Some entire solutions to the competitive reaction diffusion system

This manuscript focuses on the study of the classical Lotka–Volterra competitive reaction diffusion system. Morita and Tachibana (2009) [19] proved the existence of entire solutions of this system. They used traveling front solutions, the lower bound of the proportion between two components of which is positive, to construct these entire solutions. In order to reasonably explain this positive lower bound, we investigate it based on the eigenvalues of the linearized system and show some sufficient and necessary conditions. Particularly, if the parameters in the system satisfy some conditions, then the positive lower bound spontaneously holds. When the ratio is zero, we also construct new entire solutions based on the super(sub)-solution method, the comparison theorem and a pair of exact traveling front solutions. In addition, another kind of entire solutions is revealed by the combination of traveling front solutions, their reflects and the solutions of the diffusion-free system.