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

\AtBeginDocument{{\noindent\small \emph{
Electronic Journal of Differential Equations}, 
Vol. 2007(2007), No. 168, 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-2007/168\hfil Periodicity in a delayed system]
{Periodicity in a delayed  ratio-dependent
 predator-prey system with exploited term}

\author[Z. Zeng, L. Zhai\hfil EJDE-2007/168\hfilneg]
{Zhijun Zeng, Li Zhai}  % in alphabetical order

\address{Zhijun Zeng \newline
Institute of Systems Science, Academy of Mathematics and Systems
Sciences, Chinese Academy of Sciences, Beijing 100080,  China}
\email{zthzzj@amss.ac.cn}

\address{Li Zhai \newline
 School of Mathematics and Statistics,
 Southwest University, Chongqing 400715, China} 
\email{zl6168@swu.edu.cn}

\thanks{Submitted December 2, 2006. Published December 3, 2007.}
\thanks{Supported by grant SWUQ 2006032 from the Youth Foundation of
Southwest University} 
\subjclass[2000]{34K45, 34K13, 92D25}
\keywords{Ratio-dependent predator-prey system;
 periodic solutions; \hfill\break\indent coincidence degree}

\begin{abstract}
 With the help of the coincidence degree and the related
 continuation theorem, we explore the existence of at least two
 periodic solutions of a delayed ratio-dependent predator-prey
 system with exploited term. Some easily verifiable sufficient
 criteria are established for the existence of at least two
 positive periodic solutions.
\end{abstract}

\maketitle 
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}

\section{Introduction}

As is well-known, the traditional Lotka-Volterra type predator-prey
model with prey-dependent functional response fails to model the
interference among predators. To overcome the shortcoming, Arditi
and Ginzburg \cite{a1} proposed the following ratio-dependent
predator-prey model
\begin{equation}\label{e1.1}
 \begin{gathered}
 x'(t)=x( a-bx)-\frac{cxy}{my+x},\\
y'(t) =-dy+\frac{fxy}{my+x},
\end{gathered}
\end{equation}
which incorporates mutual interference by the predatorss. For a
detailed justification of \eqref{e1.1} and its merits versus the
prey-dependent functional response model, we  refer to \cite{a1}. In
addition, system \eqref{e1.1} and its non-autonomous variation have
been studied by many authors and seen great progress, see, for
example, \cite{f2, f3, h2, w1} and the references therein. Beretta
and Kuang \cite{b1} introduced a single discrete time delay into the
predator equation in system \eqref{e1.1}, namely
\begin{equation}\label{e1.2}
 \begin{gathered}
  x'(t)=x( a-bx)-\frac{cxy}{my+x},\\
y'(t)=y\Big[-d+\frac{fx(t-\tau)}{my(t-\tau)+x(t-\tau)}\Big],
\end{gathered}
\end{equation}
and carried out systematic work on the global qualitative analysis
of \eqref{e1.2}. In paper \cite{f1}, Fan and Wang studied a more
general delayed ratio-dependent predator-prey model
\begin{equation}\label{e1.3}
 \begin{gathered}
x'(t)=x\Big[a(t)-b(t)\int_{-\infty}^t
k(t-s)x(s)ds\Big]-\frac{c(t)xy}{my+x},\\
y'(t)=y\Big[-d(t)+\frac{f(t)x(t-\tau(t))}{my(t-\tau(t))+x(t-\tau(t))}\Big],
\end{gathered}
\end{equation}
where $x, y$ denote prey and predator density, respectively. $m$ is
a constant that denotes the half capturing saturation constant,
$a\in C(\mathbb{R},\mathbb{R})$, $ b, c,d,f,\tau$ in
$C(\mathbb{R},\mathbb{R_+})$, $\mathbb{R^+}=[0,+\infty),k(s):
\mathbb{R^+}\to \mathbb{R^+}$ is a measurable, normalized function
such that $\int_0^{+\infty}k(s)ds=1$. For a detailed discussion of
the biological significance of the parameters in \eqref{e1.3}, refer
to \cite{c1, f1, h1, l1, s1}.

 A very basic and important ecological problem associated with the
study of multi-species population interactions is the existence of
positive periodic solutions due to  various seasonal effects present
in real life situations. Although much progress has been seen in the
study of such problems, there are relatively fewer results on the
models with exploited term. Therefore, the major objective of this
paper is to investigate the existence of periodic solutions of the
following system
\begin{equation}\label{e1.4}
 \begin{gathered}
x'(t)=x\Big[a(t)-b(t)
 \int_{-\infty}^t k(t-s)x(s)ds\Big]-\frac{c(t)xy}{m(t)y+x}-h(t),\\
 y'(t)=y\Big[-d(t)+\frac{f(t)x(t-\tau(t))}{m(t)y(t-\tau(t))+x(t-\tau(t))}\Big],
\end{gathered}
\end{equation}
 where $h$ is an exploitation term standing for harvesting or hunting.

