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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.

enter image description here

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.

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    $\begingroup$ The extraneous problem with this nice question is that when the lions get hungry they may eat the sheep themselves. $\endgroup$ Apr 6 at 13:54
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    $\begingroup$ For lion = N-2, can they just go after the closest wolf first to reduce the problem to N-1? $\endgroup$
    – Paichu
    Apr 6 at 16:42
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    $\begingroup$ On p.$135$ of his A Mathematician's Miscellany, J.E.Littlewood gives a plausible, but erroneous, argument purporting to show that a single lion can always catch a single wolf (which he calls a "man"). After noting that the argument is wrong, he gives A.S.Besicovitch's strategy for the wolf that enables it to avoid capture indefinitely $\endgroup$ Apr 8 at 14:57
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    $\begingroup$ Four clarifications seem necessary to answer: (i) Is the "top speed" of each entity the same? (This may be a convention for pursuit problems, but it's advisable to be explicit.) (ii) Does each entity move differentiably, or just continuously? (iii) Can the sheep and lion be at the same point, as described verbally? (If so, my first guess would be that they should stay together, possibly modulo the fourth clarification.) (iv) What happens if the sheep, one or more wolves, and the lion meet at a point? (Hopefully not a universe-creating explosion.... :) $\endgroup$ Apr 12 at 13:45
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    $\begingroup$ @AndrewD.Hwang Reaction is instantaneous, and the situation is symmetric: there's no "just before" to which the sheep can react but the wolves can't. $\endgroup$
    – Eric
    Apr 12 at 15:46

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