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

\section*{Problem 35.3}

\textbf{Consider a species in which both no individuals live to three years old
  and only one-year olds reproduce.}

\textbf{(a) Show that $b_0 = 0$, $b_2 = 0$, $d_2 = 1$ satisfy both conditions.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(b) Let $b_1 = b$. What is the $A$ matrix?}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(c) Determine (a computer is not necessary) the time development of a
  population initially consisting of $N_0 = 100$, $N_1 = 100$, $N_2 =
  100$. Explicitly calculate the following three cases of birth and death rates:
\begin{enumerate}
\item $b=1, d_0=\frac{1}{2}, d_1=\frac{1}{2}$
\item $b=2, d_0=\frac{1}{2}, d_1=\frac{1}{2}$
\item $b=3, d_0=\frac{1}{2}, d_1=\frac{1}{2}$
\end{enumerate}
}
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\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 37.2}

\textbf{F.E. Smith suggested a different simple model of the population growth
  of a species limited by the food supply based on experiments on a type of
  water bug. As in the logistic model, the growth rate is proportional to the
  difference between the available food $f_a$ and the subsistence level of food
  consumption $f_c$:
  $$ \frac{1}{N} \frac{dN}{dt} = \alpha(f_a - f_c).$$
  However, previously $f_c$ was assumed proportional to the number in
  individuals of the species. Smith instead assumed that more food is necessary
  for survival during the growing phase of a population. Consequently a simple
  model would be
  $$ f_c = \beta N + \gamma \frac{dN}{dt} $$
  with $\gamma > 0$. What differential equation describes this model? What are
  the equilibrium populations?}
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\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 37.4}

\textbf{Which of the following are reasonable models of the spread of a disease
  among a finite number of people: \\
  (1) $dN/dt=\alpha N$ \\
  (2) $dN/dt=\alpha (N_T - N)$ \\
  (3) $dN/dt=\alpha (N - N_T)$ \\
  where $N$ is the number of infected individuals and $N_T$ is the total
  population.}
\vspace{1cm}

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

\section*{Problem 37.5}

\textbf{A certain species has an instantaneous growth rate of 27 percent per
  year when not affected by crowding.  Experimentally, for each 1000 of the
  species the birth rate drops by 12 per 1000 per year, and the death rate
  increases by 50 per 1000 per year. Determine the parameters of a logistic
  equation which models this species. What is the expected nonzero equilibrium
  population?}
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\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 38.3}

\textbf{Consider Smith's model of population growth (see exercise 37.2)
$$ \frac{dN}{dt} = \frac{N \alpha(f_a - \beta N)}{1 + N \alpha \gamma}. $$
Are the equilibrium populations stable?}
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\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 39.1}

\textbf{Consider $dN/dt = \alpha N^2 - \beta N$ (with $\alpha > 0$, $\beta >
  0$).}

\textbf{(a) How does the growth rate depend on the population?}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(b) Sketch the solution in the phase plane.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(c) Obtain the exact solution.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(d) Show how both parts (b) and (c) illustrate the following behavior:\\
${}$\qquad\quad(i) If $N_0 > \beta/\alpha$, then $N \to \infty$. (At what time does $N \to
\infty$?)}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{${}$\qquad\quad(ii) If $N_0 < \beta/\alpha$, then $N \to 0$.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{${}$\qquad\quad(iii) What happens if $N_0 = \beta/\alpha$?}
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\textbf{\textcolor{red}{Your solution goes here}}

\section*{Problem 39.2}

\textbf{The general solution of the logistic equation, equation 39.1, has been
  shown to be equation 39.3.  Show that this exact solution has the following
  properties:}
\textbf{(1) It is defined for all $t \ge 0$.}
\vspace{1cm}

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

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\textbf{(2) As $t \to \infty$, $N \to a/b$ for all initial conditions except
  $N_0 = 0$. (Note: It takes an infinite amount of time to reach the equilibrium
  population.)}
\vspace{1cm}

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

\section*{Problem 39.4}

\textbf{Consider the following growth model:}
$$ \frac{dN}{dt} = aN + bN^2 \qquad \mbox{with $a>0$, $b>0$}. $$

\textbf{(a) How does the growth rate depend on $N$?}
\vspace{1cm}

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

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\textbf{(b) What would you expect happens to the population?}
\vspace{1cm}

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

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\textbf{(c) By using the phase plane, show that the population tends towards infinity.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(d) By considering the exact solution, show that the population reaches
  infinity in finite time; what might be called a population explosion.}
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\textbf{\textcolor{red}{Your solution goes here}}

\section*{Additional Problem}

\textbf{Consider the population model $dN/dt = -(a - b N)^2$, where $a>0$,
  $b>0$, and $N > 0$.  (At $N=0$, the population is extinct, and $dN/dt=0$.)}

\textbf{(a) Show that this model is just the logistic equation with a constant
  migration term added to it.  Are individuals migrating into the population or
  away from it?}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(b) Sketch the solution in the phase plane.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(c) Identify the equilibrium.  Is it stable?  Justify your answer.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(d) Find the exact solution in terms of $a$, $b$, and the initial value $N_0$.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(e) Show that if $N_0$ is greater than the equilibrium value, $N$
  approaches equilibrium but neither reaches it nor crosses it.}
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\textbf{\textcolor{red}{Your solution goes here}}

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\textbf{(f) Show that if $N_0$ is less than the equilibrium value, $N$
  becomes extinct in finite time.}
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\textbf{\textcolor{red}{Your solution goes here}}

\end{document}

