Chapter 6.4, Problem 11E

Interpretation Introduction

Interpretation:

To show that $\text{x + y + z = 1}$ is invariant. To analyze the long-term behavior for both positive and negative values of r. Also, interpret the results in political terms.

Concept Introduction:

Vasquez and Redner mention a highly simplified model of political opinion dynamics. It consists of a population of leftists, rightists, and centrists. The leftist and rightists never talk to each other but they do talk to centrists. When an extremist of either type talks with a centrist, one of them convinces the other to change his mind, with the winner depending on the sign of the parameter r. If $\text{r>0}$, the extremist always wins and persuades the centrist to move to that end of the spectrum. If $\text{r<0}$, the centrist always wins and pulls the extremist to the middle.

The model’s governing equations are $\dot{\text{x}}\text{ = rxz}$, $\dot{\text{y}}\text{ = ryz}$, $\dot{\text{z}}\text{ = - rxz - ryz}$ where x, y, z are the relative fractions of rightists, leftists, and centrists in the population.