\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Ninth MSU-UAB Conference on Differential Equations and Computational
Simulations.
\emph{Electronic Journal of Differential Equations},
Conference 20 (2013),  pp. 15--25.\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} \setcounter{page}{15}
\title[\hfilneg EJDE-2013/Conf/20/ \hfil S-shaped bifurcation curves]
{S-shaped bifurcation curves for logistic growth and
 weak Allee effect growth models with grazing on an interior patch}

\author[D. Butler, R. Shivaji, A. Tuck \hfil EJDE-2013/conf/20 \hfilneg]
{Dagny Butler, Ratnasingham Shivaji, Anna Tuck}  % in alphabetical order

\address{Dagny Butler \newline
Department of Mathematics \& Statistics,
Mississippi State University, Mississippi State, MS 39762, USA.\newline
Department of Mathematics \& Statistics,
University of North Carolina at Greensboro, Greensboro, NC 27412, USA}
\email{dlgrilli@uncg.edu, dg301@msstate.edu}

\address{Ratnasingham Shivaji \newline
Department of Mathematics \& Statistics,
University of North Carolina at Greensboro,
Greensboro, NC 27412, USA}
\email{shivaji@uncg.edu}

\address{Anna Tuck \newline
Department of Mathematics \& Statistics,
University of North Carolina at Greensboro,
 Greensboro, NC 27412, USA}
\email{avtuck@uncg.edu}

\thanks{Published October 31, 2013.}
\subjclass[2000]{34B18}
\keywords{Grazing on an interior patch; positive solutions;
\hfill\break\indent S-shaped bifurcation curves}

\begin{abstract}
 We study the positive solutions to the steady state reaction
 diffusion equations with Dirichlet boundary conditions of the form
 \begin{gather*}
 -u''=  \begin{cases}
      \lambda[u - \frac{1}{K}u^2 - c \frac{u^2}{1+u^2}], & x \in (L,1-L),\\
      \lambda[u - \frac{1}{K}u^2], & x \in (0,L)\cup(1-L,1),
            \end{cases} \\
           u(0)=0, \quad u(1)=0
 \end{gather*}
 and
 \begin{gather*}
 -u''=  \begin{cases}
     \lambda[u(u+1)(b-u) - c \frac{u^2}{1+u^2}], & x \in (L,1-L), \\
  \lambda[u(u+1)(b-u)], & x \in (0,L)\cup(1-L,1),
            \end{cases} \\
 u(0)=0,\quad u(1)=0.
 \end{gather*}
 Here, $\lambda, b, c, K$, and $L$ are positive constants with
 $0<L<\frac{1}{2}$.  These types of steady state equations occur in
 population dynamics; the first model describes logistic growth with grazing,
 and the second model describes weak Allee effect with grazing.
 In both cases, $u$ is the population density, $\frac{1}{\lambda}$
 is the diffusion coefficient, and $c$ is the maximum grazing rate.
 These models correspond to the case of symmetric grazing on an interior region.
 Our goal is to study the existence of positive solutions.  Previous studies
 when the grazing was throughout the domain resulted in S-shaped bifurcation
 curves for certain parameter ranges.  Here, we show that such S-shaped
 bifurcations occur even if the grazing is confined to the interior.
 We discuss the results via a modified quadrature method and
 \emph{Mathematica} computations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks


\section{Introduction}

In \cite{LSS}, the authors studied the nonlinear boundary-value problem
\begin{equation} \label{e2}
\begin{gathered}
 -\Delta u= \lambda[u - \frac{1}{K}u^2 - c \frac{u^2}{1+u^2}], \quad x \in \Omega, \\
 u=0, \quad x\in\partial\Omega.
\end{gathered}
\end{equation}
Here, $\Delta u=\operatorname{div} \big(\nabla u\big)$ is the Laplacian 
of $u$, and $\Omega$ is a smooth bounded region with $\partial\Omega\in C^2$. 
Also, $\lambda, K,$ and $c$ are positive constants where $u$ is the population
density and $\frac{1}{\lambda}$ is the diffusion coefficient.  
The term $u-\frac{1}{K}u^2$ represents logistic growth; this means that as 
a given population grows, its per capita growth rate declines linearly 
(see \cite{M} and \cite{N}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig1a} \quad% graz
\includegraphics[width=0.40\textwidth]{fig1b} % cows.pdf
\end{center} 
\caption{Grazing}
\label{pics2}
\end{figure}

