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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 88, pp. 1--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2014/88\hfil Halo-shaped bifurcation curves ]
{Halo-shaped bifurcation curves in ecological systems}

\author[J. Goddard II, R. Shivaji \hfil EJDE-2014/88\hfilneg]
{Jerome Goddard II, Ratnasingham Shivaji}  % in alphabetical order

\address{Jerome Goddard II \newline
Department of Mathematics\\
Auburn University Montgomery\\
Montgomery, AL 36124, USA}
\email{jgoddard@aum.edu}

\address{Ratnasingham Shivaji \newline
Department of Mathematics and Statistics\\
University of North Carolina Greensboro\\
Greensboro, NC 27402, USA}
\email{r\_shivaj@uncg.edu}

\thanks{Submitted December 30, 2013. Published April 2, 2014.}
\subjclass[2000]{34B18, 34B08}
\keywords{Nonlinear boundary conditions; logistic growth; positive solutions}

\begin{abstract}
 We examine the structure of positive steady state solutions for a diffusive
 population model with logistic growth and negative density dependent
 emigration on the boundary. In particular, this class of nonlinear
 boundary conditions depends on both the population density and the diffusion
 coefficient. Results in the one-dimensional case are established via quadrature
 methods. Additionally, we discuss the existence of a Halo-shaped bifurcation curve.
\end{abstract}

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


\section{Introduction}

We consider the diffusive logistic population dynamics model with nonlinear 
boundary conditions:
\begin{equation} \label{e:1}
\begin{gathered}
 u_t =  d\Delta u + au - bu^2, \quad x \in \Omega, \; t > 0\\
d \alpha(x, u)\nabla u \cdot \eta  + [1 - \alpha(x, u)] u = 0, \quad x \in \partial \Omega, t > 0
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^n$
for $n \geq 1$, $\Delta$ is the Laplace operator, $d > 0$ is the diffusion
coefficient, $a, b$ are positive parameters, $\nabla u \cdot \eta$ is the
outward normal derivative, and
$\alpha(x, u):\partial \Omega \times \mathbb{R} \to  [0, 1]$
is a nondecreasing $C^1$ function.

Spatiotemporal models have been extensively employed in population dynamics 
to describe the distribution and abundance of organisms living in a patch, 
$\Omega$.  The archetypal form of such a model is given by
\[
u_t = d\Delta u + u\widetilde{f}(x, u), \quad x \in \Omega, t > 0
\]
with $u(t, x)$ representing the population density and 
$\widetilde{f}(x, u)$ the per capita growth rate which could be influenced 
by the heterogenous environment.  These ecological models were first 
studied by Skellam in his pioneering work, \cite{Skellam1951g56}. 
 Similar models were analyzed prior to Skellam by authors such as Kolomogoroff et
 al. in \cite{Kolmogoroff1937gN6}.  One of the most classic examples is 
Fisher's equation where $\tilde{f}(x, u) = (1 - u)$, which was first studied 
by in the Skellam in \cite{Skellam1951g56}.  Reaction diffusion models have 
since been successfully applied to other spatiotemporal phenomena in disciplines 
such as physics, chemistry, and biology 
(see \cite{Cantrell2003gsb,Fife1979gNB5,Murray2003gB4,Okuba2001gNB9,Smoller2002g112}).

Throughout the literature, the logistic growth rate, given by 
$\tilde{f}(x,u) = a(x) - b(x)u$, has been extensively used to model crowding 
effects (see \cite{Oruganti2002g10}).  However, a more general logistic type 
growth rate can be characterized as having a decreasing per capita growth 
function; i.e., $\tilde{f}(x,u)$ is decreasing with respect to $u$.  
In this paper, we consider logistic growth with $\widetilde{f}(x, u) = (a - bu)$ 
where $a, b$ are positive parameters.

To date, the homogeneous Dirichlet boundary condition, 
$u = 0;\ \partial \Omega$, 
Neumann boundary condition, 
$\frac{\partial u}{\partial \eta} = 0; \ \partial \Omega$, and linear combinations 
of the two aforementioned boundary conditions (known as a Robin boundary condition) 
have been employed almost exclusively in population models. 
 Linear boundary conditions assume the behavior of the population on the boundary 
is independent of the population density itself.  But, density dependent emigration 
rates from patches of habitat have been reported by several ecologists. 
Empirical studies conducted by ecologists have even shown a negative correlation 
between density and emigration rates, in which animals have a tendency to leave 
a patch when density is low and stay in the patch when it is high.  
This fact brings into question a commonly made assumption in ecology, that 
animals exhibit positive density dependent dispersal and patch emigration.

In fact, automatic use of this assumption has been cautioned by authors such 
as Paivinen et. al. in \cite{Paivinen2005g103}.
Negative density dependent dispersal has been reported in the black-headed 
gull \textit{Larus ridibundus} (see \cite{Heinze1996gN21}), Cassin's auklet 
\textit{Ptychoramphus aleuticus} (see \cite{Pyle2001g117}), great it 
\textit{Parus major} (see \cite{Greenwood1979g115}), bighorn sheep, 
\textit{Ovis canadensis} (see \cite{LHeureux1995g116}), roe deer 
\textit{Capreolus capreolus} (see \cite{Vincent1995gN24,Wahlstrom1995gN25}), 
banner-tailed kangaroo rat \textit{Dipodomys spectabilis} 
(see \cite{Jones1988g104}), and the Glanville fritillary butterfly 
\textit{Melitaea cinxi} (see \cite{Kuussaari1998g88}) among others.

Several mechanisms have been proposed in the literature as a cause of negative 
density dependent dispersal including, range position (in which the density of 
organisms decreases while moving along a gradient from the center of the 
species distribution range toward its edge), niche breadth 
(where a particular organism that has the ability to use a wider range of 
resources is assumed to be widespread and more abundant), density dependent 
habitat selection (in particular when organisms tend to occupy more habitats 
when density is low), and dispersal ability (especially when organisms differ 
in their ability to disperse which can reduce density but increase distribution) 
(see \cite{Paivinen2005g103}).  Notably, conspecific attraction has also been 
shown to induce negative density dependent dispersal by Kuussaari et al. who 
observed emigration of the Glanville fritillary butterfly out of low density 
areas and Danielson et al. who reported a tendency for individuals to be more 
attracted to areas with conspecifics, 
see \cite{Danielson1987g102,Kuussaari1998g88,Stamps1991g114}.

In an effort to improve population models to account for this behavior, 
Cantrell and Cosner proposed the following nonlinear boundary condition which 
explicitly models conspecific attraction occurring on the boundary of a patch 
(see \cite{Cantrell2003gsb,Cantrell2006g81,Cantrell2007g33}),
\begin{equation} \label{e:2}
d \alpha(x, u)\nabla u \cdot \eta
+  [1 - \alpha(x, u)]u = 0, \quad x \in \partial \Omega
\end{equation}
where $d$ is the diffusion coefficient, $\nabla u \cdot \eta$ is the outward
normal derivative, and
$\alpha(x, u):\partial \Omega \times \mathbb{R} \to  [0, 1]$
is a nondecreasing $C^1$ function.  Only recently has the nonlinear boundary
condition \eqref{e:2} been studied in terms of population dynamics
(see \cite{Cantrell2011g363,Cantrell2002g31,Cantrell2003gsb,Cantrell2006g81,Cantrell2007g33,Cantrell2009g107}).
 Note that if $\alpha(x,u) \equiv 0$, then \eqref{e:2} becomes the
Dirichlet boundary condition and all organisms leave the patch upon reaching
the boundary.  For the case when $\alpha(x,u) \equiv 1$, \eqref{e:2}
becomes the Neumann boundary condition implying that all organisms remain on
the boundary when reached.  If $\alpha(x, u) = \alpha_0 \in (0, 1)$ then only
a fraction of the organisms will remain on the boundary when reached.

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig1} % R8_alpha_real
\end{center}
\caption{Typical graph of $\alpha(x, u)$}
\label{fig1}
\end{figure}

The class of $\alpha(x, u)$'s which model negative density dependent emigration
on the boundary have the structure exemplified in Figure \ref{fig1},
where $\alpha(x, 0) = 0$ and $\alpha(x,u)$ is increasing to one as 
$u \to \infty$.  With this in mind we will be interested in $\alpha(x, u)$'s 
of the form:
\[
\alpha(x, u) = \alpha(u) := \frac{u}{u + g(u)}, \quad x \in \partial \Omega
\]
where $g \in C^1([0, \infty), [\delta, \infty))$ for some 
$\delta > 0$, $\frac{g(u)}{u}$ tends to $0$ as $u \to \infty$.

The dynamics of the population model \eqref{e:1} are completely determined
by the model's steady state solutions.  Thus, we are interested in obtaining 
the structure of positive steady state solutions of \eqref{e:1}, namely
we consider
\begin{gather}
\label{e:4a} -\Delta u  =  \lambda [ au - bu^2 ], \quad x \in \Omega,\\
\label{e:4} u [\frac{1}{\lambda}\nabla u \cdot \eta + g(u) ]  =  0, \quad x \in \partial \Omega
\end{gather}
where $\lambda = 1/d$ and $d > 0$ is the diffusion coefficient.
In the case when $\lambda = 1$, the authors have studied 
\eqref{e:4a} - \eqref{e:4} with the addition of constant yield harvesting
in \cite{Goddard2009gM4,goddard2009gM3} for one-dimension and higher dimensions, 
respectively.  See also \cite{Goddard2012gM1,Goddard2011gM5} where the authors 
have also considered \eqref{e:4a} - \eqref{e:4} with $\lambda = 1$,
 strong Allee effect, and constant yield harvesting in both one-dimension 
and higher dimensions, respectively.

In this paper, we are interested in the case when $\lambda > 0$ is allowed to vary. 
 Notice that the diffusion parameter will be present in the nonlinear boundary
 condition.  In particular, we consider the case when $n = 1$, $\Omega = (0, 1)$, 
and $g(u) \equiv 1$.  Thus, we study the nonlinear boundary value problem,
\begin{gather}
\label{e:5}-u''  =  \lambda [au - bu^2 ], \quad x \in(0, 1)\\
\label{e:6}[-\frac{1}{\lambda}u'(0) + 1 ]u(0)  =  0\\
\label{e:7}[\frac{1}{\lambda}u'(1) + 1 ]u(1)  =  0.
\end{gather}
It is clear that analyzing the positive solutions of 
\eqref{e:5} - \eqref{e:7} is equivalent to studying the four boundary value
problems
\begin{gather}
\label{e:8}-u''  =  \lambda [au - bu^2 ], \quad x \in(0, 1)\\
\label{e:9}u(0)  =  0\\
\label{e:10}u(1)  =  0,
\end{gather}
\begin{gather}
\label{e:11}-u''  =  \lambda [au - bu^2 ], \quad x \in(0, 1)\\
\label{e:12}u(0)  =  0\\
\label{e:13}u'(1) =  -\lambda,
\end{gather}
\begin{gather}
\label{e:14}-u''  =  \lambda [au - bu^2 ], \quad x \in(0, 1)\\
\label{e:15}u'(0) =  \lambda\\
\label{e:16}u(1)  =  0,
\end{gather}
and
\begin{gather}
\label{e:17}-u''  =  \lambda [au - bu^2 ], \quad x \in(0, 1)\\
\label{e:18}u'(0) =  \lambda\\
\label{e:19}u'(1) =  -\lambda.
\end{gather}

We note that if $u(x)$ is a solution of \eqref{e:11} - \eqref{e:13}
then $v(x) := u(1 - x)$ is also solution of \eqref{e:14} - \eqref{e:16}.
It suffices to only consider \eqref{e:8} - \eqref{e:10},
\eqref{e:11} - \eqref{e:13}, and \eqref{e:17} - \eqref{e:19}.
 The structure of positive solutions for \eqref{e:8} - \eqref{e:10}
has been established in the literature (even for the higher dimensional case), 
see \cite{Cantrell2003gsb,Shi2003g163}.  For completeness, we detail the structure 
of positive solutions of \eqref{e:8} - \eqref{e:10} in Section 2.1 via the
quadrature method introduced by Laetsch in \cite{Laetsch1970g23}.  
In Sections 2.2 and 2.3, we extend the quadrature method to study 
\eqref{e:11} - \eqref{e:13} and \eqref{e:17} - \eqref{e:19}, respectively.
Section 3 is concerned with employing the mathematics software package 
Mathematica to generate the bifurcation curve of positive solutions of 
\eqref{e:5} - \eqref{e:7} as the parameter $\lambda > 0$ is varied.

