\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 164, 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}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/164\hfil Nonexistence of periodic orbits]
{Nonexistence of periodic orbits for predator-prey system
with strong Allee effect in prey populations}
\author[J. Wang, J. Shi, J. Wei \hfil EJDE-2013/164\hfilneg]
{Jinfeng Wang, Junping Shi, Junjie Wei} % in alphabetical order
\address{Jinfeng Wang \newline
School of Mathematical Science,
Harbin Normal University, Harbin, Heilongjiang, 150025, China}
\email{jinfengwangmath@163.com}
\address{Junping Shi \newline
Department of Mathematics, College of William and Mary,
Williamsburg, VA 23187-8795, USA}
\email{shij@math.wm.edu}
\address{Junjie Wei \newline
Department of Mathematics,
Harbin Institute of Technology,
Harbin, Heilongjiang, 150001, China}
\email{weijj@hit.edu.cn}
\thanks{Submitted March 25, 2013. Published July 19, 2013.}
\subjclass[2000]{34C25, 34D23, 92D25}
\keywords{Predator-prey system; nonexistence of periodic orbits;
\hfill\break\indent Dulac criterion; global bifurcation}
\begin{abstract}
We use Dulac criterion to prove the nonexistence of periodic orbits
for a class of general predator-prey system with strong Allee effect
in the prey population growth. This completes the global bifurcation
analysis of typical predator-prey systems with strong Allee effect
for all possible parameters.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks
\section{Introduction}
The importance of limit cycles in predator-prey systems has been recognized by
ecologists since the observation of Rosenzweig \cite{R} and May
\cite{M}. The existence and uniqueness of the limit cycle
in planar systems is mathematically quite non-trivial, and there are many
important work on that direction in the last 30 years, see for example
\cite{C,KF,XZ,Z}.
On the other hand, the nonexistence of limit cycles of some planar systems
is also useful for excluding oscillatory behavior, and it often implies the
global stability of an equilibrium point.
It is well known that the Dulac criterion \cite{D} is an efficient method for
proving the nonexistence of closed orbits. However, in general it is
difficult to find a suitable Dulac function for specific systems.
Many work on the existence (nonexistence) and
uniqueness of limit cycles are carried out, for example in
\cite{C,KF,XZ,Z}, by translating a
planar system into a Li\'enard system. But the conditions
for the nonexistence of limit cycles are usually difficult to
verify (\cite{WSW,XZ}). In this paper, we prove the nonexistence of limit
cycles for a class of general predator-prey systems with strong Allee effect,
as well as a Rosenzweig-MacArthur predator-prey model \cite{CS,H2}
(or Gause type predator-prey model \cite{H1,RM})
by constructing a suitable Dulac function.
A differential equation model of predator-prey interaction was first
formulated by Lotka \cite{Lo} and Volterra \cite{Vo}
in 1920s, hence it is called Lotka-Volterra
equation:
\begin{equation}\label{lv}
\begin{gathered}
\frac{du}{dt}=au-buv, \\
\frac{dv}{dt}=cuv-dv,
\end{gathered}
\end{equation}
where $a,b,c,d>0$. A more realistic predator-prey model assumes that the
prey grows following a logistic law, and the interaction rate between the
prey and predator species saturates to a finite limit when the prey
population tends to infinity (Holling type II functional response).
This was the basis of the Rosenzweig-MacArthur predator-prey model \cite{R,RM}:
\begin{equation}\label{rm}
\begin{gathered}
\frac{du}{dt}=ru\big(1-\frac{u}{K}\big)-\frac{muv}{a+u}, \\
\frac{dv}{dt}=\frac{cmuv}{a+u}-dv,
\end{gathered}
\end{equation}
where $a,c,d,r,K>0$. For some biological growth, a minimal threshold value
for the growth exists then instead of the logistic type growth, one may
assume a growth pattern of Allee effect \cite{AW}, in which the growth
rate per capita is initially increasing for the low density.