It is interesting to study natural phenomena affecting the population
 such as grazing.  Here, the term $c \frac{u^2}{1+u^2}$ corresponds
 to the grazing rate by a fixed number of grazers on the population,
 where the coefficient $c$ is the maximum grazing rate (see Figure \ref{pics2}). 
 This type of model can be used to describe several ecological systems such 
as the dynamics of fish (see \cite{M} and \cite{SH}) and spruce budworm 
populations (see \cite{VS} and Figure \ref{pics1}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig2a} \quad % schoolfish.pdf
\includegraphics[width=0.48\textwidth]{fig2b} % worm
\end{center} 
\caption{Examples of Fish and Spruce Budworms}
\label{pics1}
\end{figure}

The authors in \cite{LSS} proved the existence of at least one positive 
solution for all $\lambda>\lambda_1(\Omega)$, where $\lambda_1(\Omega)$ 
is the principal eigenvalue of the operator $-\Delta$ with Dirichlet
 boundary conditions.  Also, the authors discuss the existence of at 
least three positive solutions for certain ranges of $\lambda$.

In \cite{PRS}, the authors studied the one-dimensional reaction diffusion model:
\begin{equation}\label{e12}
\begin{gathered}
  -u''=  \lambda[u(u+1)(b-u) - c \frac{u^2}{1+u^2}], \quad x \in (0,1),\\
u(0)=0, \quad u(1)=0.
\end{gathered}
\end{equation}
Here, $\lambda$, $b$, and $c$ are all positive parameters where $u$ 
is the population density and $\frac{1}{\lambda}$ is 
the diffusion coefficient.  The term $u(u+1)(b-u)$ represents weak
 Allee effect (see \cite{A} and \cite{SS}).  
Under a weak Allee effect, for small populations the per capita growth
begins positive and initially increases.  This differs from logistic
growth whose per capita growth rate is decreasing.  
The initial increase in population growth can be caused by a number 
of factors such as shortage of mates or predator saturation.
Two examples of populations that experience weak Allee effect are the 
apple snail and the smooth cordgrass plant (see \cite{BFL} and 
\cite{DK} and Figure \ref{pics100}).  
Furthermore, the term $c \frac{u^2}{1+u^2}$ is the same grazing rate 
described previously.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.52\textwidth]{fig3a}\quad  % applesnail.pdf
\includegraphics[width=0.45\textwidth]{fig3b} % grass.pdf
\end{center} 
\caption{Examples of an Apple Snail Shell with Eggs and Smooth Cordgrass}
\label{pics100}
\end{figure}

The authors in \cite{PRS} were able to show the evolution of the bifurcation 
curve over a range of $c$-values, for a fixed value of $b$.  In particular, 
they discuss S-shaped bifurcation curves for certain ranges of $b$ and $c$.

We are interested in extending some of the results in \cite{LSS} and 
\cite{PRS} regarding the S-shaped bifurcation curve in the one dimensional 
case when the grazing is confined to an interior patch. 
 Previous studies have been done examining population dynamics on split domains; 
by a split domain, we mean that phenomena such as grazing or harvesting are 
only allowed on part of the domain. In \cite{APS},  the authors studied logistic 
growth with constant yield harvesting in both the symmetric and asymmetric 
cases.  We are interested in pursuing a similar study by analyzing models which 
describe grazing of a fixed number of grazers within the interior of the domain 
on a logistically growing species and a species subject to weak Allee effect, 
respectively.

