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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due January 26 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}


% Allman/Rhodes 1.2.5
\item Four of the many common ways of writing the discrete logistic
  growth equation are 
  \[
  \begin{array}{llll}
    \Delta P=rP(1-P/K),&&& \Delta P=sP(K-P), \\
    \Delta P=tP-uP^2,&&& P_{t+1}=vP_t-wP_t^2\,.
  \end{array}
  \]
  Write each of the following in all four of these forms.
  \begin{enumerate}
  \item $P_{t+1}=P_t+.2P_t(10-P_t)$
  \item $P_{t+1}=2.5P_t-.2P_t^2$
  \end{enumerate}



% Made up myself (section 1.2)
\item Consider the difference equation $P_{t+1}=P_t(2-P_t)$. The
  graphs of $F(x)=x(2-x)$, and $x=y$ intersect at $(x,y)=(1,1)$, which
  is the local maximum of the parabola.
  \begin{enumerate}
  \item Draw two cobweb plots, one corresponding to an initial condition
    $P_0<1$ and the other to the initial condition $P_0>1$. Recall
    that these lie on the $(P_t,P_{t+1})$-plane.
  \item For each of these, plot the points $(t,P_t)$ for a few small 
    integer values of $t$. 
  \item How do the previous parts compare to the case when the point
    of intersection is to the left of the local maximum? To the right
    of the local maximum?
  \end{enumerate}



% Made up myself (section 1.3)
\item Consider the following instance of the discrete logistic
  equation:
  \[
  P_{t+1}=P_t(1+r(1-P_t/K))
  \]
  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 non-linear terms.
 


% Allman/Rhodes 1.4.2 (modified)
\item The discrete logistic and the Ricker population models when
  written as $P_{t+1}=F(P_t)$ have the property that for small values
  of $P_t$, the graph of $F(x)$ lies \emph{above} the line $y=x$ This means
  that $F(P_t)>P_t$ for small value of $P_t$. Consider a model for
  which $F(P_t)<P_t$ for small values of $P_t$. Explain the affect of
  this feature on population dynamics. Why might this be a
  biologically important feature? (The resulting behavior is sometimes
  known as an \emph{Allee effect.})



% Allman/Rhodes 1.4.3 (modified)
\item Construct a simple model showing an Allee effect as follows. 
  \begin{enumerate}
  \item Explain why for some $0<L<K$, the per-capita grown should be 
    \begin{align*}
      &\frac{\Delta P}{P}<0,\;\text{ when } 0<P<L\; \text{ or } P>K\,, \\
      &\frac{\Delta P}{P}>0,\;\text{ when } L<P<K\,.
    \end{align*}
    Sketch a possible graph of $\Delta P/P$ vs.~$P$.
  \item Explain why $\Delta P/P=P(K-P)(P-L)$ has the qualitative
    features desired. 
  \item Investigate the resulting model using the MATLAB programs {\tt
    onepop} and {\tt cobweb} (available on the Math 4500 webpage) for
    some choices of $K$ and $L$. Print out, or sketch the results of a
    few sample trials. Is the behavior as expected?
  \item What features of this modeling equation are unrealistic? How
    might the model be improved? 
  \end{enumerate}


\end{enumerate}

\end{document}