Moreover it is called a strong Allee effect if the per capita growth rate of
low density is negative, and a weak Allee effect means that the per
capita growth rate is positive at low density. A predator-prey model under
the assumption of strong Allee effect and Holling type II functional
response is in form (\cite{CS,WSW}):
\begin{equation}\label{allee}
\begin{gathered}
\frac{du}{dt}=ru\big(1-\frac{u}{K}\big)\big(\frac{u}{M}-1\big)-\frac{muv}{a+u}, \\
\frac{dv}{dt}=\frac{cmuv}{a+u}-dv,
\end{gathered}
\end{equation}
where $a,c,d,r,K>0$ and $00$ on
$[A,\bar{\lambda})$, $f'(u)<0$ on
$(\bar{\lambda},K]$;
\item[(A2)] $g\in C^1(\mathbb{R}^+)$, $g(0)=0$; $g(u)>0$ for $u>0$ and $g'(u)> 0$
for $u\ge 0$, and there exists $\lambda>0$ such that $g(\lambda)=d$.
\item[(A3)] $f(u)$ and $g(u)$ are $C^3$ near $\lambda=\bar{\lambda}$, and
$f''(\bar{\lambda})<0$.
\end{itemize}
Here the function $g(u)$ is the predator functional response, and
$g(u)f(u)$ is the net growth rate of the prey. The graph of $v=f(u)$
is the prey isocline on the phase portrait. In the absence of the
predator, the prey $u$ has a strong Allee effect growth which can
been seen from the assumptions (A1). The carrying capacity of the
prey is $K$, while $A$ is the survival threshold
of the prey. The predator isocline is a vertical line $u=\lambda$ solved
from $g(\lambda)=d$. The condition (A2) on the functional response
$g(u)$ includes the commonly used Holling types II and III as
well as the linear Lotka-Volterra one. When the functional response
$g(u)=u$, then $f(u)$ is the growth rate per capita. The parameter
$d$ is the mortality rate of predator; the number $\lambda$ can also be
thought as a measure of the predator mortality as $\lambda$ increases
with $d$, and $\lambda$ is also the stationary prey population density
coexisting with predator. The $C^3$ conditions in (A3) is only to
fulfill the standard condition for a Hopf bifurcation \cite{W}.
It is known
that $\lambda=\bar{\lambda}$ is the Hopf bifurcation point, and the
bifurcation is supercritical if $f'''(\bar{\lambda})\le 0$ and
$g''(\bar{\lambda})\le 0$. We note that system \eqref{allee} satisfies
the assumptions (A1)-(A3), and more examples satisfying (A1)-(A3)
can be found in Section 3 where applications of our main results are given.
On the other hand, we will also consider predator-prey systems of
Rosenzweig-MacArthur type in Section 4, where we define a parallel
set of assumptions (A1')-(A2') which are satisfied by \eqref{lv}
and \eqref{rm}.
The dynamical properties of some special cases of system
\eqref{general} have been obtained by numerical simulation in recent
studies \cite{DML,Ma,GA}. The rigorous global dynamics and
bifurcation of \eqref{general} has been thoroughly investigated in
our previous paper \cite{WSW}, by utilizing phase portrait analysis and
performing global bifurcation analysis, the
existence/uniqueness of point-to-point heteroclinic orbit and limit
cycle are obtained. One of the main results in \cite{WSW} is as follows
(see \cite[Theorem 5.2]{WSW}, and we use the same numbering of assumptions
in \cite{WSW}).
\begin{theorem}\label{final-thm}
Suppose that $f(u)$ satisfies {\rm (A1), (A3)} and
\begin{itemize}
\item[{\rm (A6)}] $uf'''(u)+2f''(u)\le 0$ for all $u\in (A,K)$;
\end{itemize}
and $g(u)$ is one of the following:
\begin{equation}\label{gu}
g(u)=u, \quad \text{or } \quad g(u)=\frac{mu}{a+u}, \quad a,m>0.
