If any officer can move more than $\delta$, then that officer can simply chase down the fugitive.  Thus, I propose modifying the question to ask: If each officer can move at most distance $\delta$, but they all pull from a given pool of movement of size $c\delta$, then how small can $c$ be such that they can always catch the fugitive?

I claim that $c\leq 2\sqrt{2}$, and we only need four officers to reach this bound.  To see this, orient the playing field and place each officer at the four cardinal directions, with the partner at the opposite side.  Think of them as living on a rectangle.  Using their joint speed, they can always remain on that rectangle and be opposite their partner, with the fugitive directly between them and their partner.  If they ever have any extra movement, they can squeeze in the rectangle.

This shows that any constant $c>2\sqrt{2}$ works, but what about $2\sqrt{2}$ exactly?  In that case, the only time that there is no extra speed that can be used to squeeze in is when the fugitive goes perfectly diagonal.  In that case the officers can lag just a little behind the fugitive, and use that extra lag to squeeze in a bit.  The fugitive has to start backtracking when he gets too close to a corner (where the two close officers could just capture him directly), and the officers once again get some extra speed to use.

Now, the race is on to improve this constant!