\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\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 $0<M<K$.

In this article we consider the following predator-prey system with strong 
Allee effect under very general conditions, following \cite{WSW}:
\begin{equation}\label{general}
\begin{gathered}
\frac{du}{dt}=g(u)(f(u)-v), 
\frac{dv}{dt}=v(g(u)-d),
\end{gathered}
\end{equation}
where $f,g$ satisfy the following assumptions:
\begin{itemize}
  \item[(A1)] $f\in C^2(\mathbb{R}^+)$, $f(A)=f(K)=0$, where $0<A<K$; $f(u)$ 
is positive for   $A<u<K$, and $f(u)$ is negative otherwise; 
there exists $\bar{\lambda}\in (A,K)$ such that $f'(u)>0$ 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 $\bar{\lambda}<\lambda<K$, and if there is no periodic orbit, then
  the system is globally bistable with respect to
  the coexistence equilibrium $(\lambda,v_{\lambda})$ and $(0,0)$;
\item If $\lambda>K$, 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 $\bar{\lambda}<\lambda<K$, the
nonexistence of periodic orbit is assumed rather than proved in
Theorem \ref{final-thm}. For several special cases, the nonexistence
of periodic orbit is established by applying a general result on
Li$\acute{\rm e}$nard equation \cite{XZ}.

In this article we provide this missing link in our studies in
\cite{WSW} by proving a general  nonexistence result of limit cycles
for \eqref{general} with direct application of the Dulac criterion, and we will
prove that under the conditions of Theorem \ref{final-thm}, indeed there are
no periodic orbits for \eqref{general}. Hence the nonexistence of
periodic orbits in the part 3 in Theorem \ref{final-thm} can be
\textit{proved} instead of \textit{assumed}.
Our result is proved under the conditions (A1)-(A2) on $f$ and $g$,
as well as one of two additional but natural conditions, see Theorem
\ref{thm:dulac}. Our result is motivated by earlier ones in
\cite{HS,H1} for Rosenzweig-MacArthur model with logistic type growth.


The rest of the paper is structured in the following way. In Section
2, we prove our main result of the nonexistence of limit cycles of
\eqref{general} by constructing a suitable Dulac function. In
Section 3 we apply the main results to some typical predator-prey
systems with strong Allee effect, following the same line as
\cite{WSW}. We discuss the corresponding result for
Rosenzweig-MacArthur model without strong Allee effect in Section 4,
which includes the cases of logistic or weak Allee effect growth.

\section{Nonexistence of periodic orbits}

Recalling from \cite{WSW}, there are four possible equilibrium points of 
\eqref{general}:
\begin{equation*}
(0,0),\quad (K,0),\quad (A,0),\quad (\lambda,v_{\lambda})=(\lambda,f(\lambda)),
\end{equation*}
where $\lambda$ is defined in (A2).
 The coexistence equilibrium point $(\lambda,v_{\lambda})$  is the
intersection of the prey isocline $v=f(u)$ and the predator isocline
$g(u)=d$ (or $u=\lambda$), and it is a positive equilibrium only when
$A<\lambda<K$ (see Figure \ref{fig1} left). Otherwise there are only three
equilibrium points in the positive quadrant or boundary.

We construct a bounded region that contains the limit cycle. 

\begin{lemma}\label{lem:dulac}
Suppose that $f,g$ satisfy {\rm (A1)-(A2)}, and
\begin{itemize}
\item [{\rm (A7)}] $f''(u) \leq 0$ for all $u\in(A,K)$,
\end{itemize}
 then all the closed orbits
of \eqref{general} in the first quadrant lie in
$\Omega=\Omega_1\cup\Omega_2$ (see Figure \ref{fig1} right), where $\Omega_1$ and 
$\Omega_2$ are defined by
\begin{equation}\label{omega}
\begin{gathered}
\Omega_1=\{(u,v):A\leq u\leq \lambda,\;0\leq v\leq (1-f'(K))(K-\lambda)\},\\
\Omega_2=\{(u,v):\lambda\leq u\leq K,\;0\leq v\leq (1-f'(K))(K-u)\}.
\end{gathered}
\end{equation}
\end{lemma}

\begin{figure}[ht]
\centering
\includegraphics[width=0.49\textwidth]{fig1a} %phase.eps
\includegraphics[width=0.49\textwidth]{fig1b} %closed.eps
\caption{
(Left): The phase portrait of \eqref{general}; (Right): The bound of closed orbits. 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 $u=\lambda$}
\label{fig1}
\end{figure}