 An outline of this paper is given as follows. In section 2,
we present some preliminaries including the famous coincidence
degree theory and a basic lemma. In section 3, by using the
coincidence degree theory, we will establish some sufficient
conditions for the existence of positive periodic solutions of
system (1.4). At last, an example is given to verify and support our
theoretical result.

\section{Preliminaries}

 Let us begin by introducing some
terminology and results.

If $g$ is a real continuously bounded function defined on
$\mathbb{R}$, we set
\begin{equation*}
\bar{g}=\frac{1}{\omega}\int_0^\omega g(t)dt,\quad g^L=\min_{t\in
[0,\omega]}g(t),\quad g^M=\max_{t\in [0,\omega]}g(t).
\end{equation*}
In system \eqref{e1.4}, we always assume that $a,\,d: \mathbb{R}\to
 \mathbb{R}$ and $ b,\, c, \,m,\, f, \,h, \tau: \mathbb{R}\to  \mathbb{R^+}$ are
$\omega$-periodic and $\bar{a}>0,\, \bar{d}>0$, where $\omega$, a
fixed positive integer, denotes the prescribed common period of the
parameters in system \eqref{e1.4}. Moreover, for biological reasons,
we only consider solutions $(x(t),y(t))$ with $x(0)>0, y(0)>0$.

For the reader's convenience, we now recall Mawhin's coincidence
degree  which our study is based upon. Let $X,\,Z$ be normed vector
spaces,  $L: \mathop{\rm Dom}L\subset X\to Z$  a linear mapping, $N:
X\to Z$ is 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 $Z$.  If $L$ is a Fredholm mapping of index zero there exist
 continuous projectors $P:  X\to X$ and $Q:
  Z\to Z$ such that $ImP=\ker L,  \mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q)$.  It follows that
  $L|\mathop{\rm Dom}L\cap \ker P:  (I-P)X\to \mathop{\rm Im}L$ is invertible.  We denote the inverse of
  that map by $K_P$.  If $\Omega$ be 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 an
  isomorphism $J:  \mathop{\rm Im} Q\to \ker L$.


\begin{lemma}[Continuation Theorem [8]]  \label{lem2.1}
 Let $L$ be a Fredholm mapping of index zero and let $N$ be $L$-compact
on $\bar{\Omega}$. Suppose
\begin{itemize}
\item[(i)]For each $\lambda\in (0,  1)$,  every solution $x$ of
$Lx=\lambda Nx$ is such that $x\not\in
\partial\Omega;$

\item[(ii)] $QNx\not=0$ for each $x\in \partial\Omega\cap \ker L$ and
$$deg\{JQN,  \Omega\cap \ker L,  0\}\not=0.$$
\end{itemize}
Then the equation $Lx=Nx$ has at least one solution lying in
$\mathop{\rm Dom}L\cap\bar{\Omega}$.
\end{lemma}


\begin{lemma}\label{lem2.2}
If $\bar{f}>\bar{d}$ \,and\,
$\bar{a}-\overline{(c/m)}>2\sqrt{\bar{b}\bar{h}}$, then the
following algebraic equations
\begin{equation}\label{e2.1}
 \begin{gathered}
\bar{a}-\bar{b}\exp\{u\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{c(t)
\exp\{v\}}{m(t)\exp\{v\}+
\exp\{u\}}dt-\frac{\bar{h}}{\exp\{u\}}=0\\
-\bar{d}+
\frac{1}{\omega}\int_{0}^{\omega}\frac{f(t)\exp\{u\}}{m(t)\exp\{v\}+
\exp\{u\}}dt=0
\end{gathered}
\end{equation}
have two solutions.
\end{lemma}

