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 "In this project we will use ",
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equations are currently used to model a wide range of phenomena from biology, \
physics, chemistry, computer science, economic analysis, etc. The theory of \
differential equations has become an essential tool in all areas of science, \
particularly since computers became commonly available."
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The fish and game department in a certain state is planning to issue hunting \
permits to control the deer population (one deer per permit). It is known \
that if the deer population falls below a certain level m, the deer will \
become extinct. It is also known that if the deer population rises above the \
carrying capacity M, the population will decrease back to M through disease \
and malnutrition. Consider the following model for the growth rate of the \
deer population P as a function of time t: \
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where P is the deer population and r=0.00003 is a constant of \
proportionality. The values of the other parameters are M=100 and m=55.\
\>", "Text",
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Cell["\<\
Plot the vector field of the differential equation using VectorPlot[..]. Then \
plot the vector field using StreamPlot[..]; identify all the constant \
solutions and color them in red using StreamPlot[..]. Try a window size of \
100x130 (in the txP plane). What happens to the solutions as time t \
increases, t\[RightArrow]\[Infinity]? Color a couple of trajectories to \
illustrate the possible behaviors, then explain.\
\>", "ItemNumbered",
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 "Solve the system of differential equations for the initial condition \
P(20)=110. If DSolve[...] does not work (and most likely it won\
\[CloseCurlyQuote]t), then a numeric approximation may be the next best thing \
to hope for, so use ",
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Cell["\<\
 If the initial deer population is 140, about how many hunting permits should \
be issued so that the deer population does not become extinct?\
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 "Write a small interactive model using Manipulate[..] and Locator[..], that \
initially plots the StreamPlot[..] from part ",
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point where the user clicked on.  "
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Cell["\<\
Solve the differential equation given in the problem. It is a separable \
differential equation, so you can separate the variables and integrate to \
obtain an implicit equation involving P.  When integrating, you may need to \
use partial fraction decomposition (the Mathematica function Apart[..] might \
come in handy). Plot a couple of relevant level curves of the implicit \
equation you have obtained, and compare them to the trajectories of the \
PhasePortrait from part 1. \
\>", "ItemNumbered",
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Cell[TextData[{
 "In this problem, we will visualize a famous phenomenon in dynamics of \
nonlinear differential equations, called the Poincar\[EAcute]-Andronov-Hopf \
bifurcation, or in short a Hopf bifurcation. It exhibits the birth of a limit \
cycle through a change in the stability of the equilibrium point. This type \
of bifurcation is widely present in physical systems. ",
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\[OpenCurlyQuote]spontaneous\[CloseCurlyQuote] oscillations such as airfoil \
flutter and other wind-induced oscillations (e.g., those that caused the \
Tacoma-Narrows bridge collapse) in structural engineering systems, vortex \
shedding in fluid flow around a solid body at sufficiently high stream \
velocity, LCR oscillations in electrical circuits, relaxation oscillations \
(e.g., as described by the Van der Pol oscillator), the periodic firing of \
neurons in nervous systems (e.g., in the FitzHugh-Nagumo equation modelling \
these phenomena), oscillations in autocatalytic chemical reactions (e.g., the \
Belousov-Zhabotinsky reaction) as described by the Brusselator and similar \
models, oscillations in fish populations (as described by predator-prey \
models), periodic fluctuations in the number of individuals suffering from an \
infectious disease (as described by epidemic models)\[CloseCurlyDoubleQuote] \
Gert van der Heijden",
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in economics (for example, ",
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bifurcations (for example, ",
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pattern that occurs across the equatorial Pacific Ocean roughly every three \
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air pressure difference between the central and western Pacific Ocean",
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differential equations which exhibits a Hopf bifurcation. \n\n",
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give you many starting points for your investigation. Once you find such an \
example, describe its mathematical model (the system of differential \
equations) and explain for what values of the parameters the system undergoes \
a Hopf bifurcation. No mathematical proof is necessary or required, but \
StreamPlot pictures are encouraged. Explain the relevance of all quantities \
modeled by the system of differential equations (for example, x is the debt \
at time t, y is the investment at time t, etc). Then describe the relevance \
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