We study the positive solutions to the steady state reaction diffusion 
equations with Dirichlet boundary conditions of the form
\begin{equation} \label{ee1}  
\begin{gathered}
   -u''=  \begin{cases}
\lambda[u - \frac{1}{K}u^2 - c \frac{u^2}{1+u^2}], & x \in (L,1-L), \\
\lambda[u - \frac{1}{K}u^2], & x \in (0,L)\cup(1-L,1),
            \end{cases} \\
 u(0)=0, \quad u(1)=0
\end{gathered}
\end{equation}
and
\begin{equation}\label{ee2}
\begin{gathered}
 -u''=  \begin{cases}
 \lambda[u(u+1)(b-u) - c \frac{u^2}{1+u^2}], & x \in (L,1-L),\\
 \lambda[u(u+1)(b-u)], & x \in (0,L)\cup(1-L,1),
  \end{cases} \\
 u(0)=0, \quad u(1)=0.
\end{gathered}
\end{equation}
Here, $\lambda, b, c, K$, and $L$ are positive constants with $0<L<\frac{1}{2}$.
These models correspond to the case of symmetric grazing on an interior region.
We follow the ideas used in \cite{APS}, that is, modify the quadrature method 
discussed in \cite{L} to analyze these problems.

In Section 2, we will discuss the quadrature method in the case of a 
split domain.  Section 3 is concerned with presenting our computational 
results via \emph{Mathematica} for the logistic growth model.  
In Section 4, we will discuss similar results for the weak Allee effect model.


\section{Preliminaries}

We consider a general model of the form
\begin{equation}  \label{e15} 
\begin{gathered}
     -u''=  \begin{cases}
\lambda \tilde{f}(u), & x \in (L,1-L),\\
\lambda f(u), & x \in (0,L)\cup(1-L,1),
  \end{cases} \\
 u(0)=0, \quad u(1)=0.
\end{gathered}
\end{equation}
We will study solutions  $u$ that are symmetric about 
$x=1/2$ such that $u(L^-)=u(L^+)$ and $u'(L^-)=u'(L^+)$ 
where $\|u\|_{\infty}=\rho$ and $\sigma(\rho)=u(L)$.
 A typical solution to \eqref{e15} can be seen in Figure \ref{pic85}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig4} % sol.pdf
\end{center} 
\caption{Typical Solution $u$}
\label{pic85}
\end{figure}

 First, we will focus on the region $(L,\frac{1}{2})$. Suppose $u$ 
is a positive solution to \eqref{e15}. On the interval $(L, \frac{1}{2})$,
we have $-u''=\lambda \tilde f(u)$.

 Multiplying by $u'$, integrating, and using the fact that
 $u'(1/2)=0$ and $u(1/2)=\rho$, we obtain
\begin{equation*}
u'(x)=\sqrt{2\lambda[\tilde F(\rho)-\tilde F(u)]} \quad L<x\leq \frac{1}{2}
\end{equation*}
where $\tilde F(s):= \int^s_0 \tilde f(t)dt$.
Integrating again, we see that on the interval $(L, \frac{1}{2})$ 
solutions will satisfy
\begin{equation*}
\int^{\rho}_{u(x)}\frac{1}{\sqrt{\tilde F(\rho)-\tilde F(v)}}dv
=\sqrt{2\lambda}[\frac{1}{2}-x].
\end{equation*}
Let $\sigma:=u(L)$.  Then, if we evaluate the equation above at $x=L$, we have
\begin{equation*}
\int^{\rho}_{\sigma}\frac{1}{\sqrt{\tilde F(\rho)-\tilde F(v)}}dv=\sqrt{2\lambda}[\frac{1}{2}-L].
\end{equation*}
Simplifying we obtain
\begin{equation}\label{g1}
\lambda=\Big[\frac{1}{\sqrt{2}(\frac{1}{2}-L)}
\int^{\rho}_{\sigma}\frac{1}{\sqrt{\tilde F(\rho)-\tilde F(v)}}dv \Big]^2
=: G_1(\sigma,\rho).
\end{equation}

