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\begin{document}

\title{Math 142-2, Homework 4}
\author{Your name here}
\date{\today}
\maketitle

\section*{Problem 50.3}

\textbf{Reconsider exercise 50.2. In the phase plane, sketch the isocline(s)
  along which $dF/dS = 0$. Also sketch the isocline(s) along which $dF/dS =
  \infty$. Indicate arrows as in Fig. 50-3. Explain on your diagram where all
  possible equilibrium populations are. (If needed, sketch all possible cases.)}
\vspace{1cm}

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

\section*{Problem 50.4}

\textbf{Reconsider the predator-prey model of exercise 50.2 ($b \ne
  0$). Determine the constant coefficient linear system of differential
  equations which governs the small displacements from the equilibrium
  populations,
$$ F = \frac{k}{\lambda}, S = \frac{a}{c} - \frac{b k}{c \lambda} > 0. $$
Do this in two ways:}

\textbf{1. Using the Taylor series for a function of two variables (equation
44.5).}
\vspace{1cm}

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

\vspace{1cm}
\textbf{2. Using perturbation methods as described in Sec. 50.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{Show that the two are equivalent. Do not attempt to solve the resulting
system. [Hint: See exercise 50.5 for the answer.]}
\vspace{1cm}

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

\section*{Problem 50.5}

\textbf{Show that your answer to exercise 50.4 can be put in the form:
$$ \frac{dF_1}{dt} = \frac{k}{\lambda} (-b F_1 - c S_1) \qquad\qquad
  \frac{dS_1}{dt} = \gp{\frac{a}{c} - \frac{b k}{c \lambda}} \lambda F_1, $$
where $F_1$, and $S_1$, are the displacements from the equilibrium populations of
the fish and sharks respectively. Eliminate $F_1$ (from the second equation)
to derive a second-order constant coefficient differential equation for $S_1$.
Analyze that equation and determine the conditions under which the
equilibrium population is stable.}
\vspace{1cm}

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

\section*{Problem 50.7}

\textbf{Refer to exercise 50.4. Suppose that the equilibrium number of prey
  without predators $a/b$ is smaller than the prey necessary to sustain the
  predators $k/\lambda$. Sketch the solution in the phase plane. [Hint: Use the
    behavior of the phase plane in the neighborhood of all equilibrium
    populations.]}
\vspace{1cm}

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

\section*{Problem 50.12}
\textbf{Consider the following two-species population growth model:
$$ \frac{dF}{dt} = F(a - bF - cS) \qquad\qquad \frac{dS}{dt} = S(-k + \lambda F
  - \sigma S). $$
How does this model differ from equation 50.1. Without explicitly determining
the equilibrium population, assume that one exists (with both species nonzero)
and analyze its linear stability.}
\vspace{1cm}

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

\section*{Problem 50.13}
\textbf{Reconsider exercise 50.12.}

\textbf{(a) Sketch the phase plane in the neighborhood of the equilibrium
  population in which both populations are nonzero.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(b) Sketch the phase plane in the neighborhood of $S = 0$, $F = a/b$. (What
  is the ecological significance of this population?)}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(c) Sketch the phase plane in the neighborhood of $S = 0, F = 0$.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(d) Use the information gained from parts (a)-(c) to sketch the entire
  phase plane. Describe the predator-prey interaction. Sketch typical time
  dependence of predators and prey.}
\vspace{1cm}

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

\section*{Problem 54.4}
\textbf{Consider the competing species model, equation 54.1. Suppose $b = d = 0$.}

\textbf{(a) What are all possible equilibrium solutions? Show that $x = c/\sigma$,
$y = a/k$ is an equilibrium population.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(b) Show that the system of differential equations describing populations
near the equilibrium ($x = c/\sigma$, $y = a/k$) is
$$ \frac{dx_1}{dt} = - \alpha y_1 \qquad \frac{dy_1}{dt} = -\beta x_1, $$
where $x_1$ and $y_1$ are the displacements from equilibrium. What are
$\alpha$ and $\beta$? Show that $\alpha>0$, $\beta>0$.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(c) By eliminating either $x_1$ or $y_1$, determine whether this equilibrium
solution is stable or unstable.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(d) Sketch the phase plane solution of equation 54.1 with $b = d = 0$.
[Hint: Are there any approximately straight line solutions in the
neighborhood of the equilibrium solution $x = c/\sigma$, $y = a/k$?]}
\vspace{1cm}