 Throughout this paper we will consider the case when $a > b$.  
Computational results indicate that for certain ranges of $a$ and $b$, 
the bifurcation curve consists of three different connected components.  
One component forms a closed-loop connecting two distinct points on the 
$\|u\|_\infty$-axis, while another component forms an ellipse or ``halo''. 
 Moreover, for certain ranges of the parameter $\lambda > 0$, 
\eqref{e:5} - \eqref{e:7} has exactly 7 positive solutions,
see Figure \ref{fig2}.  Note that \eqref{e:8} - \eqref{e:10} is portrayed
in red, \eqref{e:11} - \eqref{e:13} in green, and
\eqref{e:17} - \eqref{e:19} in blue.

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig2} % R8_LvsP_a_3.084_b_1_NOLINES_MM.eps
\end{center}
\caption{Halo-shaped bifurcation curve of positive solutions for
\eqref{e:5} - \eqref{e:7}}
\label{fig2}
\end{figure}


\section{Results via the quadrature method}

\subsection{Positive solutions of \eqref{e:8} - \eqref{e:10}}

Here we recall some of the one-di\-mensional results of \cite{Cantrell2003gsb} 
for positive solutions of \eqref{e:8} - \eqref{e:10} via the quadrature method.
Evidently, a positive solution, $u(x)$, of \eqref{e:8} - \eqref{e:10}:
\begin{gather*}
-u''  =  \lambda [au - bu^2] =: \lambda f(u), \quad x \in(0, 1)\\
u(0)  =  0\\
u(1)  =  0,
\end{gather*}
must resemble Figure \ref{fig3}, where $\rho := \|u\|_\infty$.

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig3} % R8_pde_u1_MM
\end{center}
\caption{Typical solution of \eqref{e:8} - \eqref{e:10}} \label{fig3}
\end{figure}

We now present the main result for positive solutions of 
\eqref{e:8} - \eqref{e:10} in Theorem \ref{Thm2.1}.

\begin{theorem}[\cite{Brown1981g5,Laetsch1970g23}] 
\label{Thm2.1}
Problem \eqref{e:8} - \eqref{e:10} has a positive solution,
 $u(x)$, with $\|u\|_\infty = \rho$ if and only if
 $G_1(\rho) := \sqrt{2}\int_0^{\rho}\frac{ds}{\sqrt{F(\rho) - F(s)}}
 = \sqrt{\lambda}$ for some $\lambda > 0$, where $F(s) := \int_0^s f(s)ds$.
\end{theorem}

\noindent\textbf{Remark} (see \cite{Brown1981g5})
$G_1(\rho)$ is well defined and the included improper integral is convergent 
on $S := (0, \frac{a}{b})$, since $f(\rho) > 0$, $\rho \in S$ and 
$F(s)$ is strictly increasing on $S$.  Moreover, $G_1(\rho)$ is a continuous 
and differentiable function on $S$.

\begin{proof}[Proof of Theorem \ref{Thm2.1}]
$(\Rightarrow:)$ Assume that $u(x)$ is a positive solution to 
\eqref{e:8} - \eqref{e:10} with $\rho := \|u\|_\infty$.
Since \eqref{e:8} is an autonomous differential equation,
if there exists an $x_0 \in (0, 1)$ such that $u'(x_0) = 0$ then 
$v(x) := u(x_0 + x)$ and $w(x) := u(x_0 - x)$ will both satisfy the initial
 value problem,
\begin{equation} \label{e:23}
\begin{gathered}
-z''  =  \lambda f(z)\\
z(0)  =  u(x_0)\\
z'(0) =  0
\end{gathered}
\end{equation}
for all $x \in [0, d)$ with $d = \min\{x_0, 1 - x_0\}$.
Picard's Existence and Uniqueness Theorem asserts that
$u(x_0 + x) \equiv u(x_0 - x)$.  Hence, $u(x)$ must be symmetric about
$x_0 = \frac{1}{2}$, $u'(x) \geq 0; x \in [0, x_0]$, and
$u'(x) \leq 0; x \in [x_0, 1]$.  Multiplying \eqref{e:8} by $u'(x)$ , gives
\begin{equation} \label{e:24}
-\big[ \frac{[u'(x)]^2}{2}\big]' = \lambda[F(u(x))]'.
\end{equation}
Integration of \eqref{e:24} from $x$ to $1/2$ yields,
\begin{equation} \label{e:25}
\frac{u'(x)}{\sqrt{F(\rho) - F(u(x))}} = \sqrt{2\lambda}, \quad
x \in [0, \frac{1}{2}).
\end{equation}
Integrating \eqref{e:25} from $0$ to $x$, we have
\begin{equation} \label{e:26}
\int_0^{u(x)}\frac{ds}{\sqrt{F(\rho) - F(s)}} = \sqrt{2\lambda}x, \quad x \in [0, \frac{1}{2}].
\end{equation}
Substitution of $x = 1/2$ into \eqref{e:26} and use of the fact that
$u(1/2) = \rho$, yields,
\begin{equation} \label{e:27}
G_1(\rho) := \sqrt{2}\int_0^{\rho}\frac{ds}{\sqrt{F(\rho) - F(s)}} = \sqrt{\lambda}.
\end{equation}

\noindent $(\Leftarrow:)$ Now, suppose that there exists a $\lambda > 0, \rho \in S$ 
such that $G_1(\rho) = \sqrt{\lambda}$.
Define $u:[0, \frac{1}{2}] \to \mathbb{R}$ by
\begin{equation} \label{e:28}
\int_0^{u(x)}\frac{ds}{\sqrt{F(\rho) - F(s)}} = \sqrt{2\lambda}x.
\end{equation}
It remains to be seen that $u(x)$ is well defined and a positive solution of
\eqref{e:8}.  It follows that the left-hand side of \eqref{e:28} is
a differentiable function of $u$, strictly increasing from $0$ to $\frac{1}{2}$
as $u$ increases from $0$ to $\rho$.  Hence, for each $x \in [0, \frac{1}{2})$
there exists a unique $u(x)$ that satisfies \eqref{e:28}.
Now, use of the Implicit Function Theorem establishes that $u(x)$ is differentiable
as a function of $x$.  Differentiating \eqref{e:28}, we have
\begin{equation} \label{e:29}
u'(x) = \sqrt{2\lambda[F(\rho) - F(u(x))]}, \quad x \in (0, \frac{1}{2}).
\end{equation}
Rearranging \eqref{e:29}, it  yields
\begin{equation} \label{e:30a}
-\frac{[u'(x)]^2}{2} = \lambda[F(u(x)) - F(\rho)], \quad x \in (0, \frac{1}{2}).
\end{equation}
Differentiating \eqref{e:30a}, we have
\[
-u''(x) = f(u(x)), \quad x \in (0, 1).
\]
Hence, $u(x)$ satisfies the differential equation in \eqref{e:8}.
It is also easy to see that $u(0) = 0$.  Finally, defining $u(x)$ as
a symmetric function on (0, 1) yields a positive solution to
\eqref{e:8} - \eqref{e:10} with $\|u\|_\infty = \rho$ and $u(0) = 0 = u(1)$.
\end{proof}

To close this subsection, we recall a result about the global behavior of 
the bifurcation curve of positive solutions for \eqref{e:8} - \eqref{e:10}.
Figure \ref{fig4} exemplifies the behavior of the bifurcation curve of positive 
solutions.

\begin{theorem}[\cite{Cantrell2003gsb}]
\label{Thm2.1a}
Problem \eqref{e:8} - \eqref{e:10} has no positive solution for
 $\lambda \leq \frac{\pi^2}{a}$.  Furthermore, \eqref{e:8} - \eqref{e:10}
has a unique positive solution for $\lambda > \frac{\pi^2}{a}$ and this
 branch of positive solutions approaches infinity in the $\lambda$-direction
 as $\rho = \|u\|_\infty \to \frac{a}{b}$.
\end{theorem}

\begin{figure}[ht]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig4} % R8_LvsP_Gen_DBC_MM
\end{center}
\caption{Bifurcation curve of positive solutions
for \eqref{e:8} - \eqref{e:10}} \label{fig4}
\end{figure}


\subsection{Positive solutions of \eqref{e:11} - \eqref{e:13}}

In this subsection, we extend the quadrature method to study the structure 
of positive solutions of \eqref{e:11} - \eqref{e:13}, namely
\begin{gather*}
-u''  =  \lambda [au - bu^2] =: \lambda f(u), \quad x \in(0, 1)\\
u(0)  =  0\\
u'(1) =  -\lambda.
\end{gather*}
It is clear that a positive solution, $u(x)$, of 
\eqref{e:11} - \eqref{e:13} will resemble Figure \ref{fig5}
 with $\rho := \|u\|_\infty$, $u'(x_0) = 0$ for some $x_0 \in (\frac{1}{2}, 1)$, 
and $q := u(1)$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig5} % R8_pde_u2_MM
\end{center}
\caption{Typical solution of \eqref{e:11} - \eqref{e:13}} \label{fig5}
\end{figure}

 We now state and prove the main result for positive solutions of 
\eqref{e:11} - \eqref{e:13} in Theorem \ref{Thm2.2}.

\begin{theorem}
\label{Thm2.2}
Problem \eqref{e:11} - \eqref{e:13} has a positive solution, $u(x)$,
with $\rho = \|u\|_\infty$ and $q = u(1)$ if and only if
\[
G_2(\rho, q) := 2[F(\rho) - F(q)] = \lambda
\]
for some $\lambda > 0$, where $q = q(\rho) \in [0, \rho)$ satisfies
\[
\widetilde{G_2}(\rho,q) := 2\int_0^\rho \frac{ds}{\sqrt{F(\rho) - F(s)}} 
- \int_0^q \frac{ds}{\sqrt{F(\rho) - F(s)}} - 2\sqrt{F(\rho) - F(q)} = 0
\]
and $F(s) := \int_0^s f(s)ds$.
\end{theorem}

As in the previous subsection, the improper integral in 
$\widetilde{G_2}(\rho, q)$ is well-defined and convergent for $\rho \in S$ 
and $q = q(\rho) \in [0, \rho)$.  Now we prove Theorem \ref{Thm2.2}.