Next, we consider the region $(0,L)$.  On the interval $(0,L)$,
 we have $-u''=\lambda f(u)$. If we use similar calculations as before 
and if $u'(0)=m$, then we obtain
\begin{equation*}
u'(x)=\sqrt{2}\sqrt{\frac{m^2}{2}-\lambda F(u(x))}.
\end{equation*}
Integrating again we obtain
\begin{equation*}
\int^{u(x)}_0 \frac{1}{\sqrt{\frac{m^2}{2}-\lambda F(v)}}dv=\sqrt{2}x.
\end{equation*}
Letting $\sigma=u(L)$, we find that
\begin{equation}
\label{e10}
\int^{\sigma}_0 \frac{1}{\sqrt{\frac{m^2}{2}-\lambda F(v)}}dv=\sqrt{2}L.
\end{equation}
Recall $u'(L^-)=u'(L^+)$ which implies
\begin{equation*}
\frac{m^2}{2}=\lambda [\tilde F(\rho)- \tilde F(\sigma)+F(\sigma)].
\end{equation*}
Substituting the above into \eqref{e10}, we obtain
\begin{equation}\label{g2}
\lambda=\Big[\frac{1}{\sqrt{2}L}\int^{\sigma}_{0}
\frac{1}{\sqrt{\tilde F(\rho)-\tilde F(\sigma) + F(\sigma)-F(v)}}dv \Big]^2
=: G_2(\sigma,\rho).
\end{equation}
Therefore,
\begin{equation*}
G_1(\sigma,\rho)=G_2(\sigma,\rho).
\end{equation*}
Thus, if $u$ is a solution of \eqref{e15} with $u(L)=\sigma(\rho)$ and
$\|u\|_{\infty}=\rho$, then $\rho$ and $\sigma$ must satisfy
 $G_1(\sigma,\rho)=\lambda$ and $G_2(\sigma,\rho)=\lambda$.

Suppose that given a $\rho$, there exists $\sigma(\rho)$ such that 
$G_1(\sigma(\rho),\rho)=G_2(\sigma(\rho),\rho)$.  Then
\begin{equation}\label{e50}
\begin{aligned}
 \sqrt{\lambda}
&= \frac{1}{\sqrt{2}(\frac{1}{2}-L)}\int^{\rho}_{\sigma(\rho)}
 \frac{1}{\sqrt{\tilde F(\rho)-\tilde F(v)}}dv\\
&= \frac{1}{\sqrt{2}L}\int^{\sigma(\rho)}_{0}
 \frac{1}{\sqrt{\tilde F(\rho)-\tilde F(\sigma(\rho)) 
 + F(\sigma(\rho))-F(v)}}dv.
\end{aligned}
\end{equation}
In fact, given $\lambda$ and $\rho$ such that \eqref{e50}
is satisfied where $\sigma(\rho)$ satisfies 
$G_1(\sigma,\rho)=G_2(\sigma,\rho)$, we can back track and use the 
Implicit Function Theorem to obtain a solution of the form seen 
in Figure \ref{pic85}.  
Further, $\lambda=G_1(\sigma(\rho),\rho)$ (or $\lambda=G_2(\sigma(\rho),\rho)$) 
provides the bifurcation diagram for these positive solutions.
The following theorem summarizes the above discussion:

\begin{theorem} \label{thm2.1}
Let $\rho>0$ and $\sigma(\rho)\in(0,\rho)$ be such that 
$G_1(\rho,\sigma)=G_2(\rho,\sigma)$.  Then, \eqref{e15} has a positive
solution symmetric about $x=\frac{1}{2}$ with $\|u\|_{\infty}=\rho$ 
and $u(L)=\sigma(\rho)$ if and only if 
$\sqrt{\lambda}=G_1(\rho,\sigma(\rho))$ $(=G_2(\rho,\sigma(\rho)))$.
\end{theorem}

\section{Results for logistic growth with grazing}

In this section, we consider a model with logistic growth with grazing 
only in an interior region (see Figure \ref{pic1}),
\begin{equation}\label{e4}
\begin{gathered}
           -u''= \begin{cases}
\lambda\tilde{f}(u)=\lambda[u - \frac{1}{K}u^2 - c \frac{u^2}{1+u^2}], &
 x \in (L,1-L),\\
\lambda f(u)=\lambda[u - \frac{1}{K}u^2], & x \in (0,L)\cup(1-L,1),
            \end{cases} \\
 u(0)=0, \quad u(1)=0.
\end{gathered}
\end{equation}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig5a} \quad % f
\includegraphics[width=0.46\textwidth]{fig5b} \\ % ftilde
(a) $f(u)=u-\frac{1}{K}u^2$ \hfil
(b) $\tilde{f}(u)=u-\frac{1}{K}u^2-c \frac{u^2}{1+u^2}$
\end{center} 
\caption{Logistic growth with grazing only in an interior region}
\label{pic1}
\end{figure}

