\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 3 $|$ Due February 3, 2017 (Monday)}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling $|$ Spring 2017 $|$ M.~Macauley}} \begin{document} %$\;$ \begin{enumerate} % Allman/Rhodes Exercise 2.3.5 \item Consider a structured population model with matrix $P=\begin{bmatrix}.3&2\\.4&0\end{bmatrix}$ (called a \emph{Leslie matrix}): \begin{enumerate} \item By thinking about the biological meaning of each entry in this matrix, do you think it describes a growing or declining population. Would you guess the population size would change rapidly or slowly? Explain your reasoning. \item Compute the eigenvalues and eigenvalues of the model. (Use a computer.) %\item What is the intrinsic growth rate? \item Express the initial vector $\x_0=(5,5)$ as a sum of the eigenvectors. \item Use your answer in the previous part to give a formula for the population vector $\x_t$. \item What is the long-term behavior, $\displaystyle\lim_{t\to\infty}\x_t$? \end{enumerate} % Allman/Rhodes Exercise 2.2.8 \item A model given in (Cullen, 1985), based on data collected in (Nellis and Keith, 1976), describes a certain coyote population. The population is stratified in three classes: pup, yearling, and adult, and the matrix \[ P=\begin{bmatrix}.11&.15&.15\\.3&0&0\\0&.6&.6\end{bmatrix} \] describes changes over a time step of $1$ year. \begin{enumerate} \item Carefully explain what each entry in this matrix is saying about the population. %Be careful in explaining the $P_{1,1}=.11$ entry. \item Find the intrinsic growth rate (dominant eigenvalue) and corresponding eigenvector. Feel free to use a computer. \item Will the population grow or decline? Quickly or slowly? \end{enumerate} % Allman/Rhodes Exercise 3.1.5 \item In class, we saw that the model $ \left\{\begin{array}{l} P_{t+1}=P_t(1+1.3(1-P_t))-.5P_tQ_t \\ Q_{t+1}=.3Q_t+1.6P_tQ_t \end{array}\right. $ has a steady-state equilibrium that is approached through oscillations. Because the discrete logistic model $P_{t+1}=P_t(1+1.3(1-P_t))$ on which it is based has $r=1.3$, we know that it alone would produced underdamped dynamics (=damped oscillations) rather than the overdamped dynamics that arise when $r<1$. Thus, it is not clear whether the oscillations in the model above are inherient to the model or, simply due to $r>1$. \indent Explore this using the MATLAB program {\tt twopop} with a number of values of $r$ -- less than and greater than 1.3 in the predator--prey model. Can find a value or $r<1$ that yields oscillations in the predator--prey model? If so, can you find a value of $r$ that yields no oscillations, and where is the ``threshold'' between these two dynamical regimes? Are there any other ``thresholds'' where the qualitative dynamics changes? Include print-outs for a few different values of $r$. % Allman/Rhodes Exercise 3.1.7 \item Imagine a predator--prey interaction in which a certain number of the prey population cannot be eaten because of a refuge in their environment that the predator cannot enter. \begin{enumerate} \item Give a real-life example two populations that might exhibit this feature. \item Why might interaction terms like $-s(P-w)Q$ and $v(P-w)Q$ be reasonable in the modeling equation? \item What is the meaning of $w$? Would you expect $w>P$ or $w