  We study a boundary value problem with an integral constraint that arises
from the modelings of species competition proposed by Lou and Ni in \cite{LN2}.
Through bifurcation theories, we obtain the existence of non-constant positive
solutions over one-dimensional domain, which are small perturbations from the
positive constant solution of the system. Moreover, we determine the stability
of the bifurcating solutions. Finally, for the diffusion rate being
sufficiently small, we construct infinitely many positive solutions with single
transition layer, which is represented as an approximation of a step function
that. The transition-layer solution can be used to model the phenomenon of
species segregation through competitions.