\begin{proof}[Proof of Theorem \ref{Thm2.2}]
$(\Rightarrow:)$ Assume that $u(x)$ is a positive solution to
 \eqref{e:11} - \eqref{e:13} with $\rho := \|u\|_\infty$ and $q := u(1)$.
Through a similar argument to the one used in the proof of Theorem \ref{Thm2.1},
 it is easy to show that if there exists an $x_0 \in (0, 1)$ such that 
$u'(x_0) = 0$ then $u(x)$ will be symmetric about $x_0$ with $u'(x) > 0;\ [0, x_0)$ 
and $u'(x) < 0;\ (x_0, 1]$.  Now, multiplying \eqref{e:11} by $u'$ and integrating
with respect to $x$ yields,
\begin{equation}
\label{e:33}
-\frac{[u'(x)]^2}{2} = \lambda F(u(x)) + C, \quad x \in [0,1].
\end{equation}
Substituting $x = x_0$ and $x = 1$ into \eqref{e:33} gives
\begin{gather}
\label{e:34}C  =  -\lambda F(\rho)\\
\label{e:35}C  =  -\lambda F(q) - \frac{\lambda^2}{2}.
\end{gather}
Combining \eqref{e:34} with \eqref{e:35} we have,
\begin{equation}
\label{e:36}F(\rho)  =  F(q) + \frac{\lambda}{2}.
\end{equation}
Now, substitution of \eqref{e:34} into \eqref{e:33} yields
\begin{equation}
\label{e:37}\frac{[u'(x)]^2}{2} = \lambda [F(\rho) - F(u(x))], \quad x \in [0,1].
\end{equation}
Solving for $u'(x)$ in \eqref{e:37} and using the fact that
$u'(x) > 0;\ [0, x_0)$ and $u'(x) < 0;\ (x_0, 1]$ we have
\begin{gather}
\label{e:38}u'(x)  =  \sqrt{2\lambda} \sqrt{F(\rho) - F(u(x))}, \quad x \in [0, x_0]\\
\label{e:39}u'(x)  =  -\sqrt{2\lambda} \sqrt{F(\rho) - F(u(x))}, \quad x \in [x_0, 1].
\end{gather}
Integration of \eqref{e:38} from $0$ to $x$ and \eqref{e:39} from $x_0$
to $x$ yields
\begin{gather}
\label{e:40}\int_0^{x} \frac{u'(x)dx}{\sqrt{F(\rho) - F(u(x))}}
 =  \sqrt{2\lambda}x, \quad x \in [0, x_0]\\
\label{e:41}\int_{x_0}^{x} \frac{u'(x)dx}{\sqrt{F(\rho) - F(u(x))}}
  =  -\sqrt{2\lambda}(x - x_0), \quad x \in [x_0, 1].
\end{gather}
Through a change of variables and using the fact that $u(0) = 0$ and
$u(x_0) = \rho$ we have
\begin{gather}
\label{e:42}\int_0^{u(x)} \frac{ds}{\sqrt{F(\rho) - F(s)}}
  =  \sqrt{2\lambda}x, \quad x \in [0, x_0]\\
\label{e:43}\int_\rho^{u(x)} \frac{ds}{\sqrt{F(\rho) - F(s)}}
 =  -\sqrt{2\lambda}(x - x_0), \quad x \in [x_0, 1].
\end{gather}
Substituting $x = x_0$ into \eqref{e:42} and $x = 1$ into \eqref{e:42} gives
\begin{gather}
\label{e:44}\int_0^{\rho} \frac{ds}{\sqrt{F(\rho) - F(s)}}
 =  \sqrt{2\lambda}x_0\\
\label{e:45}\int_\rho^{q} \frac{ds}{\sqrt{F(\rho) - F(s)}}
  =  -\sqrt{2\lambda}(1 - x_0).
\end{gather}
Now, subtraction of \eqref{e:45} from \eqref{e:44} yields,
\begin{equation}
\label{e:46}2\int_0^{\rho} \frac{ds}{\sqrt{F(\rho) - F(s)}}
- \int_0^{q} \frac{ds}{\sqrt{F(\rho) - F(s)}} -\sqrt{2\lambda} = 0.
\end{equation}
Solving for $\sqrt{2\lambda}$ in \eqref{e:36}, we have
\begin{equation}
\label{e:47}\sqrt{2\lambda} = 2\sqrt{F(\rho) - F(q)}.
\end{equation}
Finally, combining \eqref{e:46} and \eqref{e:47}, gives
\begin{eqnarray*}
\widetilde{G_2}(\rho,q) := 2\int_0^\rho \frac{ds}{\sqrt{F(\rho)
- F(s)}} - \int_0^q \frac{ds}{\sqrt{F(\rho) - F(s)}} - 2\sqrt{F(\rho) - F(q)} = 0.
\end{eqnarray*}
It is now readily apparent from \eqref{e:47} that
\[
G_2(\rho,q) := 2[F(\rho) - F(q)] = \lambda.
\]

\noindent$(\Leftarrow:)$
Suppose $G_2(\rho, q) = \lambda$ for some $\rho \in S$ and $\lambda > 0$
 where $q = q(\rho) \in [0, \rho)$ is a solution of $\widetilde{G_2}(\rho, q) = 0$. 
Now, define $u(x):[0, 1] \to \mathbb{R}$ by
\begin{gather}
\label{e:48}\int_0^{u(x)} \frac{ds}{\sqrt{F(\rho) - F(s)}}
 =  \sqrt{2\lambda}x, \quad x \in [0, x_0]\\
\label{e:49}\int_\rho^{u(x)} \frac{ds}{\sqrt{F(\rho) - F(s)}}
=  -\sqrt{2\lambda}(x - x_0), \quad x \in [x_0, 1].
\end{gather}
We will show that $u(x)$ is a positive solution to \eqref{e:11} - \eqref{e:13}.
It is easy to see that the turning point given by 
$x_0 = \frac{1}{\sqrt{2\lambda}}\int_0^{\rho}\frac{ds}{\sqrt{F(\rho) - F(s)}}$ 
is unique for fixed $\lambda$- and $\rho$-values.  The function,
\[
\frac{1}{\sqrt{2\lambda}}\int_0^{u}\frac{ds}{\sqrt{F(\rho) - F(s)}},
\]
is a differentiable function of $u$ which is strictly increasing from 
$0$ to $x_0$ as $u$ increases from $0$ to $\rho$.  Thus, for each 
$x \in [0, x_0]$, there is a unique $u(x)$ such that
\begin{equation}
\label{e:50}\int_0^{u(x)} \frac{ds}{\sqrt{F(\rho) - F(s)}}  =  \sqrt{2\lambda}x.
\end{equation}
Moreover, by the Implicit Function theorem, $u(x)$ is differentiable with
respect to $x$.  Differentiating \eqref{e:50}, gives
\begin{equation}
\label{e:51}
u'(x)  =  \sqrt{2[F(\rho) - F(u(x))]}, \quad x \in (0, x_0).
\end{equation}
Through a similar argument, $u(x)$ is a differentiable, decreasing function
of $x$ for $x \in (x_0, 1)$ with
\begin{equation}
\label{e:52} u'(x)  =  -\sqrt{2[F(\rho) - F(u(x))]}, \quad x \in (x_0, 1).
\end{equation}
This implies that we have,
\[
\frac{-[u'(x)]^2}{2}  =  F(\rho) - F(u(x)), \quad x \in (0, 1).
\]
Differentiating again, we have
\[
-u''(x) = f(u(x)), \quad x \in (0, 1).
\]
Thus, $u(x)$ satisfies \eqref{e:11}.  It only remains to be seen that
$u(x)$ satisfies \eqref{e:12} and \eqref{e:13}.
However, from \eqref{e:48} it is clear that $u(0) = 0$.
Since $G_2(\rho, q) = \lambda$, we have
\begin{equation}
\label{e:53}2[F(\rho) - F(q(\rho))] = \lambda.
\end{equation}
Substituting $x = 1$ into \eqref{e:52}, gives
\begin{equation}
\label{e:54}u'(1) = -\sqrt{2\lambda}\sqrt{F(\rho) - F(q)}.
\end{equation}
Combining \eqref{e:53} and \eqref{e:54}, we have
\[
u'(1) = -\lambda.
\]
Hence, $u(x)$ satisfies both \eqref{e:12} and \eqref{e:13}.
\end{proof}

With Theorem \ref{Thm2.2}, it is imperative that we study the existence 
and possible multiplicity of $q$-values for a given $\rho \in S$.  
We see that the sign of 
$$
[\widetilde{G_2}(\rho, q)]_q = \frac{f(q) - 1}{\sqrt{F(\rho) - F(q)}}
$$ 
is completely determined by $f(q) - 1 = aq - bq^2 -1$.  
Let $h_1(q) := f(q) - 1$ and denote its roots by 
$q_1(a,b) : = \frac{a - \sqrt{a^2 - 4b}}{2b}$ and 
$q_2(a,b) : = \frac{a + \sqrt{a^2 - 4b}}{2b}$.  Clearly, when $q_1(a,b)$ 
and $q_2(a,b)$ are real we must have that 
$0 < q_1(a,b) \leq q_2(a,b) < a/b$.  Lemma \ref{lem1} below gives a detailed 
description of the sign of $[\widetilde{G_2}(\rho, q)]_q$ for 
all possible parameter values.  Its proof is just elementary algebra 
and is omitted.

\begin{lemma} \label{lem1}
\begin{itemize}
  \item[(1)] Let $b < 4$ and $a \leq 2\sqrt{b}$.
     \begin{itemize}
         \item[(a)] If $a < 2\sqrt{b}$ then $[\widetilde{G_2}(\rho, q)]_q < 0$ 
 for all $\rho \in S$ and $q \in [0, \rho)$.
         \item[(b)] If $a = 2\sqrt{b}$ then
            \begin{itemize}
               \item[(i)] for $\rho \in (0, q_1(a,b)]$, 
$[\widetilde{G_2}(\rho, q)]_q < 0$ when $q \in [0, \rho)$
               \item[(ii)] for $\rho \in (q_1(a,b), \frac{a}{b})$, 
$[\widetilde{G_2}(\rho, q)]_q < 0$ when $q \in [0, q_1(a,b)) \cup\ (q_1(a,b), \rho)$ 
 and $[\widetilde{G_2}(\rho, q_1(a,b))]_q = 0$.
            \end{itemize}
      \end{itemize}

   \item[(2)] Let $a > 2\sqrt{b}$. Then
            \begin{itemize}
               \item[(a)] for $\rho \in (0, q_1(a,b)]$, 
$[\widetilde{G_2}(\rho, q)]_q < 0$ when $q \in [0, \rho)$
               \item[(b)] for $\rho \in (q_1(a,b), q_2(a,b)]$
                  \begin{itemize}
                     \item[(i)] $[\widetilde{G_2}(\rho, q)]_q < 0$ when 
$q \in [0, q_1(a,b))$
                     \item[(ii)] $[\widetilde{G_2}(\rho, q_1(a,b))]_q = 0$
                     \item[(iii)] $[\widetilde{G_2}(\rho, q)]_q > 0$ when 
$q \in (q_1(a,b), \rho)$
                  \end{itemize}
  \item[(c)] for $\rho \in (q_2(a,b), \frac{a}{b})$
   \begin{itemize}
                     \item[(i)] $[\widetilde{G_2}(\rho, q)]_q < 0$ when 
$q \in [0, q_1(a,b))$
                     \item[(ii)] $[\widetilde{G_2}(\rho, q_1(a,b))]_q = 0$
                     \item[(iii)] $[\widetilde{G_2}(\rho, q)]_q > 0$ when 
$q \in (q_1(a,b), q_2(a,b))$
                     \item[(iv)] $[\widetilde{G_2}(\rho, q_2(a,b))]_q = 0$
                     \item[(v)] $[\widetilde{G_2}(\rho, q)]_q < 0$ when 
$q \in (q_2(a,b), \rho)$.
                  \end{itemize}
            \end{itemize}
\end{itemize}
\end{lemma}

The above lemma  gives sufficient conditions for nonexistence of positive 
solutions of \eqref{e:11} - \eqref{e:13}, which are outlined in the
following theorem.

\begin{theorem} \label{Thm2.4}
If $b < 4$ and $a \leq 2\sqrt{b}$ then \eqref{e:11} - \eqref{e:13}
 has no positive solution for any $\lambda > 0$.  Moreover, 
if $a > 2\sqrt{b}$ then \eqref{e:11} - \eqref{e:13} has no positive solution,
$u(x)$, whenever $\|u\|_\infty \leq q_1(a,b)$ for any $\lambda > 0$.
\end{theorem}

\begin{proof}
Let $\rho \in S$.  Clearly, $\widetilde{G_2}(\rho, \rho) > 0$.  But, 
if $a \leq 2\sqrt{b}$ then Lemma \ref{lem1} gives that $[\widetilde{G_2}(\rho, q)]_q \leq 0$ 
for all $q \in [0, \rho)$.  Hence, $\widetilde{G_2}(\rho, q) \neq 0$ for all 
$q \in [0, \rho)$ and Theorem \ref{Thm2.2} guarantees that 
\eqref{e:11} - \eqref{e:13} will not have a positive solution for any
$\lambda > 0$. Now, if $a > 2\sqrt{b}$ and $\rho = \|u\|_\infty \leq q_1(a,b)$ 
then Lemma \ref{lem1} gives that $[\widetilde{G_2}(\rho, q)]_q < 0$ for all 
$q \in [0, \rho)$ and it follows as in the previous case that 
\eqref{e:11} - \eqref{e:13} will not have a positive solution for any
$\lambda > 0$.
\end{proof}