\begin{proof} Consider the function
\begin{equation*}
 f(z)=-\bar{d}+
\frac{1}{\omega}\int_{0}^{\omega}\frac{f(t)}{m(t)z+1}dt.
\end{equation*}
 It is easily seen that $f(z)$ is decreasing with $z$ and
\begin{equation*}
f(0)=\bar{f}-\bar{d}>0,\quad \lim_{z\to +\infty}f(z)=-\bar{d}<0,
\end{equation*}
then it follows that there exists a unique $z^*$ such that
$f(z^*)=0$. Substituting $z^*=\exp\{v-u\}$ into the first equation
in \eqref{e2.1}, we have
\begin{equation}
\label{e2.2}
\bar{a}-\bar{b}\exp\{u\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{c(t)
z^*}{m(t)z^*+1}dt-\frac{\bar{h}}{\exp\{u\}}=0.
\end{equation}
Obviously, it is a quadratic equation with respect to $\exp\{u\}$,
then it has two solutions, denoted by $u_1$ and $u_2$ with
$u_1<u_2$. Moreover, one can easily see that
\begin{equation*}
\bar{a}-\bar{b}\exp\{u\}-\overline{(\frac{c}{m}}-\frac{\bar{h}}{\exp\{u\}}<0.
\end{equation*}
Solving the inequality, produces
\begin{gather*}
 \exp\{u_1\}<\frac{\bar{a}-\overline{(c/m)}-
\sqrt{[\bar{a}-\overline{(c/m)}]^2-4\bar{b}\bar{u}}}{2\bar{b}},\\
\exp\{u_2\}>\frac{\bar{a}-\overline{(c/m)}+
\sqrt{[\bar{a}-\overline{(c/m)}]^2-4\bar{b}\bar{u}}}{2\bar{b}},
\end{gather*}
which implies \eqref{e2.1} has two solutions and this completes the
proof.
\end{proof}

\section{Main results}

In this section, we devote ourselves to establishing easily
verifiable sufficiency criteria for the existence of at least two
positive periodic solutions of  system \eqref{e2.1} by employing the
coincidence degree and the related continuation theorem introduced
in the previous section.

\begin{theorem}\label{thm3.1}
If $\bar{f}>\bar{d}$ and $(a-c/m)^L>2\sqrt{b^M h^M}$, then system
\eqref{e1.4} has at least two positive $\omega$-periodic solutions.
\end{theorem}

\begin{proof}
 Let $x(t)=\exp\{u(t)\},~ y(t)=\exp\{v(t)\}$. Then system
\eqref{e1.4} can be written as
\begin{equation}\label{e3.1}
 \begin{aligned}
  u' (t)&=a(t)-b(t)\int_{-\infty}^t
k(t-s)\exp\{u(s)\}ds\\
&\quad -\frac{c(t)\exp\{v(t)\}}{m(t)
\exp\{v(t)\}+\exp\{u(t)\}}-\frac{h(t)}{\exp\{u(t)\}},\\
v' (t)&=-d(t)+\frac{f(t)\exp\{u(t-\tau(t))\}}{m(t)
\exp\{v(t-\tau(t))\}+\exp\{u(t-\tau(t))\}}.
\end{aligned}
\end{equation}
 It is easy to see that if system \eqref{e3.1} has an $\omega$-periodic solution
$(u^*,v^*)^T$, then $(x^*,y^*)^T=(\exp\{u^*\},\exp\{v^*\})^T$ is a
positive $\omega$-periodic solution of system \eqref{e1.4}. To this
end, it suffices to prove that system \eqref{e3.1} has at least two
$\omega$-periodic solutions.