We know from \cite{LSS} that for certain parameter ranges the authors
 were able to establish the occurrence of an S-shaped bifurcation curve 
for \eqref{e2} when grazing was allowed over the entire domain 
(case when $L=0$).  In fact, they found that given $c<2$ then 
for $K>>1$ the bifurcation curve for model \eqref{e2} will be
S-shaped (see Figure \ref{pic7}).  That is, there exist $m_1, m_2, m_3>0$ 
such that \eqref{e2} has:
\begin{itemize}
\item  no positive solution for $\lambda\in(0,m_1]$
\item exactly one positive solution for $\lambda\in(m_1,m_2)$
\item exactly two positive solutions for $\lambda=m_2$
\item exactly three positive solutions for $\lambda\in(m_2,m_3)$
\item exactly two positive solutions for $\lambda=m_3$
\item exactly one positive solution for $\lambda\in(m_3,\infty)$
\end{itemize}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig6} % lpicgen}\\
\end{center} 
\caption{Symmetric grazing: $\lambda$ versus $\rho$}
\label{pic7}
\end{figure}

We hope to obtain similar results even when grazing is restricted 
to an interior patch.  To plot the bifurcation curve for our model, 
we use \emph{Mathematica}.  For a fixed $K, c,$ and $L$, we input a 
$\rho$ value and use \emph{Mathematica} to solve for the value of 
$\sigma$ where $G_1(\sigma,\rho)=G_2(\sigma,\rho)$.

Given a $\rho$, does there exist a unique $\sigma=\sigma(\rho)$ such 
that $G_1(\sigma,\rho)=G_2(\sigma,\rho)$? From our computations, 
we observe that for any given $\rho$-value there will be a unique 
$\sigma$ (see Figure \ref{pic2}).  Our calculations imply that $G_1$ 
and $G_2$ will intersect only once for any given $\rho$-value; 
the value where the intersection occurs is $\sigma$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig7} % g1g2 
\end{center} 
\caption{Graph of $G_1$ and $G_2$}
\label{pic2}
\end{figure}

Once we find $\sigma$, we substitute the values of $\sigma$ and $\rho$ 
back into either $G_1$ or $G_2$.  Thus, for a given $\rho$, we can 
find the corresponding value of $\lambda$ which we then use to plot 
the bifurcation curve. From our \emph{Mathematica} computations, 
we observe that even when the grazing is confined to a small interior 
patch an S-shaped bifurcation curve similar to the results from \cite{LSS} 
still occurs.

For example, we fix $c=1.5$, and we let $L=.05, .25, .4,$ and $.4999$ 
to show the evolution of the bifurcation curve as grazing is restricted 
to a smaller interior region.  For each case, we find $K>>1$ such that 
an S-shape bifurcation curve occurs (see Figures \ref{pics3} and \ref{pics4}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig8a} \quad % lpicl05
\includegraphics[width=0.46\textwidth]{fig8b} \\ % lpicl25
(a) $K=10$, $c=1.5$, and $L=.05$ \hfil
(b) $K=25$, $c=1.5$, and $L=.25$
\end{center} 
\caption{Asymmetric grazing: $\lambda$ versus $\rho$}
\label{pics3}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig9a} \quad % lpicl40.pdf
\includegraphics[width=0.46\textwidth]{fig9b} \\ %  lpicl4999.pdf
(a) $K=50$, $c=1.5$, and $L=.40$ \hfill
(b) $K=16000$, $c=1.5$, and $L=.4999$
\end{center} \caption{Asymmetric grazing: $\lambda$ versus $\rho$}
\label{pics4}
\end{figure}


