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\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 19, 2025
    (Wednesday)}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical
    Modeling $|$ Spring 2025 $|$ M.~Macauley}}


\begin{document}

$\;$

\begin{enumerate}


\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. Give several real-life examples
  of two populations that might exhibit this feature, and propose a model
  of this. Either an ODE of difference equation framework is fine. \\


  
\item Consider the following model of two species:
  \begin{align*}
    \frac{dX}{dt}&=r_1X\left(1-\frac{X}{M_1+b_{12}Y}\right)  \\
    \frac{dY}{dt}&=r_1Y\left(1-\frac{Y}{M_2+b_{21}X}\right).
  \end{align*}
  \begin{enumerate}
  \item Describe what interactive behavior between species $X$ and $Y$ is implied by the model, and an example of two species that this might model. Include a discussion of what the parameters might represent. 
  \item Find the nullclines, $X'=0$ and $Y'=0$, and sketch them on the
    $XY$-plane.
  \item Find the steady states of this model, and determine their
    stability by linearization.
  \item Summarize the ecological implications of your results. \\
  \end{enumerate}

  

\item Consider the system of difference equations
  $
  \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.
  $
  \begin{enumerate}
  \item Describe what type of interacting species this models, and
    why.
  \item Find the steady-states, and linearize them to determine their
    stability.
  \item Plot the solutions using the MATLAB program {\tt twopop},
    which is available on the course website. 
  \item The discrete logistic model $P_{t+1}=P_t(1+1.3(1-P_t))$ has
    underdamped dynamics (=damped oscillations) when $r=1.3$, but
    overdamped dynamics when $r<1$. Using the {\tt twopop} program,
    can you find a value of $r$ that yields no oscillations in the
    interacting species model above? If so, what is the approximate
    threshold of $r$ between these two dynamical regimes? Include
    printouts, screenshots, or accurate sketches to support your hypothesis.
  \end{enumerate}
 


\end{enumerate}

\end{document}
