\documentclass[11pt]{article} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amstext} \usepackage{amsopn} %\usepackage{graphicx} \usepackage[pdftex]{graphicx} \DeclareGraphicsExtensions{.eps, .pdf, .jpg, .tif} \oddsidemargin 0in \evensidemargin 0in \topmargin=0in \headsep=0.1in \headheight=0.1in \textwidth=6.5in \textheight=8.5in \footskip=0.2in \parskip=6pt \parindent=20pt \newcommand{\ds}{\displaystyle} \newcommand{\la}{\lambda} \newcommand{\ep}{\varepsilon} \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 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{a} \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} 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} \begin{figure} \begin{center} \includegraphics[height=2in]{panda.jpg} \caption{A panda picture} \end{center} \end{figure} \begin{figure}\begin{center} \includegraphics[height=2in, width=3in]{bifurcation.jpg} \caption{A bifurcation diagram} \end{center}\end{figure} \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: There are various reaction-diffusion models with nonlinear density dependent diffusion. The general form of such equation is \begin{equation}\label{1} \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{3a} [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 \bibliographystyle{amsalpha} \begin{thebibliography}{MMMM} \bibitem[AT1]{AT1} Alama, Stanley; Tarantello, Gabriella, On semilinear elliptic equations with indefinite nonlinearities. {\sl Calc. Var. Partial Differential Equations} {\bf 1} (1993), no. 4, 439--475. \bibitem[AT2]{AT2}Alama, Stanley; Tarantello, Gabriella, Elliptic problems with nonlinearities indefinite in sign. {\sl J. Funct. Anal.} {\bf 141} (1996), no. 1, 159--215. \bibitem[Al]{Al} Allee, W.C., The social life of animals. 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