  We investigate the location and structure of the Birkhoff center for
competitive dynamical systems, and give a comprehensive description of
recurrence and statistical behavior of orbits. An order-structure dichotomy is
established for any connected component of the Birkhoff center, that is, either
it is unordered, or it consists of strongly ordered equilibria. Moreover, there
is a canonically defined countable disjoint family $\mathcal{F}$ of invariant
$(n-1)$-cells such that each unordered connected component of the Birkhoff
center lies on one of these cells. We further show that any connected component
of the supports of invariant measures either consists of strongly ordered
equilibria, or lies on one element of $\mathcal{F}$. In particular, any
$3$-dimensional competitive flow has topological entropy $0$.
