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?
 A: 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."
A: Maybe I'm just grouchy today, but the example in the question strikes me as an example of misuse of a mathematical model.  The model begins reasonably enough, analyzing a simlified version of the dynamics of one army fighting another, finding an invariant quantity, and describing the trajectories of this dynamical system.  Then, it jumps to a conclusion about a notion of "strength", which hasn't been defined. Finally, this notion is applied to an entirely different situation, where one army is split in two, and the strength is just assumed to be additive.  The conclusion, that a single unified army is better than two halves of it, is probably true in some situations and false in others (depending on what capabilities armies and their parts have, for example pincer movements), but I would definitely not call this argument a mathematical proof of it.
A: The mathematics of evolution is full of them, see for example Lotka-Volterra systems of equations, encapsulating succinctly the a priori surprising phenomenon of periodic phase trajectories.
A: 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^{'}} &=& \frac{dy}{dx} \\
                    &=& \frac{-bx}{-ay} \\
                    &=& \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}  &=& \frac{b}{a}\frac{x}{y}     \\
\end{eqnarray}
\begin{eqnarray} \notag
\Rightarrow  ydy                &=& \left( \frac{b}{a} \right) xdx \\
\text{(Integrating)} \int ydy   &=& \left(\frac{b}{a}\right)\int xdx \\
\Rightarrow   \frac{1}{2}y^{2}  &=& \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.