For $\lambda\in (0,1)$, we consider the following system
\begin{equation}\label{e3.2}
 \begin{aligned}
 u' (t)&=\lambda\Big[
a(t)-b(t)\int_{-\infty}^t
k(t-s)\exp\{u(s)\}ds\\
&\quad -\frac{c(t)\exp\{v(t)\}}{m(t)
\exp\{v(t)\}+\exp\{u(t)\}}-\frac{h(t)}{\exp\{u(t)\}}\Big],\\
v' (t)&=\lambda\Big[-d(t)+\frac{f(t)\exp\{u(t-\tau(t))\}}{m(t)
\exp\{v(t-\tau(t))\}+\exp\{u(t-\tau(t))\}}\Big].
\end{aligned}
\end{equation}
Suppose that $(u(t),v(t))^T$ is an arbitrary $\omega$-periodic
solution of system \eqref{e3.2} for a certain $\lambda\in (0,1)$.
Integrating on both sides of \eqref{e3.2} over the interval
$[0,\omega]$, leads to
\begin{gather}\label{e3.3}
\begin{aligned}
\bar{a}\omega &=\int_0^\omega\Big[
b(t)\int_{-\infty}^t k(t-s)\exp\{u(s)\}d s\\
&\quad +\frac{c(t)\exp\{v(t)\}}{m(t)
\exp\{v(t)\}+\exp\{u(t)\}}+\frac{h(t)}{\exp\{u(t)\}}\Big] dt,
\end{aligned}\\
\label{e3.4}
\bar{d}\omega=\int_0^\omega\Big[\frac{f(t)\exp\{u(t-\tau(t))\}}{m(t)
\exp\{v(t-\tau(t))\}+\exp\{u(t-\tau(t))\}}\Big] dt.
\end{gather}
From these two equations, it follows that
\begin{equation}\label{e3.5}
\begin{aligned}
\int_0^\omega|u' (t)|dt &\leq \int_0^\omega
|a(t)|dt+\int_0^\omega\Big[  b(t)\int_{-\infty}^t
k(t-s)\exp\{u(s)\}ds \\
 &\quad+\frac{c(t)\exp\{v(t)\}}{m(t)
\exp\{v(t)\}+\exp\{u(t)\}}+\frac{h(t)}{\exp\{u(t)\}}\Big] dt\\
&=:(\bar{A}+\bar{a})\omega,
\end{aligned}
\end{equation}
and
\begin{equation}\label{e3.6}
\begin{aligned}
&\int_0^\omega|v' (t)|dt \\
&\leq\int_0^\omega
|d(t)|dt+\int_0^\omega\Big[\frac{f(t)\exp\{u(t-\tau(t))\}}{m(t)
\exp\{v(t-\tau(t))\}+\exp\{u(t-\tau(t))\}}\Big]
dt:=(\bar{D}+\bar{d})\omega.
\end{aligned}
\end{equation}
Choose $\xi_i, \eta_i\in I_\omega, i=1, 2$, such that
\begin{gather}\label{e3.7}
u(\xi_1)=\min_{t\in [0,\omega]}\{u(t)\},\quad
u(\eta_1)=\max_{t\in [0,\omega]}\{u(t)\}, \\
\label{e3.8} v(\xi_2)=\min_{t\in [0,\omega]}\{v(t)\},\quad
v(\eta_2)=\max_{t\in [0,\omega]}\{v(t)\}.
\end{gather}
 By \eqref{e3.3} and \eqref{e3.7}, we obtain
\begin{equation*}
\bar{a}\omega> \int_0^\omega
b(t)\exp\{u(\xi_1)\}dt=\exp\{u(\xi_1)\}\bar{b}\omega,
\end{equation*}
which reduces to $u(\xi_1)<\ln\{\frac{\bar{a}}{\bar{b}}\}$.
 This inequality and \eqref{e3.5} give
\begin{equation}\label{e3.9}
u(t)\leq u(\xi_1)+ \int_0^\omega|u'(t)|dt<
\ln\{\frac{\bar{a}}{\bar{b}}\}+(\bar{A}+\bar{a})\omega:=\rho_1.
\end{equation}
Multiplying the first equality of \eqref{e3.2} by $\exp\{u(t)\}$,
and integrating over $[0,\omega]$, we have
\begin{align*}
\int_0^\omega a(t)\exp\{u(t)\}dt
&=\int_0^\omega \Big[ b(t)\exp\{u(t)\}\int_{-\infty}^t k(t-s)\exp\{u(s)\}ds\\
&\quad +\frac{c(t)\exp\{v(t)\}}{m(t)
\exp\{v(t)\}+\exp\{u(t)\}}+h(t)\Big] dt
\end{align*}
 Again from \eqref{e3.3} and
\eqref{e3.7}, it follows that
\begin{equation*}
\exp\{u(\eta_1)\}\bar{a}\omega\geq\int_0^\omega
a(t)\exp\{u(t)\}dt>\int_0^\omega h(t)dt=\bar{h}\omega,
\end{equation*}
which implies $u(\eta_1)>\ln\{\frac{\bar{h}}{\bar{a}}\}$. Therefore,
by \eqref{e3.5} and \eqref{e3.7}, we obtain
\begin{equation}\label{e3.10}
u(t)\geq u(\eta_1)-\int_0^\omega|u' (t)|dt>
\ln\{\frac{\bar{h}}{\bar{a}}\}-(\bar{A}+\bar{a})\omega:=\rho_2.
\end{equation}