\begin{proof}
Define
\begin{equation*}
    f_1(u,v)=g(u)(f(u)-v), \quad f_2(u,v)=v(g(u)-d)\,.
\end{equation*}
Since the positive equilibrium $(\lambda,v_{\lambda})$ only exists when $A<\lambda<K$,
then \eqref{general} can only have a  periodic orbit in the first quadrant 
when $A<\lambda<K$.
Hence we always assume that $A<\lambda<K$ in the following. In this case,
the boundary equilibria $(A,0)$ and $(K,0)$ are both saddle points.
Thus the stable manifold
 of $(A,0)$ (denoted by $\Gamma_{\lambda}^{s}$) and the unstable manifold
 of $(K,0)$ (denoted by $\Gamma_{\lambda}^{u}$) are the separatrices to the
 dynamical behavior of \eqref{general}. From 
\cite[Propositions 2.2 and  2.4]{WSW}, if there exists a periodic orbit, 
it must be below  both $\Gamma_{\lambda}^s$ and $\Gamma_{\lambda}^u$,  
and it is in the region $\{(u,v):A< u< K, v>0\}$.

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 $\lambda<u<K$. 
Then the claim is equivalent to
\begin{equation*}
|\frac{dv}{du}|\le 1-f'(K), \quad (u,v)=\big(u,v_2(u)\big).
\end{equation*}
Let $M=1-f'(K)$, then for $(u,v)=\big(u,v_2(u)\big)$, $\lambda\le u<K$,
\begin{equation*}
  |\frac{dv}{du}|=\frac{M(K-u)(g(u)-d)}{|f(u)-M(K-u)|g(u)}
                          \le \frac{M(K-u)}{|f(u)-M(K-u)|}.
\end{equation*}
 The condition (A7) implies that $f'(u)$ is non-increasing for $u\in[\lambda,K]$. 
Then from the mean-value theorem, we have
\begin{equation*}
 f(u)= f(u)- f(k)= f'(\xi)(u-K)\le f'(K)(u-K)=(1-M)(u-K)
\end{equation*}
for some  $\xi\in (u,K)$. Hence
$f(u)-M(K-u)\le (1-M)(u-K)-M(K-u)=u-K$. Therefore $|f(u)-M(K-u)|\geq K-u$ 
and
\begin{equation*}
 |\frac{dv}{du}| \le \frac{M(K-u)}{|(1-M)(u-K)+M(u-K)|}=M,
\end{equation*}
which proves that $v_1(u)\le v_2(u)=(1-f'(K))(K-u)$. It is easy to see that 
the other sides of the boundary of $\Omega$ are
invariant for the vector field $(f_1,f_2)$, hence $\Omega$ is invariant for 
\eqref{general}, and  the periodic must lie inside $\Omega$.
\end{proof}

We recall the following well-known  Dulac criterion \cite{D}, see
for example, \cite[Theorems 6.1.2, 6.1.3]{H2} and 
\cite[Theorem 1.1.5]{W}.

\begin{lemma} \label{lem2.2}
Consider a planer system
\begin{equation}\label{planer}
\frac{du}{dt} =f(u,v),\quad \frac{dv}{dt} =g(u,v),
\end{equation}
where $f,g$ are continuously differentiable functions defined on a
simply-connected region $D\subset \mathbb{R}^2$.  Let $h(u,v)$ be another
continuously differentiable function on $D$.
 For the system \eqref{planer}, if 
$\frac{\partial (fh)}{\partial u}+\frac{\partial (gh)}{\partial v}$
 is of one  sign in $D$, then \eqref{planer} has no closed orbits in $D$.
\end{lemma}

Our main result on the nonexistence of periodic orbits is as
follows (here we continue the numbering of assumptions in \cite{WSW}).

\begin{theorem}\label{thm:dulac}
Suppose that  $f,g$ satisfy {\rm (A1)--(A3)}, and one of the
following holds:
\begin{itemize}
\item [{\rm (A8)}]  $f\in C^3 (\mathbb{R}^+)$ and $g\in C^2 (\mathbb{R}^+)$, $(uf'(u))''\le 0$ 
and $\left(u/g(u)\right)''\geq 0$ for
$u\in[A,K]$, and $(uf'(u))'\le 0$ for $u\in (\bar{\lambda},K)$; or