Another consequence of Lemma \ref{lem1} is that given a $\rho \in S$, 
the number of positive solutions of \eqref{e:11} - \eqref{e:13}
having $\|u\|_\infty = \rho$ can be easily ascertained by computing the values 
of both $\widetilde{G_2}(\rho, 0)$ and $\widetilde{G_2}(\rho, q_1(a,b))$, 
as exemplified in the following Lemma.

\begin{lemma} \label{lem2}
Suppose that $a > 2\sqrt{b}$ and $\rho \in (q_1(a,b), \frac{a}{b})$.
\begin{itemize}
  \item[(1)] Let $\widetilde{G_2}(\rho, q_1(a,b)) > 0$. 
Then \eqref{e:11} - \eqref{e:13} has no positive solution with
$\|u\|_\infty = \rho$ for any $\lambda > 0$.
  \item[(2)] Let $\widetilde{G_2}(\rho, q_1(a,b)) = 0$. 
Then \eqref{e:11} - \eqref{e:13} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q = q_1(a,b)$ for some $\lambda > 0$.
  \item[(3)] Let $\widetilde{G_2}(\rho, q_1(a,b)) < 0$.
     \begin{itemize}
         \item[(i)] If $\widetilde{G_2}(\rho, 0) > 0$ then 
\eqref{e:11} - \eqref{e:13} has two positive solutions both having
$\|u\|_\infty = \rho$ and the first with $u(1) = q \in (0, q_1(a,b))$ and the 
second has $u(1) = q \in (q_1(a,b), \min\left\{\rho, q_2(a,b)\right\})$ 
corresponding to two different $\lambda$-values.
         \item[(ii)] If $\widetilde{G_2}(\rho, 0) = 0$ then 
\eqref{e:11} - \eqref{e:13} has two positive solutions both having
$\|u\|_\infty = \rho$ and the first with $u(1) = 0$ and the second with 
$u(1) = q \in (q_1(a,b), \min\left\{\rho, q_2(a,b)\right\})$ corresponding to 
two different $\lambda$-values.
         \item[(iii)] If $\widetilde{G_2}(\rho, 0) < 0$ then 
\eqref{e:11} - \eqref{e:13} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q \in (q_1(a,b), \min\{\rho, q_2(a,b)\})$ 
for some $\lambda > 0$.
     \end{itemize}
\end{itemize}
\end{lemma} 

\begin{proof}
Let $a > 2\sqrt{b}$.
Notice that $\widetilde{G_2}(\rho, q)$ has a unique local minimum at 
$q = q_1(a,b)$ and clearly $\widetilde{G_2}(\rho, \rho) > 0$.  
Whenever $\rho \in (q_2(a,b), \frac{a}{b})$, $\widetilde{G_2}(\rho, q)$ 
strictly decreasing (from Lemma \ref{lem1}) combined with $\widetilde{G_2}(\rho, \rho) > 0$ 
implies that $\widetilde{G_2}(\rho, q) \neq 0$ for any $q \in [q_2(a,b), \rho)$.

(1)
Since $\widetilde{G_2}(\rho, q_1(a,b)) > 0$ and $\widetilde{G_2}(\rho, \rho) > 0$ 
we have that $\widetilde{G_2}(\rho, q) \neq 0$ for all $q \in [0, \rho)$.  
Theorem \ref{Thm2.2} immediately gives that \eqref{e:11} - \eqref{e:13}
has no positive solution with $\|u\|_\infty = \rho \in (q_1(a,b), \frac{a}{b})$ 
for any $\lambda > 0$.

(2)
Theorem \ref{Thm2.2} and the fact that $\widetilde{G_2}(\rho, q_1(a,b)) = 0$ 
implies that \eqref{e:11} - \eqref{e:13} has a positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q = q_1(a,b)$.  Since $q = q_1(a,b)$ 
is the unique local minimum for $\widetilde{G_2}(\rho, q)$ and 
$\widetilde{G_2}(\rho, \rho) > 0$ we have that this $q$ is the only solution 
of $\widetilde{G_2}(\rho, q) = 0$.  Hence, the aforementioned positive solution 
must be unique.

(3)
For (i), we note that since $\widetilde{G_2}(\rho, 0) > 0$, 
$\widetilde{G_2}(\rho, q_1(a,b)) < 0$, and $\widetilde{G_2}(\rho, q)$ 
is strictly decreasing on $q \in [0, q_1(a,b))$ we have that 
$\widetilde{G_2}(\rho, q) = 0$ for a unique $q \in (0, q_1(a,b))$.  
Also, $\widetilde{G_2}(\rho, \rho) > 0$ and $\widetilde{G_2}(\rho, q)$ is 
strictly increasing on $q \in (q_1(a,b), \\ \min\{\rho, q_2(a,b)\})$.  
Thus, $\widetilde{G_2}(\rho, q) = 0$ for a unique 
$q \in (q_1(a,b), \min\{\rho, q_2(a,b)\})$.  Theorem \ref{Thm2.2} then 
guarantees that \eqref{e:11} - \eqref{e:13} has two positive solutions both
with $\|u\|_\infty = \rho$ where the first solution is such that 
$u(1) = q \in (0, q_1(a,b))$ and the second is such that 
$u(1) = q \in (q_1(a,b), \min\{\rho, q_2(a,b)\})$ for two different 
$\lambda$-values. A similar argument proves (ii).  For (iii),
 $\widetilde{G_2}(\rho, 0) < 0$ and $\widetilde{G_2}(\rho, q_1(a,b)) < 0$ 
combined with the fact that $\widetilde{G_2}(\rho, q)$ is strictly increasing 
on $q \in (q_1(a,b), \min\{\rho, q_2(a,b)\})$ and $\widetilde{G_2}(\rho, \rho) > 0$ 
implies that there can be only one solution of $\widetilde{G_2}(\rho, q) = 0$, 
namely, some unique $q \in (q_1(a,b), \min\{\rho, q_2(a,b)\})$.  
Theorem \ref{Thm2.2} again gives that \eqref{e:11} - \eqref{e:13} has a
unique positive solution with $\|u\|_\infty = \rho$ and 
$u(1) = q \in (q_1(a,b), \min\{\rho, q_2(a,b)\})$ for some $\lambda > 0$.
\end{proof}

Next, we compute $\widetilde{G_2}(\rho, 0)$ and $\widetilde{G_2}(\rho, q_1(a,b))$ 
values using Mathematica in order to conclude the shape of the bifurcation curve
 of positive solutions for \eqref{e:11} - \eqref{e:13}.  In particular, we
are interested in the case when $b = 1$ and $a > 2$ is varied 
(note that from Theorem \ref{Thm2.4} we must have $a > 2$ to have the possibility 
of a positive solution).  Our computational results indicate the following cases:

\noindent\textbf{Case 1.}
For $b = 1$, if $a \in (2, a_1)$ (some $a_1 > 0$) then $\widetilde{G_2}(\rho, 0)$ 
and $\widetilde{G_2}(\rho, q_1(a,b))$ have the structure displayed in 
Figure \ref{fig6}.  Computations indicate that $a_1 \approx 3.072$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig6a} % R_8_G2_P_0_a_3_MM
\includegraphics[width=0.45\textwidth]{fig6b} % R_8_G2_P_Q1_a_3_MM
\end{center}
\caption{ (left) $\rho$ vs $\widetilde{G_2}(\rho, 0)$ for
$a \in (2, a_1)$. 
(right) $\rho$ vs $\widetilde{G_2}(\rho, q_1(a,b))$ for $a \in (2, a_1)$} 
\label{fig6}
\end{figure}

Note that Lemma \ref{lem2} gives that \eqref{e:11} - \eqref{e:13} has no
 positive solution when $a \in (2, a_1)$ for any $\lambda > 0$.

\noindent\textbf{Case 2.}
For $b = 1$, if $a = a_1$ then $\widetilde{G_2}(\rho, 0)$ and 
$\widetilde{G_2}(\rho, q_1(a,b))$ have the structure displayed in Figure \ref{fig7}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig7a} % R_8_G2_P_0_a_3.078_MM.eps
\includegraphics[width=0.45\textwidth]{fig7b}  % R_8_G2_P_Q1_a_3.078_MM.eps
\end{center}
\caption{ (left) $\rho$ vs $\widetilde{G_2}(\rho, 0)$ for $a = a_1$.
(right) $\rho$ vs $\widetilde{G_2}(\rho, q_1(a,b))$ for
$a = a_1$} \label{fig7}
\end{figure}

Denote $M_1 > 0$ as the $\rho$-value for which $\widetilde{G_2}(M_1, q_1(a,b)) = 0$. 
In this case, Lemma \ref{lem2} gives that \eqref{e:11} - \eqref{e:13} has only
one positive solution with $\|u\|_\infty = M_1$, $u(1) = q_1(a,b)$,  
and a corresponding unique $\lambda > 0$.  In this case, the bifurcation curve 
of positive solutions would consist of a single point.

\noindent\textbf{Case 3.}
For $b = 1$, if $a \in (a_1, a_2)$ (for some $a_2 > a_1$), then 
$\widetilde{G_2}(\rho, 0)$ and \\ $\widetilde{G_2}(\rho, q_1(a,b))$
 have the structure displayed in Figure \ref{fig8}.   Computations suggest 
that $a_2 \approx 3.1123$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig8a} % R_8_G2_P_0_a_3.084_MM.eps
\includegraphics[width=0.45\textwidth]{fig8b} % R_8_G2_P_Q1_a_3.084_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_2}(\rho, 0)$ for $a \in (a_1, a_2)$.
(right)  $\rho$ vs $\widetilde{G_2}(\rho, q_1(a,b))$ for $a \in (a_1, a_2)$}
 \label{fig8}
\end{figure}

 Denote $M_i > 0$ as the $\rho$-values for which 
$\widetilde{G_2}(M_i, q_1(a,b)) = 0$ where $i = 1,2$. Using Lemma \ref{lem2}, we can 
describe the structure of positive solutions for \eqref{e:11} - \eqref{e:13}
as $\rho$ varies from $M_1$ to $M_2$.  Note that Lemma \ref{lem2} implies 
that \eqref{e:11} - \eqref{e:13} has no positive solution with
$\|u\|_\infty = \rho$ for $\rho \in [q_1(a,b), M_1)$ and 
$\rho \in (M_2, \frac{a}{b})$. When $\rho = M_1$ then 
\eqref{e:11} - \eqref{e:13} has only one positive solution with
 $\|u\|_\infty = M_1$, $u(1) = q_1(a,b)$, and a corresponding unique 
$\lambda > 0$.  For $\rho \in (M_1, M_2)$, \eqref{e:11} - \eqref{e:13}
 has exactly two positive solutions both with $\|u\|_\infty = M_1$ but the 
first having $u(1) \in (0, q_1(a,b))$ and the second having 
$u(1) \in (q_1(a,b), \min\{q_2(a,b), \rho\})$ for corresponding $\lambda$-values. 
 Finally, when $\rho = M_2$ then \eqref{e:11} - \eqref{e:13} has only one
positive solution with $\|u\|_\infty = M_2$, $u(1) = q_1(a,b)$, and a corresponding 
unique $\lambda > 0$. Thus, we have a closed loop or halo-shaped bifurcation 
curve of positive solutions for \eqref{e:11} - \eqref{e:13}.