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

\section*{Problem 54.5}
\textbf{Formulate one system of differential equations describing all of the
  following interactions in some region between species $x$ and species $y$:}

\textbf{(a) The nutrients for both species are limited.}

\textbf{(b) Both species compete with each other for the same nutrients.}

\textbf{(c) There is a migration (from somewhere else) of species $x$ into the region
of interest at the rate $W$ per unit time.}
\vspace{1cm}

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

\section*{Problem 54.11}
\textbf{Consider
$$ \frac{dF}{dt} = F (2 - 2 F + G) \qquad \frac{dG}{dt} = G (1 - G + F) $$
}

\textbf{(a) Give a brief explanation of each species' ecological
  behavior. (Account for each term on the right-hand side of the above
  differential equations.)}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(b) Determine all possible equilibrium populations.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(c) In the phase plane, draw the isoclines corresponding to the slope of
  the solution being 0 and $\infty$.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(d) Introduce arrows indicating the direction of trajectories (time
  changes) of this ecosystem. The qualitative time changes should be indicated
  everywhere ($F> 0$, $G > 0$) in the phase plane.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(e) From the phase plane in part (d), briefly explain which (if any) of
  the equilibrium populations is stable and which unstable. Do not do a
  linearized stability analysis.}
\vspace{1cm}

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

\section*{Problem 54.12}
\textbf{Two species, $A$ and $B$, are in competition and are the prey of a third
  species $C$. What differential equations describe this ecological system?}
\vspace{1cm}

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

\section*{Additional Problem}
\textbf{Consider that some quantities $x(t)$ and $y(t)$ (valid for all reals) obey the differential equations
$$ \frac{dx}{dt} = \cos(y) \qquad \frac{dy}{dt} = \sin(x). $$}
\textbf{(a) Identify all of the equilibrium configurations.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(b) Since $x$ and $y$ will be periodic with period $2 \pi$, we can
  concentrate on the region with $0 \le x < 2 \pi$ and $0 \le y < 2 \pi$.  There
  are four equilibrium locations in this region.  Compute the linearization of
  the differential equation at each equilibrium position.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(c) For each of the four equilibria from (b), identify the equilibrium's
  stability (unstable, stable, or weakly/neutrally stable) and shape (node, saddle point, spiral/loop).}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(d) Use the fact that $\frac{dy}{dx}$ is separable to compute an energy
  for the system.  Your solution $E(x,y)$ should take the form of a simple
  polynomial in terms of $\cos(x)$, $\sin(x)$, $\cos(y)$, and $\sin(y)$.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(e) Show that, when evaluated at equilibrium values, $E(x_e,y_e)$ takes
  on only three distinct values: $c_0 < c_1 < c_2$.  Show that $c_0$ and $c_2$
  are minima and maxima of $E$ and that these are the only minima and maxima of
  $E$.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(f) Show that $E(x,y) = c_1$ along straight lines and identify these
  lines.  Trajectories corresponding to $c_0 < c < c_2$ with $c \ne c_1$ are
  bounded by the $c_1$ lines and contain no equilibria and so must represent
  closed curves.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(g) Use the information gathered above to sketch the phase plane for the
  system under consideration.  Include the $c_1$ lines and sample trajectories
  in your diagram.}
\vspace{1cm}

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

\vspace{1cm}
\textbf{(h) What do you suppose will happen to a system that starts on one of
  the $c_1$ lines?  (Describe the shape of its trajectory and what happens in
  the limits $t \to \infty$ and $t \to -\infty$.)}
\vspace{1cm}

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

\end{document}