\item [{\rm (A9)}] $f\in C^3 (\mathbb{R}^+)$ and $g\in C^2 (\mathbb{R}^+)$, 
$f'''(u) \leq 0$, $g''(u)\leq 0$ for  $u\in[A,K]$, and $f''(u)\le 0$ 
for $u\in (\bar{\lambda},K)$,
\end{itemize}
then \eqref{general} has no closed orbits in the first quadrant for
$\bar{\lambda}<\lambda<K$.
\end{theorem}

\begin{proof}
From Lemma \ref{lem:dulac},  a periodic orbit of \eqref{general}
must be  inside $\Omega$. In particular the orbit satisfies $A< u(t)<
K$ (this does not require (A7)). Define $h(u,v)=[g(u)]^{\alpha}v^{\beta}$ 
for $u\ge 0$, $v\ge 0$ and some $\alpha, \beta\in \mathbb{R}$ to be determined later. 
Therefore, thanks to Dulac's criterion, we have
\begin{equation*}
\begin{split}
    \frac{\partial (h f_1)}{\partial u}+\frac{\partial (h f_2)}{\partial v}
&=h[(\alpha+1)g'(u)(f(u)-v)+g(u)f'(u)+(\beta+1)(g(u)-d)]\\
&=h[g(u)f'(u)+(\beta+1)(g(u)-d)],
\end{split}
\end{equation*}
if $\alpha=-1$.

First we assume (A8) holds. Then
 \begin{equation}\label{eq}
\frac{\partial (h f_1)}{\partial u}+\frac{\partial (h f_2)}{\partial v}
=\frac{h(u,v)g(u)}{u}F_1(u),
 \end{equation}
 where
 \begin{equation}\label{dulac}
         F_1(u)=uf'(u)+\eta \big(u-\frac{du}{g(u)}\big),
 \end{equation}
 with $\eta=\beta+1$. It is clear that $F_1(\lambda)=\lambda f'(\lambda)<0$ for
 $\lambda\in(\bar{\lambda},K)$ and any choice of $\beta$. We  prove that $F_1(u)<0$
 for all $u\in[A,K]$ for a selected $\beta$.
With direct calculation, we have
\begin{equation*}
         F_1'(u)=(uf'(u))'+\eta \Big( 1-d(\frac{u}{g(u)})'\Big),\quad
         F_1''(u)=(uf'(u))''-\eta d (\frac{u}{g(u)})''.
\end{equation*}
  From (A1), (A2) and (A8), if we choose $\eta=-\frac{(\lambda
f''(\lambda)+f'(\lambda))g(\lambda)}{\lambda g'(\lambda)}$, then
\begin{equation*}
F_1'(\lambda)=\lambda f''(\lambda)+f'(\lambda)+\eta \frac{\lambda g'(\lambda)}{g(\lambda)} =0,
\end{equation*}
and $\eta\ge 0$ from (A8). From (A8) and $\eta\ge 0$, $F_1''(u)\le 0$ for all 
$u\in[A,K]$, so $F_1(u)$ is concave on $u\in[A,K]$.
Hence $u=\lambda$ is the unique critical point of $F_1(u)$ for $u\in (A,K)$, 
$F_1(\lambda)<0$, and $u=\lambda$ is local maximum of $F_1$ for all 
$\lambda\in(\bar{\lambda},K)$.
    Then $F_1(u)<0$ for all $u\in[A,K]$. Therefore by choosing $\alpha=-1$ 
and $\beta=-\frac{\left(\lambda f''(\lambda)+f'(\lambda)\right)g(\lambda)}{\lambda g'(\lambda)}-1$, 
we have shown that
$ \frac{\partial (h f_1)}{\partial u}+\frac{\partial (h f_2)}{\partial v}<0$ 
for $u\in (A,K)$ and $v>0$.
By the Dulac criterion (Lemma \ref{planer}),
\eqref{general} has no closed orbits in  the first quadrant if 
$\bar{\lambda}<\lambda<K$.