\end{equation}
Then with a bifurcation parameter $\lambda$ defined by
\begin{equation}\label{laa}
\lambda=d \text{ if } g(u)=u, \quad \text{or}\quad
\lambda=\frac{ad}{m-d} \text{ if } g(u)=\frac{mu}{a+u},
\end{equation}
there exist two bifurcation points $\lambda^{\sharp}$ and $\bar{\lambda}$
such that the dynamics of \eqref{general} can be classified as
follows:
\begin{enumerate}
\item If $0<\lambda<\lambda^{\sharp}$, then the equilibrium $(0,0)$ is
globally asymptotically stable;
\item If $\lambda^{\sharp}<\lambda<\bar{\lambda}$, then there exists a unique
limit cycle, and the system is globally bistable with respect to the limit
cycle and $(0,0)$;
\item If $\bar{\lambda}<\lambdaK$, then the system is globally bistable with respect to
$(K,0)$ and $(0,0)$.
\end{enumerate}
\end{theorem}
For more general results on the dynamics of \eqref{general}, see
\cite{WSW}. However one can see that when $\bar{\lambda}<\lambda0\}$.
We denote the portion of $\Gamma_{\lambda}^u$ between $u=\lambda$ and
$u=K$ by $(u,v_1(u))$. We claim that $v_1(u)\leq (1-f'(K))(K-u)$.
Define $v_2(u)=\left(1-f'(K)\right)(K-u)$, we notice that the tangent
line of the unstable manifold is
\begin{equation*}
v=\Big(1-f'(K)-\frac{d}{g(K)}\Big)(K-u),
\end{equation*}
which is below $v=v_2(u)$. Hence we only need to show that the vector
field $(f_1(u,v), f_2(u,v))$ points towards the region below the line
$v=v_2(u)$ when $(u,v)=\left(u,v_2(u)\right)$ and $\lambda__0$.
By the Dulac criterion (Lemma \ref{planer}),
\eqref{general} has no closed orbits in the first quadrant if
$\bar{\lambda}<\lambda0$ and $g(u)$ is one of
the forms in \eqref{gu}, then the Hopf bifurcation at
$\lambda=\bar{\lambda}$ is subcritical and \eqref{general} has two periodic
orbits for $\lambda\in (\bar{\lambda},\bar{\lambda}+\epsilon)$ for a small
$\epsilon>0$ (see \cite{WSW} for examples). On the other hand, we only
assume some concavity condition
on $f(u)$ for $u\in (\bar{\lambda},K)$ not for all $u\in (A,K)$.
\section{Examples}
In this section we apply our results to several examples of predator-prey
system with strong Allee effect which have
been studied in \cite{WSW}.
\subsection{Bazykin-Conway-Smoller model}
The predator-prey model with Lotka-Volterra
interaction and Allee effect quadratic growth rate per capita (in dimensionless version) is:
\begin{equation}\label{BCS-linear}
\begin{gathered}
\frac{du}{dt}=u(1-u)\left(\frac{u}{b}-1\right)-muv,\\
\frac{dv}{dt}=-dv+muv.
\end{gathered}
\end{equation}
Analysis of \eqref{BCS-linear} can be found in \cite{Ba,CS,WSW}, and
we only consider the nonexistence of
periodic orbits here. For \eqref{BCS-linear}, we define
\begin{equation}\label{fg1}
f(u)=\frac{(1-u)(u-b)}{bm}, \quad g(u)=mu.
\end{equation}
One can easily verify that
\begin{equation*}
\bar{\lambda}=\frac{1+b}{2}, \quad f'(u)=\frac{-2u+(b+1)}{bm},\quad
f''(u)=\frac{-2}{bm}<0,\quad f'''(u)=0.
\end{equation*}
Then (A1), (A2) and (A8) (or (A9)) are satisfied for $f,g$ in
\eqref{fg1}. Hence the result in Theorem \ref{thm:dulac} holds.
In fact we have obtained the same result as in \cite{WSW} due to
\cite[Theorem 2.5]{XZ}
(or \cite[Theorem 4.2]{WSW}), but Theorem \ref{thm:dulac} is much
easier to apply.
The corresponding phase portrait can be found in
Figure \ref{figure-BCS-phase}(left).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig2a} %phase-BCS-linear.eps}
\includegraphics[width =0.49\textwidth]{fig2b} %phase-BCS-Holling.eps}
\end{center}
\caption{Phase portraits of \eqref{BCS-linear}(Left) and
\eqref{BCS-Holling}(Right).
