I report here on a series of joint works with Alessandro Zilio (Université de Paris) about systems of competing predators interacting with a single prey. We focus on the analysis of stationary states, stability issues, and the asymptotic behavior when the competition parameter becomes unbounded. Existence of solutions is obtained by a bifurcation theory type approach and the segregation analysis rests on a priori estimates and a free boundary problem. We discuss the classification of solutions by using spectral properties of the limiting system. Our results shed light on the conditions under which predators segregate into packs, on whether there is an advantage to have such hostile packs, and on comparing the various territory configurations that arise in this context. These questions lead us to nonstandard optimization problems.