Question

Question 7* (Similar to Strogatz 6.4.11) Vazquez and Redner (2004, p8489) mention a highly simplified model of political opinion dynamics consisting of a population of leftists, rightists, and centrists. The leftists and rightists never talk to each other; they are too far apart politically to even begin a dialogue. But they do talk to the centrists—this is how opinion change occurs. Whenever an extremist of either type talks with a centrist, one of them convinces the other to change his or her mind, with the winner depending on the sign of the parameter r. If r > 0, the extremist always wins and persuades the centrist to move to that end of the spectrum. If r < 0, the centrist always wins and pulls the extremist to the middle. The model's governing equations are x' = rxz, y' = ryz, z' = -rxz - ryz, where x, y, and z are the relative fractions of rightists, leftists, and centrists, respectively, in the population. (a) Show that the set x + y + z = 1 is invariant. (b) Analyze the long-term behavior predicted by the model for both positive and negative values of r. (c) Interpret the results in political terms.

          Question 7* (Similar to Strogatz 6.4.11)
Vazquez and Redner (2004, p8489) mention a highly simplified model of political opinion dynamics
consisting of a population of leftists, rightists, and centrists. The leftists and rightists never talk
to each other; they are too far apart politically to even begin a dialogue. But they do talk to the
centrists—this is how opinion change occurs. Whenever an extremist of either type talks with a
centrist, one of them convinces the other to change his or her mind, with the winner depending
on the sign of the parameter r. If r > 0, the extremist always wins and persuades the centrist to
move to that end of the spectrum. If r < 0, the centrist always wins and pulls the extremist to the
middle. The model's governing equations are
x' = rxz,
y' = ryz,
z' = -rxz - ryz,
where x, y, and z are the relative fractions of rightists, leftists, and centrists, respectively, in the
population.
(a) Show that the set x + y + z = 1 is invariant.
(b) Analyze the long-term behavior predicted by the model for both positive and negative values
of r.
(c) Interpret the results in political terms.

$Question 7* (Similar to Strogatz 6.4.11) Vazquez and Redner (2004, p8489) mention a highly simplified model of political opinion dynamics consisting of a population of leftists, rightists, and centrists. The leftists and rightists never talk to each other; they are too far apart politically to even begin a dialogue. But they do talk to the centrists—this is how opinion change occurs. Whenever an extremist of either type talks with a centrist, one of them convinces the other to change his or her mind, with the winner depending on the sign of the parameter r. If r > 0, the extremist always wins and persuades the centrist to move to that end of the spectrum. If r < 0, the centrist always wins and pulls the extremist to the middle. The model's governing equations are x' = rxz, y' = ryz, z' = -rxz - ryz, where x, y, and z are the relative fractions of rightists, leftists, and centrists, respectively, in the population. (a) Show that the set x + y + z = 1 is invariant. (b) Analyze the long-term behavior predicted by the model for both positive and negative values of r. (c) Interpret the results in political terms.$

Added by Carolyn M.

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