For either cases, there is no limit cycle, and there are two locally
stable equilibrium points
$(0,0)$ and $(\lambda,v_{\lambda})$.
The horizontal axis is the prey population $u$, and the
vertical axis is the predator population $v$. The dotted curve is
the $u$-isocline $v=f(u)$, and the solid vertical line is the
$v$-isocline $g(u)=d$ or $u=\lambda$. Parameters used are given: (Left)
\eqref{BCS-linear} with $m=1$, $A=0.2$, $K=1$, $d=0.7$; (Right)
\eqref{BCS-Holling} with $m=1$, $A=0.2$, $K=1$, $d=0.58$, $a=0.5$}
\label{figure-BCS-phase}
\end{figure}
\subsection{Owen-Lewis model}
A prototypical predator-prey model with Holling type II functional response
and Allee effect on the prey was proposed by Owen and Lewis
\cite{OL}, and also Petrovskii et.al. \cite{MPL1}, which in dimensionless
version is
\begin{equation}\label{BCS-Holling}
\begin{gathered}
\frac{du}{dt}=u(1-u)\left(\frac{u}{b}-1\right)-\frac{muv}{a+u}, \\
\frac{dv}{dt}=-dv+\frac{muv}{a+u}.
\end{gathered}
\end{equation}
For \eqref{BCS-Holling},
\begin{equation}\label{fg2}
f(u)=\frac{(a+u)(1-u)(u-b)}{bm}, \quad
g(u)=\frac{mu}{a+u}.
\end{equation}
The critical point $\bar{\lambda}$ of
$f(u)$ in $(b,\lambda)$ (which is also the Hopf bifurcation point) has the form
\begin{equation*}
\bar{\lambda}=\frac{b+1-a+\sqrt{(b+1-a)^2+3(ab+a-b)}}{3}
\end{equation*}
which is the larger root of $f'(\lambda)=0$. Here
\begin{gather*}
f'(u)=\frac{-3u^2+2(1+b-a)u+a(1+b)-b}{bm},\\
f''(u)=\frac{2(-3u+b+1-a)}{bm},\quad
f'''(u)=\frac{-6}{bm}<0.
\end{gather*}
Hence $f''(\bar{\lambda})=\frac{2(-3\bar{\lambda}+b+1-a)}{bm}<0$ implies that
$f''(u)<0$ for all $\bar{\lambda}\leq uA>0$, $r,B,C,n>0$ and $h\ge 0$.
With $K>A>0$, \eqref{RM} exhibits a strong Allee effect in prey
population density. If $n=1$ and $h=0$, then the functional response
is linear, and we have
\begin{equation}\label{BSB-linear}
\begin{gathered}
\frac{du}{dt}=ru\big(1-\frac{u}{K}\big)
\big(1-\frac{A+C}{u+C}\big)-Buv,\\
\frac{dv}{dt}=-dv+Buv.
\end{gathered}
\end{equation}
If $n=1$ and $h>0$, then the functional response is Holling II, and we have
\begin{equation}\label{BSB-Holling}
\begin{gathered}
\frac{du}{dt}=ru\big(1-\frac{u}{K}\big)
\big(1-\frac{A+C}{u+C}\big)-\frac{m uv}{a+u}, \\
\frac{dv}{dt}=-dv+\frac{muv}{a+u},
\end{gathered}
\end{equation}
with $a=1/(hB)$, $m=1/h$.
For \eqref{BSB-linear} with linear functional response,
\begin{equation}\label{fg3}
f(u)=\frac{r(K-u)(u-A)}{BK(u+C)}, \quad g(u)=Bu.
\end{equation}
The critical point $\bar{\lambda}$ of
$f(u)$ in $(A,K)$ (Hopf bifurcation point) has the form
\begin{equation*}
\bar{\lambda}=-C+\sqrt{N}, \quad \text{where } N=(C+A)(C+K).
