Enlightening Mathematical Models What is your favourite Mathematical Model? What features make it intuitive or elegant? 

This question is largely inspired by an example and a desire to find other's like it.

Suppose we have two military forces of sizes $x(t)$ and $y(t)$ respectively that are about to enter into a conflict with each other at time $t=0$. Then the aptitude for an army to kill should be proportional to its size. But if army $x$ is killing then army $y$ is dying so we have $$x'(t)=-ay$$ $$y'(t)=-bx$$ where $a$ and $b$ are two positive constants. 

Consider the quantity $ay^2-bx^2$. By differentiating we see it is constant. So the trajectories of the system of equations form hyperbolas in the $x,y$ plane. So the strength of an army is directly proportional the square of its size. I love this example because it is effectively a mathematical proof of divide and conquer! (To see this, consider an army of 100 men. The strength of 1 100 man unit will be 10,000 but the strength 2 50 men units will be 5,000.) This is one of Lanchester's Laws

Has anyone got an example to match this?
If you like dynamical systems, there's a nice simple model for how romantic encounters evolve over time given the dating style of both people. There is a simplified version here. The short rules of thumb are, if both people like to give and receive attention, they fall madly in love :) . It also predicts cycles of love and hate for couples that have one person shying away from too much attention. 

If you're interested in this further, John Gottman has developed a broader application of these kinds of models to the "mathematics of marriage."
I would argue that your "consider the quantity" comment in you OP is a consequence of a variables separable situation (which you show that this quantity is indeed conserved further down your thread). Thus (assume w.l.o.g that $a, b \neq 0$) thus,

\begin{eqnarray} \notag
\frac{y^{'}}{x^{'}} &amp;=&amp; \frac{dy}{dx} \\
                    &amp;=&amp; \frac{-bx}{-ay} \\
                    &amp;=&amp; \frac{b}{a}.\frac{x}{y} \\ 
\end{eqnarray}
Now, this is a first oder differential equation, which can be solved by separating out the vairables. Thus,
\begin{eqnarray} \notag
\frac{dy}{dx}  &amp;=&amp; \frac{b}{a}\frac{x}{y}     \\
\end{eqnarray}
\begin{eqnarray} \notag
\Rightarrow  ydy                &amp;=&amp; \left( \frac{b}{a} \right) xdx \\
\text{(Integrating)} \int ydy   &amp;=&amp; \left(\frac{b}{a}\right)\int xdx \\
\Rightarrow   \frac{1}{2}y^{2}  &amp;=&amp; \frac{1}{2} \left(\frac{b}{a} \right) x^{2} + C \\
 \end{eqnarray}
Assuming (w.l.o.g) that $C=0$ we have (upon re-arranging)
\begin{equation}
ay^{2} - bx^{2} = 0
\end{equation}
Which brings about your conserved quantity above.