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\noindent Math 5110/6830
\\\\
\noindent Homework 1 \\
Due: Thursday Sept. 11 (In Class)
\\\\


\begin{enumerate}

\item Read Chapters 1 and 2 in the textbook.

  \item Consider the following linear model for the population of boars in a swamp 
  \begin{equation}
  \begin{bmatrix}
  b_{n+1} \\ B_{n+1}
  \end{bmatrix} = \vec p_{n+1} = \begin{pmatrix}
  0 & m \\
  \sigma & r
  \end{pmatrix} \cdot
  \begin{bmatrix}
  b_n \\ B_n
  \end{bmatrix}
  = A \cdot \vec p_n.
  \end{equation}
  Here, $B_n$ represents the number of adult boars, while $b_n$ represents the number of immature `boarlettes'. 
  
  \begin{enumerate}
  \item Interpret the $3$ parameters $m$, $\sigma$, and $r$ in your own words. What values are reasonable for each?
  \item Find the eigenvalues of the matrix $A$. \label{stuff}
  \item Now assume that $r = \sigma = 1/2$. Find the value of $m$ which ensures that the population of boars does not die off. Interpret this result in words.
  \end{enumerate}

    
  \vspace{0.5in}




  \item We now consider a model of the same boar population, but with human imposed harvesting
  \begin{equation}
  \vec p_{n+1} = \begin{pmatrix}
  0 & m \\ \sigma  & r
  \end{pmatrix} \cdot 
  \vec p_n - \vec H,
  \end{equation}
  where $\vec H = \begin{bmatrix}
  h_b \\ h_B
  \end{bmatrix}$ encodes the harvest of boarlettes ($h_b$) and adults ($h_B$).
  
  \begin{enumerate}
  \item Using on your calculations from Problem \ref{stuff}, do you believe there will be a positive equilibrium population (hypothesize)? If so, under what conditions? \label{stuff2}
  \item Calculate the equilibrium population of juvenile and adult boars (note: you should not be assuming anything about $r$ or $\sigma$). 
  \item Under what conditions is there a positive equilibrium population? Interpret your condition in words. Is it the same as the condition that you hypothesized in problem \ref{stuff2}.
  \end{enumerate}
  
  \newpage
  \item Consider the following discrete model of medication in the blood stream of a patient,
  \begin{equation}
  M_{n+1} = M_n - f(M_n)\cdot M_n + S.
  \end{equation}
  Here, $M_n$ is the concentration of the medication on day $n$, $S$ is the concentration of medicine administered in a daily dosage, and $f(M_n)$ is the fraction of medication absorbed by the body. Suppose that 
  \begin{equation}
  f(M) = \frac{M^\alpha}{K^\alpha + M^\alpha},
    \end{equation}
    where $K$ and $\alpha$ are also parameters. Assume that $K = 2$ and $S = 1$. In addition to your calculations, use \texttt{cobweb.m} (provided on the course web page) to generate cobweb plots and answer the following questions: 
    
    \begin{enumerate}
    \item Find find the \emph{positive} equilibrium value. %(If you get stuck, you can assume that $0 < \alpha \leq 1$, it makes the algebra easier.)
    \item For which values of $\alpha$ does the equilibrium seem stable. Why? Include a cobweb diagram of an illustrative case.
    \item For which values of $\alpha$ does the concentration $M_n$ oscillate towards the equilibrium. Why? Include a cobweb diagram of an illustrative case.
    \item Are there values of $\alpha$ for which the equilibrium is unstable? What are they? Why? Include a cobweb diagram of an illustrative case.
    \end{enumerate}

  \vspace{0.5in}

 \item In 1973 Varley, Gradwell and Hassell proposed the following model of a population with a density dependent reproduction rate 
 \begin{equation}
 p_{n+1} = \frac{\lambda}{\alpha}p_n^{1-b},
 \end{equation}
which can be rewritten as 
 \begin{equation}
 p_{n+1} = \left(\frac{1}{\alpha}p_n^{-b}\right) \Big(\lambda p_n\Big). \label{stuff3}
 \end{equation}
 \begin{enumerate}
 \item Interpret the two terms in equation \ref{stuff3} in your own words.
 \item Find the equilibrium population. Under what conditions is it stable? Sketch the bifurcation diagram. 
 \item What happens to the population when the stability conditions are violated? Include a cobweb diagram of an illustrative case.
  \end{enumerate}

\end{enumerate}

\end{document}