\section{Results for weak Allee effect with grazing}

We will now consider a model with weak Allee effect with grazing only 
in an interior region (see Figure \ref{pic11}):
\begin{equation}  \label{e14}
\begin{gathered}
 -u''=  \begin{cases}
\lambda\tilde{f}(u)=\lambda[u(u+1)(b-u) - c \frac{u^2}{1+u^2}], & x \in (L,1-L),\\
\lambda f(u)=\lambda[u(u+1)(b-u)], & x \in (0,L)\cup(1-L,1),
\end{cases} \\
         u(0)=0,\quad u(1)=0.
\end{gathered}
\end{equation}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig10a} \quad % wfu 
\includegraphics[width=0.46\textwidth]{fig10b} \\ % wftilde
(a) $f(u)=u(u+1)(b-u)$ \hfill 
(b) $\tilde{f}(u)=u(u+1)(b-u)-c \frac{u^2}{1+u^2}$
\end{center} 
\caption{Weak Allee effect with grazing only in an interior region}
\label{pic11}
\end{figure}


In \cite{PRS}, the authors found the occurrence of an S-shaped 
bifurcation curve for certain parameter ranges when grazing was through 
out the entire domain (case when $L=0$).  Specifically, they noted that
 if $b>b_0$ (some) and $c\in(b-1,c_0\mbox{ (some)})$ then the bifurcation 
curve for model \eqref{e2} will be S-shaped (see Figure \ref{pic71}).
That is, there exist $m_1, m_2, m_3>0$ such that \eqref{e2} has
\vspace{2cm}
\begin{itemize}
\item  no positive solution for $\lambda\in(0,m_1)$
\item exactly one positive solution for $\lambda=m_1$
\item exactly two positive solutions for $\lambda=(m_1,m_2]$
\item exactly three positive solutions for $\lambda\in(m_2,m_3)$
\item exactly two positive solutions for $\lambda=m_3$
\item exactly one positive solution for $\lambda\in(m_3,\infty)$
\end{itemize}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig11} % wpicgen
\end{center} 
\caption{Symmetric grazing: $\lambda$ versus $\rho$}
\label{pic71}
\end{figure}

The bifurcation curve for model \eqref{e2} can be calculated using a 
similar method as in Section 4.  For a given $\rho$, we must find 
a $\sigma$ such that $G_1(\sigma,\rho)=G_2(\sigma,\rho)$; then, 
substituting $\rho$ and $\sigma$ back into $G_1$ or $G_2$ will give 
us $\lambda$.  We use this information to then plot the bifurcation 
curve of $\lambda$ versus $\rho$ with \emph{Mathematica}.  
As we observed with the logistic model, we have computational evidence 
that for a given $\rho$ there will be a unique $\sigma$ value 
(see Figure \ref{pic111}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig12} % wgg1g2
\end{center} 
\caption{Graph of $G_1$ and $G_2$}
\label{pic111}
\end{figure}

Based on our observations using \emph{Mathematica}, we find that even
 when the domain is split and grazing is only allowed within the interior 
an S-shaped bifurcation curve similar to the results from \cite{PRS} 
will still occur.

For instance, we fix $b=5$, and we let $L=.01, .15, .25,$ and $.40$ 
to show the evolution of the bifurcation curve as grazing is restricted 
to a smaller interior patch.  In each case, we find 
$c\in(b-1,c_0\mbox{(some)})$ such that an S-shape bifurcation 
curve occurs (see Figures \ref{pics5} and \ref{pics6}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig13a} \quad % wpicl01.pdf
\includegraphics[width=0.46\textwidth]{fig13b} \\ % wpicl15.pdf
(a) $b=5$, $c=5$, and $L=.01$ \hfil
(b) $b=5$, $c=7$, and $L=.15$ 
\end{center} 
\caption{Asymmetric grazing: $\lambda$ versus $\rho$}
\label{pics5}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig14a} \quad % wpicl25.pdf
\includegraphics[width=0.46\textwidth]{fig14b} \\ % wpicl40.pdf
(a) $b=5$, $c=10$, and $L=.25$ \hfil
(b) $b=5$, $c=15$, and $L=.40$ 
\end{center} 
\caption{Asymmetric grazing: $\lambda$ versus $\rho$}
\label{pics6}
\end{figure}

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