Similarly, from \eqref{e3.4} and \eqref{e3.8}, we  derive that
\begin{equation*}
\bar{d}\omega< \int_0^\omega
\frac{f(t)\exp\{u(\eta_1)\}}{m(t)\exp\{v(\xi_2)\}}dt
=\frac{1}{\exp\{v(\xi_2)\}}\overline
{\big(\frac{f}{m}\big)}\exp\{u(\eta_1)\}\omega\,.
\end{equation*}
Then by \eqref{e3.9}, we have
$v(\xi_2)<\ln\{\frac{\bar{a}}{\bar{b}\bar{d}}\overline
{(\frac{f}{m})}\}+(\bar{A}+\bar{a})\omega$. This, together with
\eqref{e3.6}, gives
\begin{equation} \label{e3.11}
 v(t)\leq v(\xi_2)+\int_0^\omega|v'
(t)|dt< \ln\{\frac{\bar{a}}{\bar{b}\bar{d}}\overline
{\big(\frac{f}{m}\big)}\}+(\bar{A}+\bar{a})\omega.
\end{equation}
Moreover, from \eqref{e3.4}, \eqref{e3.7} and \eqref{e3.8}, we get
\begin{equation*}
\bar{d}\omega>\int_0^\omega\frac{f(t)\exp\{u(\xi_1)\}}
{m(t)\exp\{v(\eta_2)\}+\exp\{u(\xi_1)\}}dt>\frac{\exp\{u(\xi_1)\}\bar{f}\omega}
{m^M\exp\{v(\eta_2)\}+\exp\{u(\xi_1)\}}\,.
\end{equation*}
Then, by \eqref{e3.10}, we obtain
\begin{equation*}
v(\eta_2)>\ln\big\{\frac{(\bar{f}-\bar{d})\bar{u}}{m^M\bar{a}\bar{d}}
\big\}-(\bar{A}+\bar{a})\omega,
\end{equation*}
which, together with \eqref{e3.6}, produces
\begin{equation}
\label{e3.12} v(t)\geq v(\eta_2)-\int_0^\omega|v' (t)|dt>
\ln\big\{\frac{(\bar{f}-\bar{d})\bar{u}}{m^M\bar{a}\bar{d}}\big\}-
(\bar{A}+\bar{a}+\bar{D}+\bar{d})\omega:=\rho_4.
\end{equation}
It follows from \eqref{e3.11} and \eqref{e3.12} that
\begin{equation}
\label{e3.13} |v(t)|< |\rho_3|+|\rho_4|+1:=B_1.
\end{equation}
 From \eqref{e3.7} and the first equality of \eqref{e3.1}, we also
have
\begin{align*}
  &a(\eta_1)-b(\eta_1)\int_{-\infty}^{\eta_1}k(\eta_1-s)\exp\{u(s)\}ds\\
 &-\frac{c(\eta_1)\exp\{v(\eta_1)\}}{m(\eta_1)
  \exp\{v(\eta_1)\}+\exp\{u(\eta_1)\}}-\frac{h(\eta_1)}{\exp\{u(\eta_1)\}}=0,
\end{align*}
  which implies
\begin{equation*}
  b(\eta_1)\exp\{2u(\eta_1)\}-(
  a(\eta_1)-\frac{c(\eta_1)}{m(\eta_1)})\exp\{u(\eta_1)\}+h(\eta_1)>0.
  \end{equation*}
  Solving the inequality, we have
\begin{equation*}
  \exp\{u(\eta_1)\}<
  \frac{(a-c/m)^L-\sqrt{[(a-c/m)^L]^2-4b^Mu^M}}{2b^M}=:\delta_-,
  \end{equation*}
   or
 \begin{equation*}
  \exp\{u(\eta_1)\}>
  \frac{(a-c/m)^L+\sqrt{[(a-c/m)^L]^2-4b^Mu^M}}{2b^M}=:\delta_+.
 \end{equation*}
 That is, $u(\eta_1)<\ln \delta_-$ or $
u(\eta_1)>\ln\delta_+$.
  Similarly, we can obtain $u(\xi_1)<\ln\delta_-$ or $
  u(\xi_1)>\ln\delta_+$. These, together with \eqref{e3.9} and \eqref{e3.10},
  we obtain
\begin{equation}
\label{e3.14}
  \rho_2<u(t)<\ln\delta_-,\quad\text{or}\quad
 \ln\delta_+<u(t)<\rho_1.
  \end{equation}
By Lemma \ref{lem2.2}, the following algebraic equations
\begin{align*}
&\bar{a}-\bar{b}\exp\{u\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{c(t)
\exp\{v\}}{m(t)\exp\{v\}+
\exp\{u\}}dt-\frac{\bar{h}}{\exp\{u\}}=0\\
&-\bar{d}+
\frac{1}{\omega}\int_{0}^{\omega}\frac{f(t)\exp\{u\}}{m(t)\exp\{v\}+
\exp\{u\}}dt=0
\end{align*}
have two solutions, denoted by
 $(u_1,v_1)^T$ and $(u_2,v_2)^T~(v_1<v_2)$ and satisfying
\begin{equation}
\label{e3.15}
 \rho_2<u_1<\ln\delta_-,\quad\mbox{or}\quad  \ln\delta_+<u_2<\rho_1.
\end{equation}
Clearly, $\rho_1, \rho_2, B_1, \delta_-, \delta_+$ are independent
of $\lambda$.

