\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym,bm}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage[all]{xy}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}

\tikzset{big arrow/.style={decoration={markings,mark=at position 0.6
      with {\arrow[scale=2]{>}}}, postaction={decorate}, shorten >=1pt}}

% Code to make an augmented matrix (Usage: \begin{bmatrix}[ccc|c])
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

% Change this line to \def\bold{\mathbf} for ``upright boldface.''
\def\bold{\bm}

\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\0{\bold{0}}
\def\A{\bold{A}}
\def\B{\bold{B}}
\def\C{\bold{C}}
\def\I{\bold{I}}
\def\M{\bold{M}}
\def\P{\bold{P}}
\def\Q{\bold{Q}}
\def\a{\bold{a}}
\def\b{\bold{b}}
\def\c{\bold{c}}
\def\e{\bold{e}}
\def\p{\bold{p}}
\def\q{\bold{q}}
\def\t{\bold{t}}
\def\u{\bold{u}}
\def\v{\bold{v}}
\def\x{\bold{x}}
\def\<{\langle}
\def\>{\rangle}
\def\hat{\widehat}

\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}

\newcommand{\ds}{\displaystyle}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due February 4 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2025 $|$ M.~Macauley}}


\begin{document}

%$\;$

\begin{enumerate}


\item Consider the following instance of the discrete logistic
  equation:
  \[
  P_{t+1}=P_t(1+r(1-P_t/M))
  \]
  Find the two equilibirum points, $P^*$. Use the technique of
  \emph{linearization} to find the stability of these points. That is,
  plug $P_t\approx P^*+p_t$ and $P_{t+1}\approx P^*+p_{t+1}$ into the
  difference equation and express the perturbation $p_{t+1}$ in terms
  of $p_t$, disregarding the nonlinear terms. \\

  
    %
\item In this problem, we will investigate $2$-cycles in the
  logistic map $f(x)=rx(1-x)$.
  \begin{enumerate}
    \item The presence of a $2$-cycle $p,q,p,q,\dots$ means that $p$
      is a fixed point of $f^2(x):=f(f(x))$, and hence a root of the
      polynomial $f^2(x)-x=0$. Find this polynomial.
    \item The fixed points of the logistic map, $x^*=0$ and
      $x^*=\frac{r-1}{r}$, are both roots of this polynomial.
      Therefore,
      \[
      f^2(x)=x\big(x-\tfrac{r-1}{r}\big)g(x).
      \]
      Find $g(x)$ (feel free to use a computer) and its roots.
    \item Determine for what values of $r$ give rise to a $2$-cycle,
      with justification.
    \item The $2$-cycle $p,q,p,q,\dots$ of $f$ is \emph{stable} if $p$ and
      $q$ are stable fixed points for $f^2$. Compute
      \[
      \lambda:=\frac{d}{dx}(f^2(x))_{x=p}.
      \]
      Then substibute $r$ back in, and determine when
      $|\lambda|<1$. This gives the values of $r$ for which the
      $2$-cycle is stable. \\
  \end{enumerate}



\item A population model has an \emph{Allee effect} is if $F(P_t)<P_t$
  for small values of $P_t$. One example of this is the following:
  \[
  \Delta P/P=rP(M-P)(P-T)
  \]
  \begin{enumerate}
  \item Sketch a graph of $P_{t+1}$ vs.\ $P_t$, and of $\Delta P/P$
    vs.~$P$ (\emph{per capita growth rate}), and compare these to the
    logistic model.
  \item Describe what features of a population this model is capturing?
  \item Investigate the resulting model using the Mathlab programs
    {\tt onepop} and {\tt cobweb} (available on the Math 4500 webpage)
    for some choices of $r$, $M$, and $T$. Can you find any values
    that lead to interesting dynamics (doubling, chaos, etc.)? Print
    out (accurately!) sketch a few examples. 
  \item What features of this modeling equation are unrealistic? How
    might the model be improved? \\
  \end{enumerate}


\item Repeat the previous problem but for the \emph{Ricker equation}
  \[
  \Delta P=P(e^{r(1-P/M)}-1)
  \]
  
\end{enumerate}

\end{document}