\end{equation*}
which is the larger root of $f'(\lambda)=0$ with
\begin{gather*}
f'(u) =\frac{r}{BK}\Big(-1+\frac{N}{(u+C)^2}\Big),\\
f''(u) =\frac{-2rN}{BK(u+C)^3}<0, \quad
f'''(u) =\frac{6rN}{BK(u+C)^4}>0.
\end{gather*}
Here (A9) is not satisfied. But it is obvious that (A1)-(A2) and
(A7) are satisfied, and if $C\ge K/2$, then for any $u\in [A,K]$,
\begin{equation*}
uf'''(u)+2f''(u)=\frac{2rN(u-2C)}{BK(u+C)^4}\leq 0.
\end{equation*}
Thus (A8) holds and the result in Theorem \ref{thm:dulac} holds
for all $\bar{\lambda}<\lambda\bar{\lambda}$ if $C$ is
sufficiently large such that $C+a-K-A\geq 0$. Moreover when $C$ is
sufficiently large such that $M_2>0$, then (A1), (A2) and (A9) are satisfied.
Hence the result in Theorem \ref{thm:dulac} holds.
The corresponding phase portrait can be found in
Figure \ref{figure-BSB-phase}(right).
For both \eqref{BSB-linear} and \eqref{BSB-Holling}, subcritical
Hopf bifurcation is possible when $C$ is small, see \cite{WSW} for
details.
\begin{figure}[ht]
\centering
\includegraphics[width=0.49\textwidth]{fig3a} % phase-BSB-linear.eps}
\includegraphics[width=0.49\textwidth]{fig3b} %phase-BSB-Holling.eps}
\caption{Phase portraits of \eqref{BSB-linear}(Left) and
\eqref{BSB-Holling}(Right). The horizontal axis is the prey population $u$,
and the vertical axis is the predator population $v$. The dotted curve is
the $u$-isocline $v=f(u)$, and the solid vertical line is the
$v$-isocline $g(u)=d$ or $u=\lambda$. Parameters used are given:
(Left)\eqref{BSB-linear} with $r=B=1$, $A=0.4$, $K=1$, $d=0.8$,
$C=0.6$; (Right) \eqref{BSB-Holling} with $r=m=1$, $A=0.4$, $K=1$,
$d=0.62$, $a=0.5$, $C=3$}\label{figure-BSB-phase}
\end{figure}
\section{Rosenzweig-MacArthur model}
Most of these work are for predator-prey model with positive prey isocline
without Allee effect, namely the Rosenzweig-MacArthur (or Gause type)
predator-prey model, which takes a similar form as \eqref{general}:
\begin{equation}\label{equ:RM}
\begin{gathered}
\frac{du}{dt}=g(u)\left(f(u)-v\right), \\
\frac{dv}{dt}=v\left(g(u)-d(u)\right).
\end{gathered}
\end{equation}
Here we assume that $f, g, d$ satisfy
\begin{itemize}
\item[(A1')] $f\in C^3(\mathbb{R}^+)$, $f(0)>0$, there exists $K>0$,
such that for any $u>0$, $u\neq K$, $f(u)(u-K)<0$ and $f(K)=0$;
there exists $\bar{\lambda}\in (0,K)$ such that $f'(u)>0$ on
$[0,\bar{\lambda})$, $f'(u)<0$ on $(\bar{\lambda},K]$;
\item[(A2')] $g, d\in C^2(\mathbb{R}^+)$, $g(0)=0$; $g(u)>0$ for $u>0$ and
$g'(u)> 0$ for $u\ge 0$; $d(0)>0$, $d'(u)\le 0$ for $u\ge 0$ and
$\lim_{u\to\infty}d(u)=d_{\infty}>0$; there exists a unique
$\lambda\in (0,K)$ such that $g(\lambda)=d(\lambda)$.
\end{itemize}
The function $g(u)f(u)$ is the net growth rate of the prey in the absence
of predators, $g(u)$ is the predator functional response,
and $d(u)$ is the mortality rate of the predator which depends on
the prey density.
The method of constructing a Dulac function to prove the nonexistence of
periodic orbits in predator-prey systems was first used in Hsu \cite{H1},
and it was modified and improved in Hofbauer and so
\cite{HS}, Kuang \cite{K}, Liu \cite{L}, Ruan and Xiao \cite{RX}.