 Now let us take $X=Y=\{(u(t),v(t))^T\in C(
\mathbb{R},\mathbb{R}^2)|u(t+\omega)=u(t),v(t+\omega)=v(t)\}$ and
$\|(u(t),v(t))^T\|=\max_{t\in [0,\omega]}|u(t)|+\max_{t\in
[0,\omega]}|v(t)|$. Then $X$ is a Banach space equipped with the
norm $\|\cdot\|$.

Let $L(u(t),v(t))^T=(u'(t),v'(t))^T$ and $N:X\to X$, where
\begin{align*}
&N\begin{pmatrix} u(t)\\v(t)\end{pmatrix}\\
&=\begin{bmatrix} a(t)-b(t)\int_{-\infty}^t
k(t-s)\exp\{u(s)\}ds-\frac{c(t)\exp\{v(t)\}}{m(t)
\exp\{v(t)\}+\exp\{u(t)\}}-\frac{h(t)}{\exp\{u(t)\}}\\
-d(t)+\frac{f(t)\exp\{u(t-\tau(t))\}}{m(t)
\exp\{v(t-\tau(t))\}+\exp\{u(t-\tau(t))\}}\end{bmatrix}.
\end{align*}
Define projectors $P$ and $Q$ by
$$
P\begin{pmatrix} u(t)\\v(t)\end{pmatrix} =Q\begin{pmatrix}
u(t)\\v(t)\end{pmatrix}
=\begin{pmatrix}\frac{1}{\omega}\int_0^\omega
u(t)dt\\\frac{1}{\omega}\int_0^\omega v(t)dt\end{pmatrix}, \quad
\begin{pmatrix} u(t)\\v(t)\end{pmatrix}\in X.
$$
Obviously, $\mathop{\rm Im} P=\ker L$ and $\mathop{\rm Im} L=\ker
Q=\{(u(t),v(t))^T\in X :\bar{u}=\bar{v}=0\}$ is closed in $X$, and
$\dim \ker L=2=codim \mathop{\rm Im} L$. Thus, $L$ is a Fredholm
operator of index zero. Furthermore, the generalized inverse (to
$L$) is as follows
$$
K_P: \mathop{\rm Im} L\to \mathop{\rm Dom}L\cap \ker P,\quad
K_P\begin{pmatrix} u(t)\\v(t)\end{pmatrix}=\begin{pmatrix}\int_0^t
u(s)ds-\frac{1}{\omega}\int_0^\omega\int_0^t u(s)dsdt\\\int_0^t
v(s)ds-\frac{1}{\omega}\int_0^\omega\int_0^t v(s)dsdt\end{pmatrix}.
$$
Now, we reach the point where we search for appropriate open bounded
subsets $\Omega_i,~i=1,2$ for the application of the continuation
theorem. To this end, we take $B_2=|v_1|+|v_2|$, and define
\begin{gather*}
\Omega_1=\{(u(t),v(t))^T\in X:\rho_2<u(t)<\ln \delta_-,
\max_{t\in [0,\omega]}|v(t)|<B_1+B_2\},\\
\Omega_2=\{(u(t),v(t))^T\in X:\ln \delta_+<u(t)<\rho_1, \max_{t\in
[0,\omega]}|v(t)|<B_1+B_2\}.
\end{gather*}
Clearly, both $\Omega_1$ and $ \Omega_2$ are open subsets of $X$ and
$\bar{\Omega}_1\cap\bar{\Omega}_2=\phi$ in view of
$\delta_-<\delta_+$. From \eqref{e3.15}, we see that $(u_1,v_1)^T\in
\Omega_1,(u_2,v_2)^T\in \Omega_2$.

By using the Arzela-Ascoli theorem, it is not difficult to show that
$QN(\Omega_i)$ and $K_P(I-Q)N(\Omega_i),$ $i=1, 2$, are compact.
Therefore, $N$ is $L$-compact on $\Omega_i, i=1, 2$.

