The concern of the paper is nonconstant positive solutions of a class of Lotka-Volterra competition systems over 1D domains. We prove the existence of a positive monotonous solution to the shadow system for each small diffusion rate epsilon >0. Our theoretical results provide a foundation for further theoretical analysis on the shadow system and give insights on how diffusion and advection rates affect the pattern formation in the advective Lotka-Volterra competition systems. The second part of this paper includes numerical simulations of the nontrivial patterns to the shadow system and its original model. It is demonstrated that nontrivial patterns can develop from small perturbations of the homogeneous solution. Our numerics suggest that this system admits very interesting and complicated spatial-temporal dynamics even over 1D domains.