\noindent\textbf{Case 4.}
For $b = 1$, if $a = a_2$ then $\widetilde{G_2}(\rho, 0)$ and 
$\widetilde{G_2}(\rho, q_1(a,b))$ have the structure displayed in Figure \ref{fig9}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig9a} %  R_8_G2_P_0_a_3.1123_MM.eps
\includegraphics[width=0.45\textwidth]{fig9b} % R_8_G2_P_Q1_a_3.1123_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_2}(\rho, 0)$ for $a = a_2$.
(right) $\rho$ vs $\widetilde{G_2}(\rho, q_1(a,b))$ for $a = a_2$}
\label{fig9}
\end{figure}

Denote $M_i > 0$ as the $\rho$-values for which $\widetilde{G_2}(M_i, q_1(a,b)) = 0$ 
where $i = 1,2$ and $N_1 > 0$ as the $\rho$-value for which 
$\widetilde{G_2}(N_1, 0) = 0$. Clearly, $q_1(a,b) < M_1 < N_1 < M_2 < \rho$.  
Using Lemma \ref{lem2}, we can easily determine that the bifurcation curve 
of positive solutions for \eqref{e:11} - \eqref{e:13} will have the same
closed loop or halo-shape as in Case 4 with one exception.  
In this case, when $\rho = N_1$, \eqref{e:11} - \eqref{e:13} has exactly
two positive solutions both with $\|u\|_\infty = N_1$ but the first having 
$u(1) = 0$ and the second having $u(1) \in (q_1(a,b), \min\{q_2(a,b), \rho\})$ 
for corresponding $\lambda$-values. Notice that for $\rho = N_1$ and $u(1) = 0$ 
this positive solution is also a solution for the Dirichlet boundary condition case, 
namely \eqref{e:8} - \eqref{e:10}.  This implies that the halo-shaped curve
 will connect to the Dirichlet boundary case bifurcation curve at one point, 
$(N_1, \lambda^*(N_1))$ for some $\lambda^*(N_1) > 0$.

\noindent\textbf{Case 5.}
For $b = 1$, if $a \in (a_2, \infty)$ then $\widetilde{G_2}(\rho, 0)$ and 
$\widetilde{G_2}(\rho, q_1(a,b))$ have the structure displayed in 
Figure \ref{fig10}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig10a} % R_8_G2_P_0_a_3.2_MM.eps
\includegraphics[width=0.45\textwidth]{fig10b} % R_8_G2_P_Q1_a_3.2_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_2}(\rho, 0)$ for
$a \in (a_2, \infty)$.
(right) $\rho$ vs $\widetilde{G_2}(\rho, q_1(a,b))$ for $a \in (a_2, \infty)$}
\label{fig10}
\end{figure}

 Denote $M_i > 0$ as the $\rho$-values for which 
$\widetilde{G_2}(M_i, q_1(a,b)) = 0$ and $N_i > 0$ as the $\rho$-values for which 
$\widetilde{G_2}(N_i, 0) = 0$ where where $i = 1,2$. Clearly, 
$q_1(a,b) < M_1 < N_1 < N_2 < M_2 < \rho$.  Again Lemma \ref{lem2} can be used 
to to determine that the bifurcation curve of positive solutions for 
\eqref{e:11} - \eqref{e:13} will have loop structure.  However, when
$\rho \in (N_1, N_2)$ Lemma \ref{lem2} gives that \eqref{e:11} - \eqref{e:13}
 will have only one positive solution with $\|u\|_\infty = \rho$ and 
$u(1) \in (q_1(a,b), \min\{q_2(a,b), \rho\})$ for a corresponding $\lambda$-value. 
Furthermore, when $\rho = N_i$ \eqref{e:11} - \eqref{e:13} has exactly two
positive solutions both with $\|u\|_\infty = N_i$ but the first having $u(1) = 0$ 
and the second having $u(1) \in (q_1(a,b), \min\{q_2(a,b), \rho\})$ for 
corresponding $\lambda$-values with $i = 1,2$. As in the previous case, 
when $\rho = N_i$ and $u(1) = 0$ this positive solution is also a solution 
for the Dirichlet boundary condition case for $i = 1,2$.  The bifurcation curve 
of positive solutions for \eqref{e:11} - \eqref{e:13} will form a loop
connecting to the Dirichlet boundary condition bifurcation curve at two different 
points, $(N_1, \lambda^*(N_1))$ and $(N_2, \lambda^{**}(N_2))$ for some 
$\lambda^*(N_1), \lambda^{**}(N_2) > 0$.  See Section 3 for the complete evolution 
of the bifurcation curve of positive solutions of \eqref{e:5} - \eqref{e:7}.

\subsection{Positive solutions of \eqref{e:17} - \eqref{e:19}}

We further extend the quadrature method in this section to study the structure 
of positive steady states of \eqref{e:17} - \eqref{e:19}, namely
\begin{gather*}
-u''  =  \lambda [au - bu^2] =: \lambda f(u), \quad x \in(0, 1)\\
u'(0) =  \lambda\\
u'(1) =  -\lambda.
\end{gather*}
It is clear that positive solutions of \eqref{e:17} - \eqref{e:19} will
resemble Figure \ref{fig11} with $\rho := \|u\|_\infty$, $u'(\frac{1}{2}) = 0$, 
and $q := u(0) = u(1)$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig11} % R8_pde_u3_MM
\end{center}
\caption{Typical solution of \eqref{e:17} - \eqref{e:19}} \label{fig11}
\end{figure}

We now state the main result for positive solutions of 
\eqref{e:17} - \eqref{e:19} in Theorem \ref{Thm2.3}.

\begin{theorem} \label{Thm2.3}
Problem \eqref{e:17} - \eqref{e:19} has a positive solution,
$u(x)$, with $\rho = \|u\|_\infty$ and $q = u(0) = u(1)$ if and only if 
\[
G_3(\rho, q) := 2[F(\rho) - F(q)] = \lambda
\]
for some $\lambda > 0$, where $q = q(\rho) \in [0, \rho)$ satisfies
\[
\widetilde{G_3}(\rho,q) := 2\int_0^\rho \frac{ds}{\sqrt{F(\rho) - F(s)}} 
- 2\int_0^q \frac{ds}{\sqrt{F(\rho) - F(s)}} - 2\sqrt{F(\rho) - F(q)} = 0
\]
and $F(s) := \int_0^s f(s)ds$.
\end{theorem}

As previously noted in the remark from Section 2.1, the improper integral 
in $\widetilde{G_3}(\rho, q)$ is well-defined and convergent for 
$\rho \in S$ and $q = q(\rho) \in [0, \rho)$.  
The proof of Theorem \ref{Thm2.3} is almost identical to that of 
Theorem \ref{Thm2.2} and is omitted.

To understand the structure of positive solutions of 
\eqref{e:17} - \eqref{e:19} it is imperative that we study the existence
and possible multiplicity of $q$-values for any given $\rho \in S$ as 
delineated in Theorem \ref{Thm2.3}.

We see that the sign of 
$[\widetilde{G_3}(\rho, q)]_q = \frac{f(q) - 2}{\sqrt{F(\rho) - F(q)}}$ 
can be completely determined by analyzing $f(q) - 2 = aq - bq^2 -2$.  
Let $h_2(q) := f(q) - 2$ and denote its roots by 
$\bar{q}_1(a,b) : = \frac{a - \sqrt{a^2 - 8b}}{2b}$ and 
$\bar{q}_2(a,b) : = \frac{a + \sqrt{a^2 - 8b}}{2b}$.  
Clearly, when $\bar{q}_1(a,b)$ and $\bar{q}_2(a,b)$ are real we must have that 
$0 < \bar{q}_1(a,b) \leq \bar{q}_2(a,b) < \frac{a}{b}$.  
A detailed description of the sign of $[\widetilde{G_3}(\rho, q)]_q$ 
for all possible parameter values is presented in Lemma \ref{lem3}. Its proof is 
just elementary algebra and is omitted.


\begin{lemma} \label{lem3}
\begin{itemize}
  \item[(1)] Let $b < 8$ and $a \leq 2\sqrt{2}\sqrt{b}$.
     \begin{itemize}
         \item[(a)] If $a < 2\sqrt{2}\sqrt{b}$ then 
$[\widetilde{G_3}(\rho, q)]_q < 0$ for all $\rho \in S$ and $q \in [0, \rho)$.
         \item[(b)] If $a = 2\sqrt{2}\sqrt{b}$ then
            \begin{itemize}
               \item[(i)] for $\rho \in (0, \bar{q}_1(a,b)]$, 
$[\widetilde{G_3}(\rho, q)]_q < 0$ when $q \in [0, \rho)$
               \item[(ii)] for $\rho \in (\bar{q}_1(a,b), \frac{a}{b})$, 
$[\widetilde{G_3}(\rho, q)]_q < 0$ when $q \in [0, \bar{q}_1(a,b)) \cup
 (\bar{q}_1(a,b), \rho)$ and $[\widetilde{G_3}(\rho, \bar{q}_1(a,b))]_q = 0$.
            \end{itemize}
     \end{itemize}

   \item[(2)] Let $a > 2\sqrt{2}\sqrt{b}$. Then
            \begin{itemize}
               \item[(a)] for $\rho \in (0, \bar{q}_1(a,b)]$, 
$[\widetilde{G_3}(\rho, q)]_q < 0$ when $q \in [0, \rho)$
               \item[(b)] for $\rho \in (\bar{q}_1(a,b), \bar{q}_2(a,b)]$
                  \begin{itemize}
                     \item[(i)] $[\widetilde{G_3}(\rho, q)]_q < 0$ when 
$q \in [0, \bar{q}_1(a,b))$
                     \item[(ii)] $[\widetilde{G_3}(\rho, \bar{q}_1(a,b))]_q = 0$
                     \item[(iii)] $[\widetilde{G_3}(\rho, q)]_q > 0$ when 
$q \in (\bar{q}_1(a,b), \rho)$
                  \end{itemize}
               \item[(c)] for $\rho \in (\bar{q}_2(a,b), \frac{a}{b})$
                  \begin{itemize}
                     \item[(i)] $[\widetilde{G_3}(\rho, q)]_q < 0$ when 
$q \in [0, \bar{q}_1(a,b))$
                     \item[(ii)] $[\widetilde{G_3}(\rho, \bar{q}_1(a,b))]_q = 0$
                     \item[(iii)] $[\widetilde{G_3}(\rho, q)]_q > 0$ when 
$q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$
                     \item[(iv)] $[\widetilde{G_3}(\rho, \bar{q}_2(a,b))]_q = 0$
                     \item[(v)] $[\widetilde{G_3}(\rho, q)]_q < 0$ when 
$q \in (\bar{q}_2(a,b), \rho)$.
                  \end{itemize}
            \end{itemize}
\end{itemize}
\end{lemma} 

The above lemma determines sufficient conditions for nonexistence of 
positive solutions of \eqref{e:17} - \eqref{e:19},
which are outlined in the following theorem.

\begin{theorem} \label{Thm2.4b}
If $b < 8$ and $a \leq 2\sqrt{2}\sqrt{b}$ then \eqref{e:17} - \eqref{e:19}
has no positive solution for any $\lambda > 0$.  Moreover, if
 $a > 2\sqrt{2}\sqrt{b}$ then \eqref{e:17} - \eqref{e:19}
has no positive solution whenever $\|u\|_\infty \leq \bar{q}_1(a,b)$ for any 
$\lambda > 0$.
\end{theorem}

\begin{proof}
Let $\rho \in S$.  Clearly, $\widetilde{G_3}(\rho, \rho) = 0$.  
But, if $a \leq 2\sqrt{2}\sqrt{b}$ then Lemma \ref{lem3} gives that 
$[\widetilde{G_3}(\rho, q)]_q \leq 0$ for all $q \in [0, \rho)$. 
 Hence, $\widetilde{G_3}(\rho, q) \neq 0$ for all $q \in [0, \rho)$ and 
Theorem \ref{Thm2.3} guarantees that \eqref{e:17} - \eqref{e:19} will not
 have a positive solution. Now, if $a > 2\sqrt{2}\sqrt{b}$ and 
$\rho = \|u\|_\infty \leq \bar{q}_1(a,b)$ then Lemma \ref{lem3} gives that 
$[\widetilde{G_3}(\rho, q)]_q < 0$ for all $q \in [0, \rho)$ and it follows 
as in the previous case that \eqref{e:17} - \eqref{e:19} will not have a
positive solution.
\end{proof}

