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I can't seem to figure this one out

Two guerilla forces, with troop strength $x(t)$ and $y(t)$, are in combat with each other without reinforcement. Suppose the territory is rather large and full of places to hide. The $y$-force needs to find the $x$-force first before it can inflict combat losses, and the higher the $x$, the easier it is for them to be found. Therefore, the combat loss rate for the $x$-force should be proportional $x$ . $y$ (this is unlike conventional warfare, where the full force of $x$ is open to be shot at by $y$, and so the combat loss rate for $x$ shouldn't depend on the total number of the $x$-force). Thus $$\frac{dx}{dt} = -axy$$ and $$\frac{dy}{dt} = -bxy$$

where $a$ is the combat effectiveness of the $y$-force and $b$ is that of the $x$-force. Suppose initially that $x_0$ and $y_0$ are the troop strengths for the $x$- and $y$-forces, and that $x_0$ is three times as numerous as $y_0$. How much more effective must the $y$-force be to stalemate it's enemy?

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    This looks like competing species model. Have you tried to find the equilibrium points? As far as I can tell, you need to find $a$ so that one "population" decays to zero as $t \to \infty$.2012-11-13

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