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\pagestyle{fancy} \lhead{{\sf Homework 3 $|$ Due February 5 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 450 $|$ Mathematical Modeling
    $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}


% Allman/Rhodes Exercise 2.3.5
\item Consider a model of a structured population with matrix
  $P=\begin{pmatrix}.3&2\\.4&0\end{pmatrix}$ (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.
    \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 steady-state population,
        $\displaystyle\lim_{t\to\infty}\x_t$?
  \end{enumerate}



  % Allman/Rhodes Exercise 2.3.6
\item Repeat the last problem for the \emph{Usher matrix}
  $P=\begin{pmatrix}0&0&73\\.04&0&0\\0&.39&.65\end{pmatrix}$ with
  $\x_0=(100,10,1)$.



  % 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
  \[
  P=\begin{pmatrix}.11&.15&.15\\.3&0&0\\0&.6&.6\end{pmatrix}
  \]
  describes changes over a time step of $1$ year. 
  \begin{enumerate}
    \item 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 growth rate and steady-state distribution of this
      population.
    \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
  \begin{align*}
    P_{t+1}&=P_t(1+1.3(1-P_t))-.5P_tQ_t \\
    Q_{t+1}&=.3Q_t+1.6P_tQ_t
  \end{align*}
  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 By using the MATLAB program {\tt twopop} with a number of
  values of $r$ less than 1.3 in the predator--prey model, see if you
  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?

  

  % 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}
