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“The three objectives of this lab are all related to the analysis of \
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“In this lab we consider a nonlinear model for the price, “,
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” (which is measured in some unit, say 100s or 1000s of objects). If we \
consider these to be interacting functions of time, we can come up with a set \
of equations that model them, \n\t”,
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” are non-negative constants (parameters of the model). These equations \
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decreases when the price is too high; and that the quantity produced \
increases when the price is large enough, but decreases when the quantity \
being produced is too large. If “,
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equilibrium price and quantity.\n\nFor the purposes of this lab, we will take \
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“and”,
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” to define functions representing the right-hand sides of the differential \
equations for these parameter values:”
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“The first step in our linear analysis is to find the equilibrium solutions \
to the system. Use “,
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“‘s “,
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” command (remember to use double equals “,
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use) to find the equilibrium solutions. “,
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” which of these are sensible in the economics context of the model? Which \
do you expect to be stable? Why?”
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“Then, to determine the linear behavior near these equilibrium solutions, we \
linearize the system near each by taking “,
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“, because that means something else in “,
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“Now that we have a linear system, we can analyze its stability. Find the \
eigenvalues and eigenvectors for the system using “,
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” may be useful here). Is the equilibrium solution stable? What type of \
point in the phase plane is it? What do the eigenvectors tell you about \
trajectories near the equilibrium solution? What if any information about \
the linearized system carries over to the nonlinear system it locally \
approximates?”
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To get a clear sense of how solutions of the nonlinear system behave, we \
should test the stability of all of the equilibrium solutions, and should \
also find the eigenvectors to be able to sketch a phase portrait at each. \
This would give a sense of the likely long-term behavior of solutions in the \
system. For brevity in this lab, however, we stop with only the eigenvalues \
we found above for one equilibrium solution; because we have an intuitive \
sense based on economic reasoning of what we expect to find, this gives some \
window on our guess for the behavior of the system.\
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“Note that to find equilibrium solutions, we look for those points where \
both “,
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” and “,
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” of the system. For this system, we will find “,
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“Find and plot the nullclines of the system in the (p,q) phase plane. Use \
different colors to distinguish the two different types of nullclines (recall \
“,
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“). You can add a vertical line to a plot by including “,
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” in the “,
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” command for suitable “,
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“,”,
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” and choice of “,
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“How are the equilibrium solutions you found in “,
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” related to the graph that you found in “,
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“?”
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