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\begin{document}

\title{ Persistence in Reaction Diffusion Models with Weak Allee
Effect \footnote{2000 subject classification: 35J65, 35B32, 92D25,
92D40, 35Q80} \footnote{Keywords: Population biology,
Reaction-diffusion equation, Allee effect, Global Bifurcation}}


\author{Junping Shi \\
{\small Department of Mathematics,
 College of William and Mary,
Williamsburg, VA 23185}\\
{\small and, } {\small Department of Mathematics}, {\small Harbin
Normal University},
{\small Harbin, Heilongjiang, P.R.China}\\
{\small Email: shij@math.wm.edu}\\
\\
Ratnasingham Shivaji \\
{\small Department of Mathematics, Mississippi State University,
Mississippi State, MS 39762}\\{\small Email:
shivaji@ra.msstate.edu}\\}
\date{}
\maketitle

\begin{center} \LARGE{Test 2}

 \large{Math 490}
 \end{center}

 \section{Introduction}

\noindent \textbf{A.} Consider a reaction diffusion equation
\begin{equation}\label{1}
\begin{cases}
 \ds\frac{\partial u}{\partial t}
=\frac{\partial^2 u}{\partial x^2}+\la\left[u(1-u)-\frac{au}{1+u}\right], \;\; t>0, \; x\in (0,1),&\\
u(t,0)=0, \; u(t,1)=0, &\\
 u(0,x)=u_0(x), \;\; x\in(0,1),&
\end{cases}
\end{equation}

\eqref{1}
where $\la>0$ and $a>0$.
\begin{enumerate}
    \item $u(t,x)=0$ is an equilibrium solution. Consider
\begin{equation}\label{3.34}
\begin{cases}
\ds  \phi''(x)+\la f'(0)\phi(x)=\mu \phi(x), & x\in (0,1),\\
\phi(0)=\phi(1)=0, &
\end{cases}
\end{equation}
where $\ds f(u)=u(1-u)-\frac{au}{1+u}$.
    Determine for
    which $(\la,a)$, $u=0$ is asymptotically stable, and for which
    $(\la,a)$, $u=0$ is unstable.
    \item For fixed $a>0$,
    we define a bifurcation point as the value of $\la$ where $u=0$ changes from stable to unstable.
    From part 1, determine the bifurcation point when $a=a_0$ (a positive
    number). (Note that for some $a$, there is no bifurcation
    point.)
    \item When $a=0.5$, the bifurcation point is $\la=2\pi^2$.
    Near the bifurcation point, if $\ep=\la-2\pi^2$, then
    $u(\ep,x)=\ep u_1(x)+\ep^2 u_2(x)+\cdots$. Find $u_1(x)$.
    (Hint: in expansion of $au/(1+u)$, you should use the Taylor
    expansion: $\ds \frac{1}{1+u}=1-u+u^2-u^3+\cdots$.)
\end{enumerate}

\noindent \textbf{B.} Suppose that $u(x)$ is a positive solution
of
\begin{equation}\label{3.44}
\begin{cases}
\ds  u''+u^2=0, & x\in (0,1),\\
u(0)=u(1)=0. &
\end{cases}
\end{equation}
Prove that $u(x)$ is unstable. (Hint: use the idea in proof of
Proposition 3.9.)

\noindent \textbf{C.} Consider a reaction diffusion equation
\begin{equation}\label{3}
\begin{cases}
 \ds\frac{\partial u}{\partial t}
=\frac{\partial^2 u}{\partial x^2}+\la\left[u(3-u)-\frac{au}{1+u}\right], \;\; t>0, \; x\in (0,1),&\\
u(t,0)=0, \; u(t,1)=0, &\\
 u(0,x)=u_0(x), \;\; x\in(0,1),&
\end{cases}
\end{equation}
where $\la>0$ and $a>0$. \begin{enumerate} \item When $a=3.5$, use
\verb"Maple" to draw the global bifurcation
    diagram of the positive equilibrium solutions. What is the
    smallest value of $u_{max}=u(1/2)$? And what is the smallest
    $\la$ such that there is a positive equilibrium solution?
    \item There is an $a_0>0$ such that when $a>a_0$, the equation
    has no positive equilibrium solution. Find the smallest $a_0$.
    \end{enumerate}

Let's enter a matrix:

$\left(%
\begin{array}{cc}
  1 & 2 \\
  3 & 4 \\
\end{array}%
\right)$

Insert a graphics:

\begin{figure}[h]
\centering
  \includegraphics[scale=0.6]{allee11.eps}
  \caption{\label{g:3.11}This is the caption}
\end{figure}