Note that Lemma \ref{lem3} also allows for the number of positive solutions 
of \eqref{e:17} - \eqref{e:19} having $\|u\|_\infty = \rho$ to be
easily ascertained by computing the values of both $\widetilde{G_3}(\rho, 0)$ 
and $\widetilde{G_3}(\rho, \bar{q}_1(a,b))$, as exemplified in the next lemma.


\begin{lemma} \label{lem4}
Suppose that $a > 2\sqrt{2}\sqrt{b}$ and $\rho \in (\bar{q}_1(a,b), \frac{a}{b})$.
\begin{itemize}
  \item[(1)] Let $\widetilde{G_3}(\rho, \bar{q}_1(a,b)) > 0$. 
Then \eqref{e:17} - \eqref{e:19} has no positive solution with
$\|u\|_\infty = \rho$ for any $\lambda > 0$.
  \item[(2)] Let $\rho \in (\bar{q}_1(a,b), \bar{q}_2(a,b)]$.
     \begin{itemize}
        \item[(i)] If $\widetilde{G_3}(\rho, \bar{q}_1(a,b)) = 0$ then 
\eqref{e:17} - \eqref{e:19} has no positive solution with $\|u\|_\infty = \rho$
for any $\lambda > 0$.
        \item[(ii)] Let $\widetilde{G_3}(\rho, \bar{q}_1(a,b)) < 0$.
           \begin{itemize}
              \item[(a)] If $\widetilde{G_3}(\rho, 0) > 0$ then 
\eqref{e:17} - \eqref{e:19} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q \in (0, \bar{q}_1(a,b))$ for some 
$\lambda > 0$.
              \item[(b)] If $\widetilde{G_3}(\rho, 0) = 0$ then 
\eqref{e:17} - \eqref{e:19} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = 0$ for some $\lambda > 0$.
              \item[(c)] If $\widetilde{G_3}(\rho, 0) < 0$ then 
\eqref{e:17} - \eqref{e:19} has no positive solution with $\|u\|_\infty = \rho$
for any $\lambda > 0$.
           \end{itemize}
     \end{itemize}
  \item[(3)] Let $\rho \in (\bar{q}_2(a,b), \frac{a}{b})$.
     \begin{itemize}
        \item[(i)] If $\widetilde{G_3}(\rho, \bar{q}_1(a,b)) = 0$ then 
\eqref{e:17} - \eqref{e:19} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q = \bar{q}_1(a,b)$ for some $\lambda > 0$.
        \item[(ii)] Let $\widetilde{G_3}(\rho, \bar{q}_1(a,b)) < 0$.
           \begin{itemize}
              \item[(a)] If $\widetilde{G_3}(\rho, 0) > 0$ then 
\eqref{e:17} - \eqref{e:19} has two positive solutions both having
$\|u\|_\infty = \rho$, the first with $u(1) = q \in (0, \bar{q}_1(a,b))$ and 
the second has $u(1) = q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$ corresponding to 
two different $\lambda$-values.
              \item[(b)] If $\widetilde{G_3}(\rho, 0) = 0$ then 
\eqref{e:17} - \eqref{e:19} has two positive solutions both having
 $\|u\|_\infty = \rho$ and the first with $u(1) = 0$ and the second has 
$u(1) = q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$ corresponding to two different 
$\lambda$-values.
              \item[(c)] If $\widetilde{G_3}(\rho, 0) < 0$ then 
\eqref{e:17} - \eqref{e:19} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$ for some 
$\lambda > 0$.
           \end{itemize}
     \end{itemize}
\end{itemize}
\end{lemma} 

\begin{proof}
Let $a > 2\sqrt{2}\sqrt{b}$.
Notice that $\widetilde{G_3}(\rho, q)$ has a unique local minimum at 
$q = q_1(a,b)$ for $\rho \in (\bar{q_1}(a,b), \frac{a}{b})$ and a unique local 
maximum at $q = \bar{q}_2(a,b)$ for $\rho \in (\bar{q}_2(a,b), \frac{a}{b})$. 
Also, it is easy to see that $\widetilde{G_3}(\rho, \rho) = 0$.

(1) Notice that for $\rho \in (\bar{q}_1(a,b), \bar{q}_2(a,b)]$, 
$\widetilde{G_3}(\rho, q)$ attains its absolute minimum value at 
$q = \bar{q}_1(a,b)$. But, $\widetilde{G_3}(\rho, \bar{q}_1(a,b)) > 0$ so 
$\widetilde{G_3}(\rho, q) \neq 0$ for any $q \in [0, \rho)$.  
In the case when $\rho \in (\bar{q}_2(a,b), \frac{a}{b})$, 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b)) > 0$ implies that 
$\widetilde{G_3}(\rho, q) = 0$ only for $q \in (\bar{q}_2(a,b), \rho)$. 
 But, $\widetilde{G_3}(\rho, \rho) = 0$ and $[\widetilde{G_3}(\rho, q)]_q < 0$ 
for $q \in (\bar{q}_2(a,b), \frac{a}{b})$.  Thus, $\widetilde{G_3}(\rho, q) \neq 0$ 
for any $q \in [0, \rho)$.  Theorem \ref{Thm2.3} gives that 
\eqref{e:17} - \eqref{e:19} has no positive solution with $\|u\|_\infty = \rho$
for any $\lambda > 0$ in either case.

(2) Fix $\rho \in (\bar{q}_1(a,b), \bar{q}_2(a,b)]$.  For (i), since 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b)) = 0$ we have that 
$\widetilde{G_3}(\rho, \rho) \neq 0$ and thus Theorem \ref{Thm2.3} gives that 
\eqref{e:17} - \eqref{e:19} has no positive solution with $\|u\|_\infty = \rho$
for any $\lambda > 0$. In (ii) we assume that 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b)) < 0$.  For (a) we have that 
$\widetilde{G_3}(\rho, 0) > 0$.  Thus, there is a unique $q \in (0, \bar{q}_1(a,b))$ 
such that $\widetilde{G_3}(\rho, q) = 0$.  Theorem \ref{Thm2.3} guarantees that 
\eqref{e:17} - \eqref{e:19} has a positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q \in (0, \bar{q}_1(a,b))$. 
Since $[\widetilde{G_3}(\rho, q)]_q > 0$ for $q \in (\bar{q}_1(a,b), \rho)$ and 
$\widetilde{G_3}(\rho, \rho) = 0$ this positive solution must be unique. 
In case (b) we have that $\widetilde{G_3}(\rho, 0) = 0$.  Theorem \ref{Thm2.3} 
again guarantees that \eqref{e:17} - \eqref{e:19} has a positive solution
with $\|u\|_\infty = \rho$ and $u(1) = 0$. But, $[\widetilde{G_3}(\rho, q)]_q < 0$ 
for $q \in [0, \bar{q}_1(a,b))$ and $[\widetilde{G_3}(\rho, q)]_q > 0$ for 
$q \in (\bar{q}_1(a,b), \rho)$ combined with $\widetilde{G_3}(\rho, \rho) = 0$ 
gives that this positive solution is again unique.  For (c) we have that 
$\widetilde{G_3}(\rho, 0) < 0$.  Again, $[\widetilde{G_3}(\rho, q)]_q < 0$ for 
$q \in [0, \bar{q}_1(a,b))$ and $[\widetilde{G_3}(\rho, q)]_q > 0$ for 
$q \in (\bar{q}_1(a,b), \rho)$ combined with $\widetilde{G_3}(\rho, \rho) = 0$ 
implies that $\widetilde{G_3}(\rho, q) \neq 0$ for any $q \in [0, \rho)$.  
Theorem \ref{Thm2.3} gives that \eqref{e:17} - \eqref{e:19} has no positive
solution with $\|u\|_\infty = \rho$ for any $\lambda > 0$.

(3) Similar arguments give the result.
\end{proof}

Next, we compute $\widetilde{G_3}(\rho, 0)$ and 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ values using Mathematica in order 
to conclude the shape of the bifurcation curve of positive solutions 
for \eqref{e:17} - \eqref{e:19}.  Again, we are interested in the case
when $b = 1$ and $a > 2\sqrt{2}$ is varied (note that from Theorem \ref{Thm2.4b} 
we must have $a > 2\sqrt{2}$ to have the possibility of a positive solution).  
Our computational results indicate the following cases:

\noindent\textbf{Case 1.}
For $b = 1$, if $a \in (2\sqrt{2}, a_2)$ ($a_2$ is the same value from 
Section 2.2) then $\widetilde{G_3}(\rho, 0)$ and 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ have the structure displayed in 
Figure \ref{fig12}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig12a} % R_8_G3_P_0_a_3.078_MM.eps
\includegraphics[width=0.45\textwidth]{fig12b} % R_8_G3_P_Q1_a_3.078_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_3}(\rho, 0)$
for $a \in (2\sqrt{2}, a_2)$.
(right) $\rho$ vs $\widetilde{G_3}(\rho,\bar{q}_1(a,b))$ for
 $a \in (2\sqrt{2}, a_2)$} \label{fig12}
\end{figure}

 Denote $\bar{M}_1 > 0$ as the $\rho$-value for which
 $\widetilde{G_3}(\bar{M}_1, \bar{q}_1(a,b)) = 0$. From these computational results, 
we have that $\bar{q}_1(a,b) < \bar{q}_2(a,b) < \bar{M}_1 < \frac{a}{b}$.  
In this case, Lemma \ref{lem4} gives that for $\rho \in (\bar{q}_1(a,b), \bar{q}_2(a,b)]$,
 \eqref{e:17} - \eqref{e:19} has only one positive solution with
$\|u\|_\infty = \rho$, $u(1) = q \in (0, \bar{q}_1(a,b))$, and a corresponding
 unique $\lambda > 0$. Also, for $\rho \in (\bar{q}_2(a,b), \bar{M}_1)$ 
Lemma \ref{lem4} 
gives that \eqref{e:17} - \eqref{e:19} has two positive solutions both
having $\|u\|_\infty = \rho$ and the first with $u(1) = q \in (0, \bar{q}_1(a,b))$ 
(which connects to the branch in the case when 
$\rho \in (\bar{q}_1(a,b), \bar{q}_2(a,b)]$) and the second has 
$u(1) = q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$ corresponding to two different 
$\lambda$-values.  Finally, when $\rho = \bar{M}_1$ Lemma\ref{lem4} implies that
 \eqref{e:17} - \eqref{e:19} has a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q = \bar{q}_1(a,b)$ for some $\lambda > 0$. 
 Computationally, we see that the bifurcation curve of positive solutions 
for \eqref{e:17} - \eqref{e:19} forms a closed loop connecting two different
values on the $\|u\|_\infty$-axis.  An interesting feature of this case is 
that the closed loop structure seems to exist for every $a > 2\sqrt{2}$.  
This fact indicates that the nonexistence result for $a \leq 2\sqrt{2}$ 
from Theorem \ref{Thm2.4b} is the best possible.

\noindent\textbf{Case 2.}
For $b = 1$, if $a = a_2$ then $\widetilde{G_3}(\rho, 0)$ and 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ have the structure displayed in 
Figure \ref{fig13}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig13a} % R_8_G3_P_0_a_3.1123_MM.eps
\includegraphics[width=0.45\textwidth]{fig13b} % R_8_G3_P_Q1_a_3.1123_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_3}(\rho, 0)$ for $a = a_2$.
(right) $\rho$ vs $\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ for $a = a_2$}
 \label{fig13}
\end{figure}


Denote $\bar{M}_1 > 0$ as the $\rho$-value for which 
$\widetilde{G_3}(\bar{M}_1, \bar{q}_1(a,b)) = 0$ and $N_1 > 0$ as the 
$\rho$-value for which $\widetilde{G_3}(N_1, 0) = 0$ 
(recall that $\widetilde{G_3}(\rho, 0) = \widetilde{G_2}(\rho, 0)$)). 
Computationally, we have that 
$\bar{q}_1(a,b) < N_1 < \bar{q}_2(a,b) < \bar{M}_1 < \frac{a}{b}$. 
Again using Lemma \ref{lem2}, we can describe the structure of positive solutions 
for \eqref{e:17} - \eqref{e:19} as $\rho$ varies from $\bar{q}_1(a,b)$ to
$\bar{M}_1$.  The bifurcation curve of positive solutions for 
\eqref{e:17} - \eqref{e:19} will be identical to that of Case 1 with one
 exception.  When $\rho = N_1$ one of the two positive solutions of 
\eqref{e:17} - \eqref{e:19} will satisfy $u(1) = 0$ and thus the
Dirichlet boundary condition will also be satisfied at this point for some 
$\lambda > 0$.  In essence, the closed loop will connect to the Dirichlet 
boundary case at the point $(N_1, \lambda^*(N_1))$ for the same $\lambda^*(N_1) > 0$ 
as in Section 2.2.

