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\title{Math 142-2, Homework 1}
\author{Your name here}
\date{\today}
\maketitle

\section*{Problem 34.4}

\textbf{Suppose the growth rate of a certain species is not constant, but
  depends in a known way on the temperature of its environment.  If the
  temperature is known as a function of time, derive an expression for the
  future population (which is initially $N_0$). Show that the population grows
  or decays with an exponential growth coefficient, ($R_E(t)$, where $N(t)
  \propto e^{R_E(t) t}$), equal to the average of the time-dependent growth
  rate.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 34.5}

\textbf{In this problem we study the effect of any time-dependent migration,
  $f(t)$.  Consider both the resulting discrete and continuous growth models:
\begin{equation}
N_{m+1} - N_m = R_0 \dt N_m + \dt f_m \label{eqn:35.5-i}
\end{equation}
\begin{equation}
\frac{dN}{dt} = R_0 N + f(t) \label{eqn:35.5-ii}
\end{equation}
}

\textbf{(a) If the growth rate is zero ($R_0 = 0$), show that the solution of
  the discrete growth model is analogous to integration.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) If the growth rate is nonzero, show that
\begin{equation}
N_{m+1} = \alpha N_m + f_m \dt \label{eqn:35.5-iii}
\end{equation}
where $\alpha = 1 + R_0 \dt$. By explicit calculation of $N_1, N_2, N_3,
\ldots$, determine $N_m$, if $N_0$ is known.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) For the differential equation (\ref{eqn:35.5-ii}), calculate the
  solution using the integrating factor $e^{-R_0 t}$.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(d) In this part of the problem, we wish to develop a technique to
  solve difference equation (\ref{eqn:35.5-i}), analogous to the integrating
  factor method used in part (c). By dividing equation (\ref{eqn:35.5-iii}) by
  $\alpha^{m+1}$, show that
  \begin{equation}
    Q_{m+1} - Q_m = \frac{f_m}{\alpha^{m+1}} \dt \label{eqn:35.5-iv}
  \end{equation}
  where $Q_n = N_m / \alpha^{m}$.  Using the result of part (a), solve equation
  (\ref{eqn:35.5-iv}), and thus determine $N_m$.  Show that your answer agrees
  with part (b), and show how your answer is analogous to the results of part
  (c).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 34.9}

\textbf{Suppose that one species has an instantaneous growth rate of $\alpha$
  percent per year while another species grows in discrete units of time at the
  annual rate of $\beta$ percent per year.  Suppose the second species has four
  growth periods a year.  What relationship exists between $\alpha$ and $\beta$
  if both species (starting with the same number) have the same number 5 years
  later?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}


\section*{Problem 34.14}

\textbf{The parameters of a theoretical population growth curve are ofter
  estimated making the best fit of this curve to some data.  If discrete
  population data is known (not necessarily measured at equal time intervals),
  $N_d(t_m)$, then the mean-square deviation between the data and a theoretical
  curve, $N(t)$, is
$$ \sum_m [N(t_m) - N_d(t_m)]^2, $$
  the sum of the squared differences. The ``best'' fit is often defined as those
  values which minimize the above mean-square deviation.}

\textbf{(a) Assume that the initial population is known with complete certainty,
so that we insist that the theoretical population curve initially agree
exactly.  Assume the theoretical curve exhibits exponential growth. By
minimizing the above mean-square deviation, obtain an equation for the best
estimate of the growth rate. Show that this is a transcendental equation.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) One way to bypass the difficulty in part (a) is to fit the natural
  logarithm of the data to the natural logarithm of the theoretical curve. In
  this way, the mean-square deviation is
$$ \sum_m [\ln N_0 + R_0 t_m - \ln N_d(t_m)]^2, $$
Show that this method now is the least squares fit of a straight line to data.
If $N_0$ is known ($N_0 = N_d(t_0)$), determine the best estimate of the growth
rate using this criteria.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) Redo part (b) assuming that best estimate of the initial population
is also desired (i.e., minimize the mean-square deviation with respect to both
$N_0$ and $R_0$).}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 34.17}

\textbf{In this section it was shown that the difference equation of discrete
  growth becomes a differential equation in the limit as $\dt \to 0$. If a
  differential equating is known, for example,
  $$ \frac{dN(t)}{dt} = R_0 N(t),$$
  we will show how to calculate a difference equation which corresponds to
  it. The idea is useful for obtaining numerical solutions on the computer of
  differential equations. If a Taylor series is used, then
  $$ N(t+\dt) = N(t) + \dt \frac{dN}{dt} + O((\dt)^2).$$
  Thus, if higher order terms are neglected:}

\textbf{(a) Show that
  $$ \frac{dN}{dt} = \frac{N(t+\dt) - N(t)}{\dt} + O(\dt).$$
  Replacing the derivative by this difference is known as Euler's method to
  numerically solve ordinary differential equations.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(b) Show that the resulting difference equation (called the
  discretization of the differential equation) is itself a discrete model of
  population growth.}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\vspace{1cm}
\textbf{(c) At the very least, for this to be a reasonable procedure, the first
  neglected term in the Taylor series $(\dt)^2 d^2N/dt^2/2$ must be much less
  than $\dt(dN/dt)$.  Using the differential equation, are there thus any
  restrictions on the discretization time $\dt$ for this problem?}
\vspace{1cm}

\textbf{\textcolor{red}{Your solution goes here}}

\end{document}