There are various reaction-diffusion models with nonlinear density
dependent diffusion. The general form of such equation is
\begin{equation}\label{1111}
   \frac{ \partial u}{\partial t}=\frac{\partial }{\partial
   x}\left( D(u) \frac{\partial u}{\partial x} \right)+f(u),
\end{equation}
where $u$ is the population density of a species, $D(u)$ is the
nonconstant diffusion rate (a differentiable function which is
non-negative), and $f(u)$ is the growth rate of $u$. We consider
the equation on a bounded interval $[0,L]$, and impose absorbing
(Dirichlet) boundary condition:
\begin{equation}\label{2}
    u(0,t)=u(L,t)=0.
\end{equation}


 The equilibrium solutions of \eqref{1} and \eqref{2} satisfy a
 nonlinear boundary value problem:
 \begin{equation}\label{3}
    [D(u)u']'+f(u)=0, \;\;\; u(0)=u(L)=0.
\end{equation}
With a nondimensionalization process, \eqref{3} can be converted
to an equation on a fixed interval $[0,1]$:
 \begin{equation}\label{4}
    [D(u)u']'+\lambda f(u)=0, \;\;\; u(0)=u(1)=0.
\end{equation}
Following the ideas in Lecture Notes Section 4.7, a time-mapping
formula can be derived for equation \eqref{4} or \eqref{3}, thus
global bifurcation diagrams for \eqref{4} can be obtained. The
idea of the formula is as follows: multiplying \eqref{4} by
$D(u)u'$, then integrating from $t=1/2$ to $t=x$, then we have
\begin{equation}\label{4.5}
    \frac{1}{2}[D(u(x))u'(x)]^2+\lambda \int_{1/2}^{x}
    f(u(t))D(u(t))u'(t) dt=0.
\end{equation}
Here we assume that $u(x)$ is a solution of \eqref{3} such that
$u(x+1/2)=u(x-1/2)$ (symmetric with respect to $1/2$),  $u'(x)<0$
in $(1/2,1)$ and $u'(1/2)=0$ (thus $x=1/2$ is the maximum point of
$u(x)$). These properties of $u(x)$ can be obtained from analyzing
the phase portrait of the dynamical system:
\begin{equation}\label{5}
D(u)u'=v, \;\; v'=-f(u).
\end{equation}
From \eqref{4.5}, we obtain
\begin{equation}\label{6}
    D(u)\frac{du}{dx}=-\sqrt{2\lambda [G(u_0)-G(u)]},
\end{equation}
where $u_0=u(1/2)$, $u=u(x)$, and  $G(u)=\int_0^u f(v)D(v) dv$.
Thus
\begin{equation}\label{7}
    \frac{dx}{du}=-\frac{1}{\sqrt{2\lambda}}\frac{D(u)}{\sqrt{G(u_0)-G(u)}},
\end{equation}
and we integrate from $u=u(1/2)$ to $u=u(1)=0$ (corresponding  to
$x=1/2$ to $x=1$)
\begin{equation}\label{8}
    1/2=\int_{u(1/2)}^{u(1)}\frac{dx}{du}
    du=-\frac{1}{\sqrt{2\lambda}}\int_{u(1/2)}^{u(1)}\frac{D(u)}{\sqrt{G(u_0)-G(u)}}du.
\end{equation}
So we obtain
\begin{equation}\label{9}
    \sqrt{\frac{\lambda}{2}}=\int_0^s \frac{D(u)}{\sqrt{G(s)-G(u)}}du,
\end{equation}
where $s=u(1/2)$.

This project is to carry out the details of the mathematical and
numerical analysis of the above ideas. To be more specific, the
research program include

\begin{enumerate}
    \item Analysis of the phase portrait of \eqref{5}, assuming
    $D(u)>0$ for all $u> 0$. But also consider the case $D(0)=0$
    (example $D(u)=u^m$).
    \item Develop \verb"Maple" program for the numerical
    calculation of the bifurcation diagrams based on \eqref{9}
    following the Maple program for the case $D(u)=1$ in Section
    4.7. Consider the cases $D(u)=u^m$ and $D(u)=au^2+bu+c$, and
    $f(u)=u(1-u)$ (logistic), $f(u)=u(1-u)(u-b)$ (Allee effect).
    \item Define  the integral on the right side of \eqref{9} as
    $T(s)$, calculate the 1st and 2nd derivatives of $T(s)$. Try
    to obtain analytic results (monotone or with only one critical point)
    for certain classes of $D(u)$ and $f(u)$. (examples in part 2)
\end{enumerate}

 {\small
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\end{document}
