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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due February 11, 2025
    (Tuesday)}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical
    Modeling $|$ Spring 2025 $|$ M.~Macauley}}


\begin{document}

%$\;$

\begin{enumerate}

  
  
  %%---------------------------------------------------------------------

\item The life stages, transitions, and reproduct data for the South American annual \emph{Hyptis suaveolens} (Schwarzkopf et. al, 2009) is shown below. Seeds germinate in late spring to early summer, and grow until the end of the wet season in December--January, when they produce seeds and die. 

  \[
  \tikzstyle{v} = [rectangle, draw, inner sep=4pt]
  \tikzstyle{e} = [draw, thick, -stealth]
  \begin{tikzpicture}
    \tikzstyle{every node}=[font=\normalsize]
    \node (seed) at (0,0) [v] {seed};
    \node (juvenile) at (4,0) [v] {juvenile};
    \node (adult) at (8,0) [v] {adult};
    \draw [e] (seed) -- (juvenile) node[midway,above]{\small $.0135$};
    \draw [e] (juvenile) -- (adult) node[midway,above]{\small $.65$};
    \draw [e,bend right] (adult) to[bend right=25] node[midway,above]{\small $113.9$} (seed);
  \end{tikzpicture}
    \]
    \begin{enumerate}
    \item Build a population matrix $A$ that describes this. 
    \item Is there the smallest power $k$ of $A$ such that every entry
      of $A^k$ is positive? (If so, what is $k$?) The existence of
      such a $k$ means that $A$ is \emph{primitive}.
    \item If a matrix is primitive, then the Perron-Frobenius theorem guarantees a unique largest positive eigenvalue, called its \emph{dominant eigenvalue}. Find the dominant eigenvalue of $A$. 
    \item Discuss the long-term behavior of this model, and the
      intrinsic growth rate.
    \item How does this answer change if seeds can remain viable for
      more than one year?
    \end{enumerate}

    %%---------------------------------------------------------------------
    
\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}
    
    %%---------------------------------------------------------------------
    
  \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}
    
    %%---------------------------------------------------------------------

  



\end{enumerate}

\end{document}