Since we are concerned with  periodic solutions $(u(t),v(t))^T$
confined in $\mathop{\rm Dom}L$, system \eqref{e3.2} can be regarded
as the following operator equation $L(u(t),v(t))^T=\lambda
N(u(t),v(t))^T$, which is system \eqref{e3.1} when $\lambda=1$.
According to the previous estimation of periodic solution of
\eqref{e3.2}, we have proved  requirement $(i)$ of Lemma
\ref{lem2.1}.

When $(u,v)^T\in \partial\Omega_i\cap \ker L, i=1, 2,$ and $(u,v)^T$
is a constant vector in $ \mathbb{R}^2$. From \eqref{e3.13} and
\eqref{e3.15} and Lemma \ref{lem2.2}, it follows that
$$
QN\begin{pmatrix} u\\v\end{pmatrix} =\begin{bmatrix}
\bar{a}-\bar{b}\exp\{u\}-\frac{1}{\omega}\int_{0}^{\omega}\frac{c(t)\exp\{v\}}{m(t)
\exp\{v\}+\exp\{u\}}dt-\frac{\bar{h}}{\exp\{u\}}\\
\\
-\bar{d}+\frac{1}{\omega}\int_{0}^{\omega}\frac{f(t)\exp\{u\}}{m(t)
\exp\{v\}+\exp\{u\}}dt \end{bmatrix}\neq 0.
$$
Moreover, direct calculation shows that
\begin{equation*}
\deg (JQN,\Omega_i\cap \ker L,0) \neq 0,\quad i=1,2,
\end{equation*}
where $\deg (\cdot)$ is the Brouwer degree and the $J$ is the
identity mapping since $\mathop{\rm Im} Q=\ker L$.

By now, we have proved that each $\Omega_i (i=1,2)$ satisfies all
the requirements of Lemma \ref{lem2.2}. Hence, system \eqref{e3.1}
has at least one $\omega$-periodic solution in each of $\Omega_1$
and $\Omega_2$. The proof is completed.
\end{proof}

Next, we consider the  ratio-dependence predator-prey system with
distributed delays
\begin{equation} \label{e3.16}
\begin{gathered}
x'(t)=x\Big[ a(t)-b(t) \int_{-\tau}^0
x(t+s)d\mu(s)\Big]-\frac{c(t)xy}{m(t)y+x}-h(t),\\
y'(t)=y\Big[-d(t)+\frac{f(t)\int_{-\sigma}^0
x(t+s)d\nu(s)}{m(t)\int_{-\sigma}^0 y(t+s)d\nu(s)+\int_{-\sigma}^0
x(t+s)d\nu(s)}\Big],
\end{gathered}
\end{equation}
where $\tau, \sigma$ are positive constants and $\mu,\nu$ are
nondecreasing functions such that
\begin{equation*}
\mu(0^+)-\mu(-\tau^-)=1,\quad  \nu(0^+)-\nu(-\sigma^-)=1.
\end{equation*}

 \begin{theorem} \label{thm3.2}
 If $\bar{f}>\bar{d}$ and $(a-c/m)^L>2\sqrt{b^M h^M}$, then system
$(3.16)$ has at least two positive $\omega$-periodic solutions.
\end{theorem}

 The proof of the above theorem is similar to that of Theorem 3.1 and hence
is omitted here.

\begin{remark} \label{rmk3.1} \rm
 From the proofs of Theorem \ref{thm3.1} and \ref{thm3.2}, it
is seen that the conclusion of Theorem \ref{thm3.2} remains valid if
some or all of the $\tau's$ and $\sigma's$ are $\infty$.
\end{remark}

In system \eqref{e1.4}, when the distributed delay in the prey
equation is replaced by the periodic delay, that is, system
\eqref{e1.4} is rewritten   as
\begin{equation}\label{e3.17}
\begin{gathered}
x'(t)=x\big[
a(t)-b(t)x(t-\tau(t))\big]-\frac{c(t)xy}{m(t)y+x}-h(t),\\y'(t)
=y\Big[-d(t)+\frac{f(t)x(t-\tau(t))}{m(t)y(t-\tau(t))+x(t-\tau(t))}\Big],
\end{gathered}
\end{equation}
  the result remains valid.

\begin{theorem} \label{thm3.3}
 If $\bar{f}>\bar{d}$ and $(a-c/m)^L>2\sqrt{b^M h^M}$, then
system \eqref{e3.17} has at least two positive $\omega$-periodic
solutions.
\end{theorem}

