On a shadow system of the S-K-T competition system

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.