\noindent\textbf{Case 3.}
For $b = 1$, if $a \in (a_2, a_3]$ (for some $a_3 > a_2$) 
then $\widetilde{G_3}(\rho, 0)$ and $\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ 
have the structure displayed in Figure \ref{fig14}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig14a} % R_8_G3_P_0_a_3.12_MM.eps
\includegraphics[width=0.45\textwidth]{fig14b} % R_8_G3_P_Q1_a_3.12_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_3}(\rho, 0)$ for 
$a \in (a_2, a_3]$. (right) $\rho$ vs $\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ for
 $a \in (a_2, a_3]$}
\label{fig14}
\end{figure}

 Denote $\bar{M}_1 > 0$ as the $\rho$-value for which 
$\widetilde{G_3}(\bar{M}_1, \bar{q}_1(a,b)) = 0$ and $N_i > 0$ as 
the $\rho$-values for which $\widetilde{G_3}(N_i, 0) = 0$ for $i = 1,2$. 
Computationally, we have that 
$\bar{q}_1(a,b) < N_1 < N_2 < \bar{q}_2(a,b) < \bar{M}_1 < \frac{a}{b}$ 
with $N_2 = \bar{q}_2(a,b)$ whenever $a = a_3$.  
For $\rho \in (\bar{q}_1(a,b), N_1]$ Lemma \ref{lem4} implies that the bifurcation 
curve of positive solutions of \eqref{e:17} - \eqref{e:19} will be
a single branch connecting a point on the $\|u\|_\infty$-axis to the Dirichlet 
boundary condition branch (at the point $(N_1, \lambda^{*}(N_1))$ 
from Section 2.2), since when $\rho = N_1$ the corresponding unique $q$-value 
is zero.  When $\rho \in (N_1, N_2)$, \eqref{e:17} - \eqref{e:19}
will have no positive solution with $\|u\|_\infty = \rho$ for any $\lambda > 0$.  
For $\rho = N_2$, \eqref{e:17} - \eqref{e:19} will have a unique positive
solution with $\|u\|_\infty = \rho$ and $u(1) = 0$ 
(this is the point on the Dirichlet boundary case, $(N_2, \lambda^{**}(N_2))$ 
from Section 2.2).  Furthermore, when $\rho \in (N_2, \bar{q}_2(a,b)]$, 
\eqref{e:17} - \eqref{e:19} will have a unique positive solution with
$\|u\|_\infty = \rho$ and $u(1) = q \in (0, \bar{q}_1(a,b))$ for some 
$\lambda > 0$.  When $\rho \in (\bar{q}_2(a,b), \bar{M}_1)$, 
\eqref{e:17} - \eqref{e:19} will have two positive solutions both with
$\|u\|_\infty = \rho$ and with one having $u(1) = q \in (0, \bar{q}_1(a,b))$ 
and the other with $u(1) = q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$ corresponding 
to two different $\lambda-$values.  Computations indicate that the bifurcation 
curve of positive solutions of \eqref{e:17} - \eqref{e:19} forms a loop
that connects the single branch mentioned in the case when 
$\rho \in (N_2, \bar{q}_2(a,b)]$ to a point on the $\|u\|_\infty$-axis.

\noindent\textbf{Case 4.}
For $b = 1$, if $a \in (a_3, \infty)$ then $\widetilde{G_3}(\rho, 0)$ and 
$\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ have the structure displayed in 
Figure \ref{fig15}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig15a} %  R_8_G3_P_0_a_3.2_MM.eps
\includegraphics[width=0.45\textwidth]{fig15b} % R_8_G3_P_Q1_a_3.2_MM.eps
\end{center}
\caption{(left) $\rho$ vs $\widetilde{G_3}(\rho, 0)$ for 
$a \in (a_3, \infty)$. (right) $\rho$ vs $\widetilde{G_3}(\rho, \bar{q}_1(a,b))$ 
for $a \in (a_3, \infty)$}
\label{fig15}
\end{figure}


 Denote $\bar{M}_1 > 0$ as the $\rho$-value for which 
$\widetilde{G_3}(\bar{M}_1, \bar{q}_1(a,b)) = 0$ and $N_i > 0$ as the 
$\rho$-values for which $\widetilde{G_3}(N_i, 0) = 0$ for $i = 1,2$. 
Computationally, we have that 
$\bar{q}_1(a,b) < N_1 < \bar{q}_2(a,b) < N_2 < \bar{M}_1 < \frac{a}{b}$.  
For $\rho \in (\bar{q}_1(a,b), N_1]$ Lemma \ref{lem4} implies that the bifurcation curve 
of positive solutions of \eqref{e:17} - \eqref{e:19} will be a single branch
connecting a point on the $\|u\|_\infty$-axis to the Dirichlet boundary condition 
branch at the point $(N_1, \lambda^{*}(N_1))$, since when $\rho = N_1$ the 
corresponding unique $q$-value is zero.  When $\rho \in (N_1, \bar{q}_2(a,b)]$, 
\eqref{e:17} - \eqref{e:19} will have no positive solution with
$\|u\|_\infty = \rho$ for any $\lambda > 0$.  For $\rho \in (\bar{q}_2(a,b), N_2)$, 
\eqref{e:17} - \eqref{e:19} has a unique positive solution with
 $\|u\|_\infty = \rho$ and $u(1) = q \in (\bar{q}_1(a,b), \bar{q}_2(a,b))$ 
for some $\lambda > 0$.  When $\rho \in [N_2, \bar{M}_1)$ the bifurcation curve 
of positive solutions of \eqref{e:17} - \eqref{e:19} has a loop that
connects the point $(N_2, \lambda^{**}(N_2))$ on the Dirichlet boundary condition 
curve to the single branch mentioned in the case when 
$\rho \in (\bar{q}_2(a,b), N_2)$.  Based on our computations, the bifurcation 
curve of positive solutions for \eqref{e:17} - \eqref{e:19} is identical
in shape for both Cases 3 and 4.

\section{Computational Results}

In this section, we present the complete evolution of the bifurcation curve 
of positive solutions of \eqref{e:5} - \eqref{e:7} as the parameter $a > 0$
is varied.  In this paper, we are particularly interested in the case when $b = 1$.  
Recalling, the Lemmas and Theorems from Section 2, we employed the mathematics 
software package Mathematica to computationally generate the bifurcation curve.  
Due to the complex nature of the formulas, these calculations were extremely 
computationally expensive.  In what follows, the bifurcation curve of positive 
solutions of \eqref{e:8} - \eqref{e:10} is portrayed in red,
\eqref{e:11} - \eqref{e:13} in green, and \eqref{e:17} - \eqref{e:19}
in blue.  Also note that the green curve represents solutions to 
\eqref{e:11} - \eqref{e:13} and \eqref{e:14} - \eqref{e:16} and thus
counts twice. Throughout this section we will denote $\lambda_0 = \frac{\pi^2}{a}$, 
the critical $\lambda$-value for \eqref{e:8} - \eqref{e:10} from Theorem 2.2.

\noindent\textbf{Case 1.} %\label{case 1}
For $b = 1$, if $a \in (0, 2\sqrt{2}]$ then there exists a $\lambda_0 > 0$ 
such that if
   \begin{itemize}
      \item[(1)] $\lambda \in (\lambda_0, \infty)$ then 
\eqref{e:5} - \eqref{e:7} has a unique positive solution.
      \item[(2)] $\lambda \in (0, \lambda_0]$ then \eqref{e:5} - \eqref{e:7}
has no positive solution.
   \end{itemize}
 Figure \ref{fig16} shows an example of Case 1.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig16} % R8_LvsP_a_1.5_b_1_MM.eps
\end{center}
\caption{ Bifurcation curve of positive solutions for
Case 1 with $a = 1.5, b = 1$} \label{fig16}
\end{figure}


\noindent\textbf{Case 2.}
For $b = 1$, if $a \in (2\sqrt{2}, a_0)$ (some $a_0 \in (0, a_1)$) then there 
exist $\lambda_0, \lambda_1 > 0$ such that if
   \begin{itemize}
      \item[(1)] $\lambda \in (0, \lambda_1)$ then \eqref{e:5} - \eqref{e:7}
has two positive solutions.
      \item[(2)] $\lambda = \lambda_1$ or $\lambda \in (\lambda_0, \infty)$ 
then \eqref{e:5} - \eqref{e:7} has a unique positive solution.
      \item[(3)] $\lambda \in (\lambda_1, \lambda_0]$ then 
\eqref{e:5} - \eqref{e:7} has no positive solution.
   \end{itemize}
Case 2 is illustrated in Figure \ref{fig17}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig17} % R8_LvsP_a_2.837_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 2 with $a = 2.837, b = 1$}
 \label{fig17}
\end{figure}

\noindent\textbf{Case 3.}
For $b = 1$, if $a = a_0$ then there exists a $\lambda_0 > 0$ such that if
   \begin{itemize}
      \item[(1)] $\lambda \in (0, \lambda_0)$ then \eqref{e:5} - \eqref{e:7}
 has two positive solutions.
      \item[(2)] $\lambda \in [\lambda_0, \infty)$ then 
\eqref{e:5} - \eqref{e:7} has a unique positive solution.
   \end{itemize}
 Figure \ref{fig18} portrays Case 3.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig18} % R8_LvsP_a_2.88_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 3 with $a = 2.88, b = 1$}
 \label{fig18}
\end{figure}

\noindent\textbf{Case 4.}
For $b = 1$, if $a \in (a_0, a_1)$ then there exist $\lambda_0, \lambda_1 > 0$ 
such that if
   \begin{itemize}
      \item[(1)] $\lambda \in (\lambda_0, \lambda_1)$ then 
\eqref{e:5} - \eqref{e:7} has three positive solutions.
      \item[(2)] $\lambda = \lambda_1$ or $\lambda \in (0, \lambda_0]$ 
then \eqref{e:5} - \eqref{e:7} has two positive solutions.
      \item[(3)] $\lambda \in (\lambda_1, \infty)$ then 
\eqref{e:5} - \eqref{e:7} has a unique positive solution.
   \end{itemize}
 Case 4 is illustrated in Figure \ref{fig19}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig19} % R8_LvsP_a_2.92_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 4 with $a = 2.92, b = 1$}
 \label{fig19}
\end{figure}


\noindent\textbf{Case 5.}
For $b = 1$, if $a = a_1$ then there exist $\lambda_0, \lambda_1, \lambda_2 > 0$ 
such that if
   \begin{itemize}
      \item[(1)] $\lambda = \lambda_1$ then \eqref{e:5} - \eqref{e:7}
has five positive solutions.
      \item[(2)] $\lambda \in (\lambda_0, \lambda_1)$ or 
$\lambda \in (\lambda_1, \lambda_2)$ then \eqref{e:5} - \eqref{e:7}
has three positive solutions.
      \item[(3)] $\lambda = \lambda_2$ or $\lambda \in (0, \lambda_0]$ 
then \eqref{e:5} - \eqref{e:7} has two positive solutions.
      \item[(4)] $\lambda \in (\lambda_2, \infty)$ then 
\eqref{e:5} - \eqref{e:7} has a unique positive solution.
   \end{itemize}
Figure \ref{fig20} exemplifies Case 5.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig20} % R8_LvsP_a_3.0728_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 5 with $a = 3.072, b = 1$}
 \label{fig20}
\end{figure}