Secondly if (A9) is satisfied, we rewrite \eqref{eq} into
 \begin{equation*}
\frac{\partial (h f_1)}{\partial u}+\frac{\partial (h f_2)}{\partial v}
=h(u,v)g(u)F_2(u),
\end{equation*}
where
\begin{equation}\label{F2}
      F_2(u)=f'(u)+\eta\big(1-\frac{d}{g(u)}\big),
\end{equation}
again  with $\eta=\beta+1$. It is clear that $F_2(\lambda)=f'(\lambda)<0$ 
for $\lambda\in(\bar{\lambda},K)$
     and any choice of $\beta$. Similarly we have
\begin{gather*}
        F_2'(u)=f''(u)+\eta d\frac{g'(u)}{[g(u)]^2},\\
       F_2''(u)=f'''(u)+\eta d \frac{[g(u)]^2g''(u)-2g(u)[g'(u)]^2}{[g(u)]^4}.
\end{gather*}
If we choose
$\eta=-f''(\lambda)g(\lambda)/g'(\lambda)$, then $F_2'(\lambda)=0$ and $\eta\ge 0$ since 
$f''(\lambda)\le 0$ from (A9). Then from (A1), (A2) and (A9), 
$F_2''(u)\le 0$ for all $u\in[A,K]$, so
$F_2(u)$ is concave on $u\in[A,K]$.
Hence $u=\lambda$ is the unique critical point of $F_2(u)$ for $u\in
(A,K)$, $F_2(\lambda)<0$, and $u=\lambda$ is local maximum of $F_2$ for all
$\lambda\in(\bar{\lambda},K)$.
Then the same conclusion holds.
\end{proof}

Note that Theorem \ref{thm:dulac} improves the result in 
\cite{WSW} (Theorem \ref{final-thm}) in the following way.

\begin{corollary} \label{coro2.4}
Suppose that $f,g$ satisfy all conditions in Theorem \ref{final-thm}.
Then part 3 of Theorem \ref{final-thm} can be changed to: if $\bar{\lambda}<\lambda<K$,
then \eqref{general} has no periodic orbit in the first quadrant, and
  the system  is globally bistable with respect to
  the coexistence equilibrium $(\lambda,v_{\lambda})$ and $(0,0)$.
\end{corollary}

\begin{proof}
We notice that if $f$ satisfies (A6), and $g(u)$ satisfies
\eqref{gu}, then the conditions on the $(uf'(u))''\le 0$ and $(u/g(u))''\ge 0$ 
in (A8) hold. In fact, $(u/g(u))''=0$ for $g(u)$ in
\eqref{gu}, thus the condition on $(uf'(u))'$ in (A8) is not needed as 
$F_1''(u)=(uf'(u))'\le 0$.
Hence the conclusion holds from Theorem \ref{thm:dulac}.
\end{proof}

 The condition (A8) is sharp for the
validity of Dulac criterion since in \cite{WSW}, we have shown that
if $f'''(\bar{\lambda})+2\bar{\lambda}f''(\bar{\lambda})>0$ 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 $\bar{\lambda}\leq u<K$.
Then
(A1), (A2) and (A9) are all satisfied for $f,g$ in \eqref{fg2}.
Again the result in Theorem \ref{thm:dulac} holds. The corresponding
phase portrait can be found in Figure  \ref{figure-BCS-phase}(right).
Note that here $f''(u)$ may be positive for $u\in (A,\bar{\lambda})$.

\subsection{Boukal-Sabelis-Berec model}

Boukal, Sabelis and Berec \cite{DML} considered the equations
\begin{equation}\label{RM}
\begin{gathered}
\frac{du}{dt}=ru\big(1-\frac{u}{K}\big)
\big(1-\frac{A+C}{u+C}\big)-\frac{B
u^n}{1+B hu^n}v,\\
\frac{dv}{dt}=-dv+\frac{B u^n}{1+B hu^n}v,
\end{gathered}
\end{equation}
where $K>A>0$, $r,B,C,n>0$ 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 $\bar{\lambda}<\lambda<K$. The corresponding phase portrait can be
found in Figure \ref{figure-BSB-phase}(left).

  For \eqref{BSB-Holling} with Holling II functional response,
\begin{equation}\label{fg4}
    f(u)=\frac{r(a+u)(K-u)(u-A)}{mK(u+C)}, \quad
    g(u)=\frac{mu}{a+u}.
\end{equation}
The Hopf bifurcation point $\bar{\lambda}$   is the larger root of
$f'(\lambda)=0$ and
\begin{gather*}
       f'(u)  =\frac{r}{mK}\Big(-2u+M_1-\frac{M_2}{(u+C)^2}\Big),\\
       f''(u)  =\frac{r}{mK}\Big(-2+\frac{2M_2}{(u+C)^3}\Big),\quad
       f'''(u) =\frac{-6rM_2}{mK(u+C)^4},
 \end{gather*}
  where
    \begin{gather*}
   M_1=K+A+C-a,\\
  M_2=C^3+(K-a+A)C^2+(-Ka+KA-Aa)C-KAa.
   \end{gather*}
 Since \begin{equation*}
 (\bar{\lambda}+C)^3-M_2=C^3+(C+a-K-A)C^2+(9C+a-K-A)
\bar{\lambda}+2(3C+a-K-A)C\bar{\lambda},
 \end{equation*}
it follows that $f''(u)<0$ for all $u>\bar{\lambda}$ 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
$\bar{\lambda}<\lambda<K$ and the positive equilibrium $(\lambda,v_{\lambda})=(\lambda, f(\lambda))$ 
is globally asymptotically stable in the first quadrant.
\end{theorem}

