# Entire Solutions for the Classical Competitive Lotka-Volterra System with Diffusion in the Weak Competition Case

## TL;DR

This paper investigates entire solutions for the classical competitive Lotka-Volterra system with diffusion under weak competition, analyzing asymptotic behaviors and constructing solutions using super-sub solutions and traveling fronts.

## Contribution

It introduces two novel methods for constructing entire solutions for the system, including solutions based on traveling fronts and their reflections.

## Key findings

- Characterized asymptotic behavior of traveling front solutions.
- Constructed two types of entire solutions using super-sub solutions.
- Demonstrated solutions' convergence to positive equilibrium over time.

## Abstract

In this paper we are concerned with the entire solutions for the classical competitive Lotka-Volterra system with diffusion in the weak competition. For this purpose we firstly analyze the asymptotic behavior of traveling front solutions for this system connecting the origin and the positive equilibrium. Then, by using two different ways to construct pairs of coupled super-sub solutions of this system, we obtain two different kinds of entire solutions. The construction of the first kind of entire solutions is based on these fronts, and their reflects as well as the solutions of the system without diffusion. One component of the solution starts from 0 at $t\approx-\infty$, and as $t$ goes to $+\infty$, the two component of the entire solution will eventually stay in a conformed region. Another kind of entire solutions is related to some traveling front solutions of scalar equations