  We study a mathematical model of environments populated by both preys and
predators, with the possibility for predators to actively compete for the
territory. For this model we study existence and uniqueness of solutions, and
their asymptotic properties in time, showing that the solutions have different
behavior depending on the choice of the parameters. We also construct
heterogeneous stationary solutions and study the limits of strong competition
and abundant resources. We then use these information to study some properties
such as the existence of solutions that maximize the total population of
predators. We prove that in some regimes the optimal solution for the size of
the total population contains two or more groups of competing predators.