In this case, the nonexistence of periodic orbits here and the local
stability of the coexistence equilibrium point together imply
the global stability of the coexistence equilibrium in the first quadrant.
Another way of proving global stability of coexistence equilibrium is to
use appropriate Lyapunov functional, see \cite{H1,RX,XZ}.
Other studies of the limit cycle of \eqref{equ:RM} can be found in
\cite{C,AGSS,GSS,HM,HS1,KY,KF}
Here we revisit the nonexistence of periodic orbits of \eqref{equ:RM},
and we modify the method in Section 2 to obtain
the following global stability result. Similar construction has been
used in \cite{HS,L,RX}, but the results are not completely same.
\begin{theorem}\label{thm:dulac-RM}
Suppose that $f,g,d$ satisfies {\rm (A1'), (A2')} and one of the followings:
\begin{itemize}
\item [(A8')] $(uf'(u))'' \leq 0$, $\left(ud(u)/g(u)\right)''\geq 0$ for all
$u\in [0,K]$, and $(uf'(u))'\le 0$ for $u\in (\bar{\lambda},K)$; or
\item [(A9')] $f'''(u) \leq 0$ and $\left(d(u)/g(u)\right)''\geq 0$ for all
$u\in [0,K]$, and $f''(u)\le 0$ for $u\in(\bar{\lambda},K)$,
\end{itemize}
then \eqref{equ:RM} has no closed orbits in the first quadrant for
$\bar{\lambda}<\lambda0.
\]
Then $F_1'(\lambda)=0$, $F_1(\lambda)=\lambda f'(\lambda)<0$. Again (A8') and $\eta > 0$
imply that $F_1''(u)\leq 0$ for all $u\in[0,K]$, so $F_1(u)$ is concave on
$u\in [0,K]$. Therefore $F_1(u)<0$ for all $u\geq 0$.
The Dulac criterion implies that \eqref{equ:RM} has no closed orbits
in first quadrant for $\bar{\lambda}<\lambda0.
\]
Then $F_2'(\lambda)=0$, $F_2(\lambda)=f'(\lambda)<0$. Again (A9') and
$\eta > 0$ imply that $F_2(u)$ is concave for $0\le u\le K$.
Therefore $F_2(u)<0$ for all $u\geq 0$, the same conclusion holds.
Moreover, (A1') shows that the unique nonnegative equilibrium
$(\lambda,v_{\lambda})$ is locally stable for $\bar{\lambda}<\lambda0$, then the growth rate per capita $f(u)$ must
be of weak Allee effect type from (A1').
An example with weak Allee effect growth rate on the
prey is given by \eqref{BSB-linear} when
$A<0$ and $C>-A$. It has been shown in \cite{WSW} that at the Hopf
bifurcation point $(\bar{\lambda},v_{\bar{\lambda}})$, the sign of
bifurcation stability is determined by
\begin{equation*}
a(\bar{\lambda})=\bar{\lambda}f'''(\bar{\lambda})+2f''(\bar{\lambda})
=\frac{2rN(\bar{\lambda}-2C)}{BK(\bar{\lambda}+C)^4}.
\end{equation*}
If we choose the parameters so that $KA+(K+A)C>8C^2$ to make
$a(\bar{\lambda})>0$, then the Hopf bifurcation is subcritical, and
there are two periodic orbits for $\lambda\in
(\bar{\lambda},\bar{\lambda}+\epsilon)$ (see Figure \ref{figure-weak-Allee}).
