Entire Solutions of Diffusive Lotka-Volterra System

This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system begin{equation}label{eq:abstract} begin{cases} u_{t}= u_{xx} + u(1-u-av), & qquad xinmathbb{R} cr v_{t}= d v_{xx}+ rv(1-v-bu), & qquad xinmathbb{R} end{cases} quad (1) end{equation} where $d,r,a$, and $b$ are positive constants with $a eq 1$ and $b eq 1$. We prove the existence of some entire solutions $(u(t,x),v(t,x))$ of $(1)$ corresponding to $(Phi_{c}(xi),0)$ at $t=-infty$ (where $xi=x-ct$ and $Phi_c$ is a traveling wave solution of the scalar Fisher-KPP defined by the first equation of $(1)$ when $a=0$). Moreover, we also describe the asymptotic behavior of these entire solutions as $tto+infty$. We prove existence of new entire solutions for both the weak and strong competition case. In the weak competition case, we prove the existence of a class of entire solutions that forms a 4-dimensional manifold.