Let $x(t)$ and $y(t)$ be measures of happiness for the husband and wife, respectively. Negative values indicate unhappiness. Let $x_0$ and $y_0$ be the "natural disposition" of the husband and wife, respectively. This is how happy they would be if they were single. During marriage, the couple develops a style of interaction that is called "validating." A model of their marriage dynamics is:
$\frac{dx}{dt} = r_1(x_0 - x) + a_1y$ $\frac{dy}{dt} = r_2(y_0 - y) + a_2x$
where $a_1$ measures how easily the husband is influenced by the wife's emotions, and $a_2$ is the corresponding quantity for his wife:
$0 < \frac{a_1}{r_1} < 1$ $0 < \frac{a_2}{r_2} < 1$
a) Find out where the marriage is heading.
b) A marriage is termed "regulated" and is low risk if the long-term happiness of each spouse is enhanced by the marital interaction. Otherwise it is called "unregulated." Give reasons for the following:
Is the marriage "regulated" if each of the spouses is naturally happy $(x_0 > 0, y_0 > 0)$? What if $x_0 < 0, y_0 < 0$?