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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due September 5
    (Monday)}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical
    Modeling $|$ Fall 2016 $|$ 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 a print-outs or detailed sketches 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 an 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 $w<P$ to
    be more reasonable?
  \end{enumerate}


\end{enumerate}

\end{document}
