Generally, in pursuit-evasion games, there's one prey and one or many pursuers. I'd like to know how extending the food chain would change the dynamics of such games.
Specifically, let's consider a closed, circular shape arena in $\mathbb{R}^2$. Three wolves are uniformly distributed at the border. The sheep and his lion friend are at the center.
If $d(w(t),s(t))=0$, the wolf eats the sheep, if $d(l(t),w(t))=0$, the lion eats the wolf, where w(t), s(t) and l(t) are the trajectories of the animals, and $d$ measures euclidean distance. The pack of wolves work as a group, their goal being to eat the sheep. The goal of the lion-sheep team is to prevent the sheep from being eaten, indefinitely. All animals move continuously in time at equal speed and are intelligent.
Can the lion protect the sheep from the wolves? In general, how many lions are necessary to protect the sheep from a pack of $N$ wolves uniformly distributed at the border?
This is a puzzle I originally asked here on puzzling.stackexchange. I know already that
- A single wolf doesn't catch the sheep.
- Two wolves will catch the sheep.
- $N-1$ lions are sufficient to fend off $N$ wolves.
See here for the proof of those claims. I'm interested in knowing whether $N-2$ or less lions can fend off $N$ wolves.