\documentclass{lizpset}

\name{\phantom{Liz Fedak}} % Remove 'phantom.' It just makes the input invisible.
\class{Math 5110}
\assignment{Assignment 2}
\duedate{09/07/17}

\begin{document}

  \begin{problem}[1.]
Consider a population consisting of juveniles and adults, where each
adult produces $m$ offspring, each juvenile has a probability $\sigma$
of maturing into an adult, and each adult has a probability $p$ of surviving.
In addition, however, a fraction $\gamma$ of the juveniles follow the 
``Peter Pan" strategy and remain alive as juveniles (a fraction 
$1-\sigma-\gamma$ die).
\begin{enumerate}[(a)]
  \item Write the matrix describing this situation.
  \item Suppose $m=3.75$ and $p=0.5$ and that $\sigma + \gamma = 0.5$.  
Find the eigenvalues when $\sigma = 0.5$.
  \item Find the eigenvalues when $\sigma = 0.25$.
  \item Find the eigenvalues when $\sigma = 0.0$.  Could you have guessed
the leading eigenvalue without doing any calculations?
  \item Which population will grow the most quickly?  How do the stable
age distributions compare (which population will be the most juvenile)?
\end{enumerate}
\end{problem}

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%  \begin{enumerate}[(a)]
%   \item 
%  \end{enumerate}
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\newpage

\begin{problem}[2.]
Suppose a population can experience two types of years. In the first,
$m=0.5$, $p=0.5$ and $\sigma=0.9$ (with $m$, $p$ and $\sigma$ defined 
as in the previous problem). In the second, $m=2.0$, $p=0.5$ and 
$\sigma=0.2$.
\begin{enumerate}[(a)]
  \item Find the leading eigenvalue for each population. What would
happen to each in the long run?
  \item Suppose instead that the two types of years {\bf alternate}. 
Find the matrix describing what the population looks like after two years.
  \item Find the leading eigenvalue of this two year matrix. Will the 
population grow or shrink?
  \item Explain how this apparently baffling paradox is possible.
\end{enumerate}
\end{problem}

% \begin{solution}
%  \begin{enumerate}[(a)]
%   \item 
%  \end{enumerate}
% \end{solution}

\newpage

\begin{problem}[3.]
Suppose individuals of age 0 have a probability $p$ of surviving to age 1,
individuals of age 1 have a probability $p$ of surviving to age 2,
and so on for all ages. Furthermore, individuals of all ages (including 
age 0) produce $m$ offspring per year.
\begin{enumerate}[(a)]
  \item Write a Leslie matrix describing this population.
  \item Summarize your Leslie matrix as a 2$\times$2 matrix.
  \item Find the leading eigenvalue of this matrix.
  \item Think of $m$ as being large, say, 1000. Suppose a mutant 
arises that produces one more offspring at age 0, but dies from the effort.
What is the long term rate of growth of a population of mutants?
Which grows faster, the population of mutants or the original population?  
  \item This result is known as Cole's Law, after Lamont Cole, a famous
ecologist and connoisseur of picnic salads. Does this result surprise you?
% Liz's note: Yes! It's rare for a scientist to be really into picnic salads!
Why would any organism bother surviving more than one year?
% Liz's note: I haven't decided whether ``we fear death" is an acceptable answer here.
\end{enumerate}
\end{problem}

% \begin{solution}
%  \begin{enumerate}[(a)]
%   \item 
%  \end{enumerate}
% \end{solution}

\end{document}
