above inequality, we get $$e^{u_2(\eta_2)}\leq \frac{1}{\alpha^L} \big[ \frac{c^Ma^M}{(b+c)^L}+\frac{D_{12}^MD_{21}^M}{(\beta+D_{21})^L}-D_{12}^L \big]:= L_2.\label{e3.9}$$ Substituting \eqref{e3.9} into \eqref{e3.7} and \eqref{e3.8}, respectively, the following estimates hold: $$e^{u_1(\eta_1)}\leq\frac{a^M}{(b+c)^L}L_2:=L_1,\quad e^{u_3(\eta_3)}\leq\frac{D_{21}^M}{(\beta+D_{21})^L}L_2:=L_3 .$$ Similarly, we also get the results: \begin{gather*} e^{u_2(\xi_2)}\geq\frac{1}{\alpha^M}\big[ \frac{c^La^L}{(b+c)^M}+\frac{D_{12}^LD_{21}^L}{(\beta+D_{21})^M}-D_{12}^M \big]:=l_2, \\ e^{u_1(\xi_1)}\geq\frac{a^L}{(b+c)^M}l_2:=l_1 \, , \quad e^{u_3(\xi_3)}\geq \frac{D_{21}^L}{(\beta+D_{21})^M}l_2:=l_3 . \end{gather*} So, we have \begin{gather*} \max_{t\in[k,k+\omega]}|u_1(t)|\leq\max\{ | \ln L_1 |,|\ln l_1| \}:=R_1, \\ \max_{t\in[k,k+\omega]}|u_2(t)|\leq\max\{ |\ln L_2 |, | \ln l_2 | \}:=R_2, \\ \max_{t\in[k,k+\omega]}|u_3(t)|\leq\max \{ | \ln L_3 |,| \ln l_3 | \}:=R_3. \end{gather*} Clearly, $R_1, R_2$ and $R_3$ are independent of $\lambda$. Let $R=R_1+R_2+R_3+R_0$, where $R_0$ is taken sufficiently large such that $R_0\geq\sum_{i=1}^3|l_i|+\sum_{i=1}^3|L_i|$. Now, we consider the algebraic equations: $$\begin{gathered} \bar{b}+\bar{c}-\bar{a}{e^{y-x}}=0, \\ \bar{D}_{12}+\bar{\alpha}e^{y}-\bar{c}e^{x-y}-\bar{D}_{12}e^{z-y}=0, \\ \bar{\beta}+\bar{D}_{21}-\bar{D}_{21}e^{y-z}=0, \end{gathered} \label{e3.10}$$ every solution $(x^\ast,y^\ast,z^\ast)^T$ of \eqref{e3.10} satisfies \$\| (x^\ast,y^\ast,z^\ast)^T \|