Some entire solutions to the competitive reaction diffusion system

Abstract

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.

Entire solutions for a delayed lattice competitive system

“…Combining with Lemmas 8-10, similar to [5,[9][10][11][12][13][14][15], we can obtain the following main result of this paper.…”

“…in [12]. Then Wang and Li in [14] investigated this technical condition and gave some sufficient conditions to ensure this technical condition. In addition, they used the exact solutions, which do not satisfy the technical condition, to construct an entire solution for (1.3).…”

Entire solutions for the classical competitive Lotka–Volterra system with diffusion in the weak competition case

“…As a result, we turn to find entire solutions similar to that in [8] and [9]. The way to construct super-sub solutions is similar to the one in [28], and different kinds of entire solutions have been obtained. One of them has the following asymptotic behavior.…”

“…In [23] and [28], the authors used some exact solutions of (1.1) to construct the super-sub solutions to obtain the existence of entire solution. Although, under the case (iv), we also can find the exact solutions of (1.1) connecting the origin and the positive equilibrium in the paper [11], we can not get any pairs of coupled super-sub solutions followed the method in [23] or [28]. Thus we here use the traveling front solutions and their reflects to construct the super-sub solutions to establish entire solutions.…”

“…for the existence of entire solutions in [23]. In [28], Wang and Li showed some sufficient and necessary conditions for this technical condition and partially proved the result still holds without this condition. In addition, except for the above mentioned papers, for the existence of entire solutions of Lotka-Volterra system one can see [18], [20] and [27] for more details.…”