\noindent\textbf{Case 6.}
For $b = 1$, if $a \in (a _1, a_2]$ then there exist $\lambda_i > 0$ for 
$i = 0,1,2,3$ such that if
   \begin{itemize}
      \item[(1)] $\lambda \in (\lambda_1, \lambda_2)$ then 
\eqref{e:5} - \eqref{e:7} has seven positive solutions.
      \item[(2)] $\lambda = \lambda_1, \lambda_2$ then \eqref{e:5} - \eqref{e:7}
has five positive solutions.
      \item[(3)] $\lambda \in (\lambda_0, \lambda_1)$ or 
$\lambda \in (\lambda_2, \lambda_3)$ then \eqref{e:5} - \eqref{e:7}
has three positive solutions.
      \item[(4)] $\lambda = \lambda_3$ or $\lambda \in (0, \lambda_0]$ then 
\eqref{e:5} - \eqref{e:7} has two positive solutions.
      \item[(3)] $\lambda \in (\lambda_3, \infty)$ then 
\eqref{e:5} - \eqref{e:7} has a unique positive solution.
   \end{itemize}
An example of Case 6 is shown in Figures \ref{fig21} and 
\ref{fig22}.  Notice in Figure \ref{fig22} that for $\lambda = \lambda^*(N_1)$ 
and the corresponding $\rho = N_1$, all four cases of the boundary conditions 
are satisfied.  This point is where the Dirichlet boundary condition branch
 bifurcates into the other cases.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig21}  % R8_LvsP_a_3.084_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 6 with $a = 3.084, b = 1$}
 \label{fig21}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig22}  % R8_LvsP_a_3.1123_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 6 with $a = 3.1123, b = 1$}
 \label{fig22}
\end{figure}


\noindent\textbf{Case 7.} 
For $b = 1$, if $a > a_2$ then there exist $\lambda_i > 0$ for $i = 0,1,2,3,4,5$ 
such that if
   \begin{itemize}
      \item[(1)] $\lambda \in (\lambda_1, \lambda_2]$ or 
$\lambda \in [\lambda_3, \lambda_4)$ then \eqref{e:5} - \eqref{e:7}
has seven positive solutions.
      \item[(2)] $\lambda = \lambda_1, \lambda_4$ then \eqref{e:5} - \eqref{e:7}
has five positive solutions.
      \item[(3)] $\lambda \in (\lambda_2, \lambda_3)$ then 
\eqref{e:5} - \eqref{e:7} has four positive solutions.
      \item[(4)] $\lambda \in (\lambda_0, \lambda_1)$ or 
$\lambda \in (\lambda_4, \lambda_5)$ then \eqref{e:5} - \eqref{e:7} has three
positive solutions.
      \item[(5)] $\lambda = \lambda_5$ or $\lambda \in (0, \lambda_0]$ then 
\eqref{e:5} - \eqref{e:7} has two positive solutions.
      \item[(6)] $\lambda \in (\lambda_5, \infty)$ then 
\eqref{e:5} - \eqref{e:7} has a unique positive solution.
   \end{itemize}
 Case 7 is illustrated in Figure \ref{fig23}.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig23}  % R8_LvsP_a_3.2_b_1_MM.eps
\end{center}
\caption{Bifurcation curve of positive solutions for Case 7 with $a = 3.2, b = 1$}
 \label{fig23}
\end{figure}


\subsection*{Acknowledgments}
 This research was partially supported through an Auburn University
 Montgomery Faculty Grant-in-Aid.

\begin{thebibliography}{00}

\bibitem{Brown1981g5}
K.~J. Brown, M.~M.~A. Ibrahim, and R.~Shivaji.
\newblock S-shaped bifurcation curves.
\newblock {\em Nonlinear Analysis, Theory, Methods, and Applications},
  5(5):475--486, 1981.

\bibitem{Cantrell2011g363}
Robert~S. Cantrell, Chris Cosner, and Salom Martnez.
\newblock Steady state solutions of a logistic equation with nonlinear boundary
  conditions.
\newblock {\em Rocky Mountain J. Math.}, 41(2):445--455, 2011.

\bibitem{Cantrell2002g31}
Robert~Stephen Cantrell and Chris Cosner.
\newblock Conditional persistence in logistic models via nonlinear diffusion.
\newblock {\em Proceedings of the Royal Society of Edinburgh}, 132A:267--281,
  2002.

\bibitem{Cantrell2003gsb}
Robert~Stephen Cantrell and Chris Cosner.
\newblock {\em Spatial Ecology via Reaction-Diffusion Equations}.
\newblock Mathematical and Computational Biology. Wiley, 2003.

\bibitem{Cantrell2006g81}
Robert~Stephen Cantrell and Chris Cosner.
\newblock On the effects of nonlinear boundary conditions in diffusive logistic
  equations on bounded domains.
\newblock {\em Journal of Differential Equations}, 231:768--804, 2006.

\bibitem{Cantrell2007g33}
Robert~Stephen Cantrell and Chris Cosner.
\newblock Density dependent behavior at habitat boundaries and the allee
  effect.
\newblock {\em Bulletin of Mathematical Biology}, 69:2339--2360, 2007.

\bibitem{Cantrell2009g107}
Robert~Stephen Cantrell, Chris Cosner, and Salome Martnez.
\newblock Global bifurcation of solutions to diffusive logistic equations on
  bounded domains subject to nonlinear boundary conditions.
\newblock {\em Proceedings of the Royal Society of Edinburgh}, 139(1):45--56,
  2009.

\bibitem{Danielson1987g102}
B.~J. Danielson and M.~S. Gaines.
\newblock The influences of conspecific and heterospecific residents on
  coloniztaion.
\newblock {\em Ecology}, 68:1778--1784, 1987.

\bibitem{Fife1979gNB5}
Paul~C. Fife.
\newblock {\em Mathematical aspects of reacting and diffusing systems},
  volume~28 of {\em Lecture Notes in Biomathematics}.
\newblock Springer-Verlag, 1979.

\bibitem{Goddard2009gM4}
Jerome Goddard~II, Eun~Kyoung Lee, and R.~Shivaji.
\newblock Population models with nonlinear boundary conditions.
\newblock {\em Electronic Journal of Differential Equations}, Conference
  19:135--149, 2010.

\bibitem{goddard2009gM3}
Jerome Goddard~II, Eun~Kyoung Lee, and R.~Shivaji.
\newblock Diffusive logistic equation with non-linear boundary conditions.
\newblock {\em Journal of Mathematical Analysis and Applications},
  375:365--370, 2011.

\bibitem{Goddard2011gM5}
Jerome Goddard~II, Eun~Kyoung Lee, and R.~Shivaji.
\newblock Population models with diffusion, strong allee effect, and nonlinear
  boundary conditions.
\newblock {\em Nonlinear Analysis: Theory, Methods, and Applications},
  74(17):6202--6208, 2011.

\bibitem{Goddard2012gM1}
Jerome Goddard~II and R.~Shivaji.
\newblock A population model with nonlinear boundary conditions and constant
  yield harvesting.
\newblock {\em Proceedings of Dynamic Systems and Applications}, 6:150--157,
  2012.

\bibitem{Greenwood1979g115}
P.~J. Greenwood, P.~H. Harvey, and C.~M. Perrins.
\newblock The role of dispersal in the great tit (parus major): the causes,
  consequences, and heritability of natal dispersal.
\newblock {\em Journal of Animal Ecology}, 48:123--142, 1979.

\bibitem{Heinze1996gN21}
G.~Heinze, R.~Pfeifer, and R.~Brandl.
\newblock Veranderte die bestandszunahme zugrichtung and zugverlauf der
  lachmowe (larus ridibundus)?
\newblock {\em Vogelwarte}, 28:146--154, 1996.

\bibitem{Jones1988g104}
W.~Thomas Jones, Peter~M. Waser, Lee~F. Elliott, and Nancy~E. Link.
\newblock Philopatry, dispersal, and habitat saturation in the banner-tailed
  kangaroo rat, diopodmys spectabilis.
\newblock {\em Ecology}, 69(5):1466--1473, 1988.

\bibitem{Kolmogoroff1937gN6}
A.~Kolmogoroff, I.~Petrovsky, and N.~Piscounoff.
\newblock Study of the diffusion equation with growth of the quantity of matter
  and its application to a biological problem.
\newblock {\em Moscow University Bulletin of Mathematics}, 1:1--25, 1937.

\bibitem{Kuussaari1998g88}
Mikko Kuussaari, Ilik Saccheri, Mark Camara, and Ilkka Hanski.
\newblock Allee effect and population dynamics in the glanville fritillary
  butterfly.
\newblock {\em Oikos}, 82(2):384--392, 1998.

\bibitem{Laetsch1970g23}
Theodore Laetsch.
\newblock The number of solutions of a nonlinear two point boundary value
  problem.
\newblock {\em Indiana University Mathematics Journal}, 20(1):1--13, 1970.

\bibitem{LHeureux1995g116}
N.~L'Heureux, M.~Lucherini, M.~Festa-Bianchet, and J.~T. Jorgenson.
\newblock Density-dependent mother-yearling association in bighorn sheep.
\newblock {\em Animal Behavior}, 49(4):901--910, 1995.

\bibitem{Murray2003gB4}
J.~D. Murray.
\newblock {\em Mathematical Biology}, volume I. An Introduction of {\em
  Interdisciplinary Applied Mathematics}.
\newblock Springer-Verlag, New York, third edition, 2003.

\bibitem{Okuba2001gNB9}
Akira Okuba and Simon Levin.
\newblock {\em Diffusion and ecological problems: modern perspectives},
  volume~14 of {\em Interdisciplinary Applied Mathematics}.
\newblock Springer-Verlag, New York, 2001.

\bibitem{Oruganti2002g10}
Shobha Oruganti, Junping Shi, and R.~Shivaji.
\newblock Diffusive logistic equation with constant yield harvesting.
\newblock {\em Transactions of the American Mathematical Society},
  354(9):3601--3619, 2002.

\bibitem{Paivinen2005g103}
Jussi Paivinen, Alessandro Grapputo, Veijo Kaitala, Atte Komonen, Janne
  Kotiaho, Kimmo Saarinen, and Niklas Wahlberg.
\newblock Negative density-distribution relationship in butterflies.
\newblock {\em BMC Biology}, 3(5), 2005.

\bibitem{Pyle2001g117}
P.~Pyle.
\newblock Age at first breeding and natal dispersal in a declining population
  of cassin's auklet.
\newblock {\em Auk}, 118:996--1007, 2001.

\bibitem{Shi2003g163}
Junping Shi and Ratnasingham Shivaji.
\newblock Global bifurcations of concave semipositone problems.
\newblock In {\em Evolution equations}, volume 234 of {\em Lecture Notes in
  Pure and Appl. Math.}, pages 385--398. Dekker, New York, 2003.

\bibitem{Skellam1951g56}
J.~G. Skellam.
\newblock Random dispersal in theoretical populations.
\newblock {\em Biometrika}, 38:196--218, 1951.

\bibitem{Smoller2002g112}
J.~Smoller and A.~Wasserman.
\newblock Global bifurcation of steady-state solutions.
\newblock {\em Journal of Differential Equations}, 7(2):237--256, 2002.

\bibitem{Stamps1991g114}
J.~A. Stamps.
\newblock The effect on conspecifics on habitat selection in territorial
  species.
\newblock {\em Behaviorial Ecology and Sociobiology}, 28:29--36, 1991.

\bibitem{Vincent1995gN24}
J.~P. Vincent, E.~Bideau, J.~M. Hewison, and J.~M. Angibault.
\newblock The influence of increasing density on body weight, kid production,
  home range and winter grouping in roe deer (capreolus capreolus).
\newblock {\em Journal of Zoology}, 236(3):371--382, 1995.

\bibitem{Wahlstrom1995gN25}
L.~K. Wahlstrom and O.~Liberg.
\newblock Patterns of dispersal and seasonal migration in roe deer (capreolus
  capreolus).
\newblock {\em Journal of Zoology}, 235:455--467, 1995.

\end{thebibliography}


\end{document}