Observing system \eqref{e3.17}, we can see that the delay is a
function of $t$. In real life, delay is not only depends on time,
but also on states. Therefore, we now propose the predator-prey
model
\begin{equation}
\label{e3.18}
\begin{gathered}
x'(t)=x\big[
a(t)-b(t)x(t-\tau(t,x(t),y(t)))\big]-\frac{c(t)xy}{m(t)y+x}-h(t),\\y'(t)
=y\Big[-d(t)+\frac{f(t)x(t-\tau(t))}{m(t)y(t-\tau(t,x(t),y(t)))
+x(t-\tau(t,x(t),y(t)))}\Big].
\end{gathered}
\end{equation}

By a similar discussion, we can obtain the following result.

\begin{theorem}\label{thm3.5}
 If $\bar{f}>\bar{d}$
and $(a-c/m)^L>2\sqrt{b^M h^M}$, then system \eqref{e3.18} has at
least two positive $\omega$-periodic solutions.
\end{theorem}

 Especially, when there is no exploited term, that is
$h(t)\equiv 0$, the system \eqref{e1.4} reduces to
\begin{equation}\label{e3.19}
\begin{gathered}
x'(t)=x\Big[ a(t)-b(t)
 \int_{-\infty}^t k(t-s)x(s)ds\Big]-\frac{c(t)xy}{m(t)y+x},\\y'(t)
=y\Big[-d(t)+\frac{f(t)x(t-\tau(t))}{m(t)y(t-\tau(t))+x(t-\tau(t))}\Big].
\end{gathered}
\end{equation}
Employing the powerful and effective coincidence degree method, we
can obtain the following theorem.

 \begin{theorem}  If $\bar{f}>\bar{d}$ and $\bar{a}>\overline{(c/m)}$,
then system \eqref{e3.19} has at least one positive
$\omega$-periodic solutions.
\end{theorem}

\begin{remark} \label{rmk3.6} \rm
In system \eqref{e3.19}, when m(t)=m is constant, system
\eqref{e3.19} reduces to the system which is studied by Fan and Wang
\cite{f1} and Theorem 3.5 reduces to the corresponding result in
\cite{f1}.
\end{remark}

At last, When $h(t)\equiv 0$, systems  \eqref{e3.16}- \eqref{e3.18}
can be reduced to corresponding simpler systems and we have the
following conclusion.

 \begin{theorem}  \label{thm3.6}
If $\bar{f}>\bar{d}$ and $\bar{a}>\overline{(c/m)}$, then systems
\eqref{e3.16}-\eqref{e3.18} have at least one positive
$\omega$-periodic solutions, respectively.
 \end{theorem}

 \section{Numerical example}

In this section, an example is given to illustrate our result.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\textwidth]{fig1}
\end{center}
\end{figure}

Take $a(t)=6+\sin(\pi t)$, $b(t)=\frac{3}{2}+\cos(\pi t)$,
$c(t)=2+\cos(\pi t)$, $m(t)=1+\frac{1}{2}\cos(\pi t)$,
$h(t)=\frac{1}{2}+\frac{1}{4}\sin(\pi t)$, $d(t)=1+\cos(\pi t)$,
$f(t)=3+\sin(\pi t)$. Moreover, the delay kernel $k(t)$ is chosen
 as a delta function in the form of $k(t)=\delta (t)$ and
$\tau(t)\equiv 0.2$, then
 system \eqref{e1.4} becomes
\begin{equation} \label{e4.1}
 \begin{aligned}
x'(t)&=x(t)\Big[ 6+\sin(\pi t)-(\frac{3}{2}+\cos(\pi t))
x(t)\Big]\\
&\quad -\frac{(2+\cos(\pi t))x(t) y(t)}{(1+\frac{1}{2}\cos(\pi
t))y(t)+x(t)}
-(\frac{1}{2}+\frac{1}{4}\sin(\pi t)),\\
 y'(t)&=y(t)\Big[-(1+\cos(\pi t))+\frac{(3+\sin(\pi t))x(t-0.2)}{(1+\frac{1}{2}
\cos(\pi t))y(t-0.2)+x(t-0.2)}\Big].
\end{aligned}
\end{equation}
In this case, all the parameters are 2-periodic functions. By simple
calculations, we have
\begin{equation}
\label{e4.2} \bar{f}=3,\quad \bar{d}=1,\quad
(a-\frac{c}{m})^L=3,\quad b^M=\frac{5}{2},\quad h^M=\frac{3}{4}.
\end{equation}
Therefore, \eqref{e4.2} shows that conditions of Theorem \ref{thm3.1}
hold and so system \eqref{e4.2} has at least two positive 2-periodic
solutions. With initial values $x(0)=10,\,y(0)=5$ and $t\in [0,30]$,
the above figure shows that the existence of periodic solutions.

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\end{document}