  In this article, we study the global dynamics of a discrete two dimensional
competition model. We give sufficient conditions on the persistence of one
species and the existence of local asymptotically stable interior period-2
orbit for this system. Moreover, we show that for a certain parameter range,
there exists a compact interior attractor that attracts all interior points
except a Lebesgue measure zero set. This result gives a weaker form of
coexistence which is referred to as relative permanence. This new concept of
coexistence combined with numerical simulations strongly suggests that the
basin of attraction of the locally asymptotically stable interior period-2
orbit is an infinite union of connected components. This idea may apply to many
other ecological models. Finally, we discuss the generic dynamical structure
that gives relative permanence.