This again shows the condition (A8') is optimal.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig4a} %weak-1-b.eps}
\includegraphics[width=0.49\textwidth]{fig4b} %weak-2.eps}
\end{center}
\caption{Phase portraits of \eqref{BSB-linear} with
weak Allee effect. (Left): The Hopf bifurcation at $\bar{\lambda}$ is
subcritical with parameters $r=B=1$,
$A=-0.028$, $K=1$, $d=0.10199$, $C=0.05$;
(Right) The Hopf bifurcation at $\bar{\lambda}$ is supercritical with parameters
$r=B=1$, $A=-0.028$, $K=1$, $d=0.6$, $C=2$}\label{figure-weak-Allee}
\end{figure}
\subsection*{Acknowledgements}
This research is supported by grants 11031002 and 11201101 from
the National Natural Science Foundation of
China, grant DMS-1022648 from the National Science Foundation of
US, grant A201106 from the Natural Science Foundation of Heilongjiang Province,
and grant 12521152 from the Scientific Research Project of Heilongjiang
Provincial Department of Education.
\begin{thebibliography}{00}
\bibitem{AW} W. C. Allee; Animal Aggregations, A Study in General Sociology.
University of Chicago Press, (1931).
\bibitem{Ba} A. D. Bazykin; Nonlinear dynamics of interacting populations.
World Scientific Series on Nonlinear Science. Series A:
Monographs and Treatises, 11.
World Scientific Publishing Co., Inc., River Edge, NJ, 1998.
\bibitem{DML} S. D. Boukal, W. M. Sabelis, L. Berec;
How predator functional responses and Allee effects in prey affect
the paradox of enrichment and population collapses. \emph{Theo.
Popu. Biol.} \textbf{72} (2007), 136--147.
\bibitem{C} K. S. Cheng,
Uniqueness of a limit cycle for a predator-prey system. \emph{SIAM
J. Math. Anal.} \textbf{12} (1981), 541--548.
\bibitem{CHL} K. S. Cheng, S. B. Hsu, S. S. Lin;
Some results on global stability of a predator-prey system.
\emph{J. Math. Bio.} \textbf{12} (1982), 115--126.
\bibitem{CS} E. D. Conway, J. A. Smoller;
Global analysis of a system of predator-prey equations.
\emph{SIAM J. Appl. Math.} \textbf{46} (1986), 630--642.
\bibitem{CBG} F. Courchamp, L. Berec, J. Gascoigne;
Allee Effects in Ecology and Conservation.
Oxford University Press, (2008).
\bibitem{D} H. Dulac; Recherche des cycles limites.
\emph{C.R. Acad. Sci. Paris} \textbf{204} (1937), 1703--1706.
\bibitem{AGSS} E. Gonz\'alez-Olivares, B. Gonz\'alez-Ya\~nez,
E. S\'aez, I. Sz\'ant\'o;
On the number of limit cycles in a predator prey model with
non-monotonic functional response. \emph{Discrete Contin. Dyn.
Syst. Ser. B} \textbf{6} (2006), no. 3, 525--534.
\bibitem{GSS} E. Gonz\'alez-Olivares, H. Meneses-Alcay,
B. Gonz\'alez-Ya\~nez; J. MenaLorca, A. Rojas-Palma, R. Ramos-Jiliberto;
Multiple stability and uniqueness of the limit cycle in a Gause predator
prey model considering the Allee effect on prey.
\emph{Nonlinear Analysis: RWA} \textbf{12} (2011), no. 6, 2931--2942.
\bibitem{HM} M. Hesaaraki, S. M. Moghadas;
Existence of limit cycles for predator prey systems with a class
of functional responses. \emph{Ecological Modelling} \textbf{142} (2001) 1--9.
\bibitem{HS} J. Hofbauer, J. So;
Multiple limit cycles for predator-prey models.
\emph{Math. Biosci.} \textbf{99} (1990), 71--75.
\bibitem{H1} S. B. Hsu;
On global stability of a predator-prey system. \emph{Math.
Biosci.} \textbf{39} (1978), 1--10.
\bibitem{H2} S. B. Hsu;
Ordinary differential equations with applications.
Series on Applied Mathematics, \textbf{16}. World Scientific
Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
\bibitem{HS1} S. B. Hsu, J. P. Shi;
Relaxation oscillator profile of limit cycle in predator-prey
system. \emph{Disc. Cont. Dyna. Syst.-B} \textbf{11} (2009) no. 4, 893--911.
\bibitem{JS} J. F. Jiang, J. P. Shi;
Bistability dynamics in some structured ecological models.
In \lq\lq Spatial Ecology\rq\rq, CRC Press, (2009).