\begin{proof}
The proof is similar to that of Theorem \ref{thm:dulac}. First it is
clear that a periodic orbit must satisfy $0<u(t)<K$, see for example
\cite{HS}. Hence we only need to show that there is no periodic
orbits in $\{(u,v):0<u<K\}$. We still use the same $h(u,v)$ and
choose $\alpha=-1$.

If (A8') is satisfied, then
 \begin{gather*}
F_1(u)=uf'(u)+\eta\Big( u-\frac{ud(u)}{g(u)}\Big),\\
F_1'(u)=uf''(u)+f'(u)+\eta \Big[1-\Big(\frac{ud(u)}{g(u)}\Big)'\Big], \\
F_1''(u)=uf'''(u)+2f''(u)-\eta \Big[\Big(\frac{ud(u)}{g(u)}\Big)''\Big].
\end{gather*}
From (A1'), (A2') and (A8'), we choose
\[
\eta=-\frac{\left(\lambda f''(\lambda)+f'(\lambda)\right)g(\lambda)}
{\lambda \left(g'(\lambda)-d'(\lambda)\right)}>0.
\]
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 $\bar{\lambda}<\lambda<K$.

If (A9') is satisfied, then
\begin{gather*}
       F_2(u)=f'(u)+\eta\Big(1-\frac{d(u)}{g(u)}\Big),\\
       F_2'(u)=f''(u)+\eta \Big(\frac{d(u)g'(u)-d'(u)g(u)}{g^2(u)}\Big),\\
       F_2''(u)=f'''(u)-\eta \Big(\frac{d(u)}{g(u)}\Big)''.
\end{gather*}

From (A1'), (A2') and (A9'), we choose  
\[
\eta=\frac{-f''(\lambda)g(\lambda)}{g'(\lambda)-d'(\lambda)}>0.
\]
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 $\bar{\lambda}<\lambda<K$;
(A8') implies that \eqref{equ:RM} undergoes
a supercritical Hopf bifurcation at $\lambda=\bar{\lambda}$ and has a unique
limit cycle for $0<\lambda<\bar{\lambda}$.  
\cite[Lemma 3.1]{H1} shows that
all solutions are bounded and Poincar\'e-Bendixson
Theorem (see \cite[Theorem 1.1.19]{W}) implies that
$(\lambda,v_{\lambda})$ is globally stable in the first quadrant for
$\bar{\lambda}< \lambda<K$.
 \end{proof}

The applications of Theorem \ref{thm:dulac-RM} to ecological models 
are discussed in the following two subsections.

\subsection{Logistic type}

Examples of $f$,  $g$ and $d$ which satisfy conditions (A8') or
(A9') can be found in \cite{CHL,H1,K,L,RX,XZ}, and two prominent examples are
 \eqref{lv} and \eqref{rm} shown in the introduction. 
A result like Theorem \ref{thm:dulac-RM} was first proved by Hsu
\cite{H1}. He claimed that there is no limit cycle for $\lambda\in
(\bar{\lambda},K)$ if $f(u)$ is concave and it has a hump at
$u=\bar{\lambda}$. But there was a gap in the proof and counterexamples
have been found \cite{CHL,HS}, and results similar to Theorem
\ref{thm:dulac-RM} have been proved in \cite{K,L,RX,XZ} and others.
Theorem \ref{thm:dulac-RM} shows that the concavity of $f(u)$ on
$[0,K]$ is neither sufficient nor necessary for the nonexistence of
periodic orbits.

\subsection{Weak Allee effect case}

 Here we point out
that the growth rate per capita corresponding to $f$ satisfying
(A1') could be of weak Allee effect type, that is, a positive
function on $[0,K)$ which is increasing in $[0,\bar{\lambda})$ and
decreasing on $(\bar{\lambda},K)$ (see \cite{CBG,JS,SS}). In fact, when
$g(u)=u$ and $d(u)=d>0$, 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.

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\end{document}