\bibitem{KY} Y. Kuang;
On the location and period of limit cycles in Gause-Type predator-prey systems.
\emph{J. Math. Anal. Appl.} \textbf{142} (1989), 130--143.
\bibitem{K} Y. Kuang;
Global stability of Gause-type predator-prey systems.
\emph{J. Math. Biol.} \textbf{28} (1990), 463--474.
\bibitem{KF} Y. Kuang, H. I. Freedman;
Uniqueness of limit cycles in Gause-type models of predator-prey systems.
\emph{Math. Biosci.} \textbf{88} (1988), 67--84.
\bibitem{L} Y. P. Liu;
Geometric criteria for the nonexistence of cycles in Gause-type
predator-prey systems.
\emph{Proc. Amer. Math. Soc.} \textbf{133} (2005), 3619--3626.
\bibitem{Lo} A. J. Lotka;
Elements of Physical Biology. Williams and
Wilkins, Baltimore, (1925).
\bibitem{Ma} H. Malchow, S. V. Petrovskii, E. Venturino;
Spatiotemporal patterns in ecology and epidemiology. Theory, models,
and simulation. Chapman \& Hall/CRC Mathematical and Computational
Biology Series. Chapman \& Hall/CRC, Boca Raton, FL, 2008.
\bibitem{M} R. M. May;
Limit cycles in predator-prey communities.
\emph{Science} \textbf{177}, (1972), 900--902.
\bibitem {MPL1} A. Morozov, S. Petrovskii, B.-L. Li;
Bifurcations and chaos in a predator-prey system with the Allee
effect. \emph{Proc. Royal Soc. London Series B-Biol. sci.}
\textbf{271}, (2004), 1407--1414.
\bibitem{OL} M. R. Owen, M. A. Lewis;
How predation can slow, stop or reverse a prey invasion.
\emph{Bull. Math. Biol.} \textbf{63}, (2001), 655--684.
\bibitem{R} M. L. Rosenzweig;
Paradox of enrichment: destabilization
of exploitation ecosystems in ecological time. \emph{Science}
\textbf{171}, (1971), no. 3969, 385--387.
\bibitem{RM} M. L. Rosenzweig, R. MacArthur;
Graphical representation and stability conditions of predator-prey interactions.
\emph{Amer. Natur.} \textbf{97} (1963), 209--223.
\bibitem{RX} S. G. Ruan, D. M. Xiao;
Global analysis in a predator-prey system with nonmonotonic
functional response. \emph{SIAM J. Appl. Math.} \textbf{61} (2001), 1445--1472.
\bibitem{SS} J. P. Shi, R. Shivaji;
Persistence in reaction diffusion models with weak Allee effect.
\emph{Jour. Math. Biol.} \textbf{52} (2006), 807--829.
\bibitem{GA} G. A. K. van Voorn, L. Hemerik, M. P. Boer, B. W. Kooi;
Heteroclinic orbits indicate overexploitation in
predator prey systems with a strong Allee effect. \emph{Math.
Biosci.} \textbf{209} (2007), 451--469.
\bibitem{Vo} V. Volterra;
Fluctuations in the abundance of species,
considered mathmatically. \emph{Nature} \textbf{118} (1926), 558.
\bibitem{WSW} J. F. Wang, J. P. Shi, J. J. Wei;
Predator-prey system with strong Allee effect in
prey. \emph{J. Math. Biol.} \textbf{62} (2011), 291--331.
\bibitem{W} S. Wiggins;
Introduction to applied nonlinear dynamical systems and chaos.
\emph{Texts in Applied Mathematics} \textbf{2}. Springer-Verlag,
New York, 1990.
\bibitem{XZ} D. M. Xiao, Z. F. Zhang;
On the uniqueness and nonexistence of limit cycles for
predator-prey system. \emph{Nonlinearity} \textbf{16} (2003), 1--17.
\bibitem{Z} Z. F. Zhang;
Proof of the uniqueness theorem of limit cycles of generalized
Li\'enard equations. \emph{Appl. Anal.} \textbf{23} (1986), no.
1-2, 63--76.
\end{thebibliography}
\end{document}
__