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 \ x\in\mathbb{R} \cr v_{t}= d v_{xx}+ rv(1-v-bu), & \qquad \ x\in\mathbb{R} \end{cases} \quad (1) \end{equation} where $d,r,a$, and $b$ are positive constants with $a\neq 1$ and $b\neq 1$. 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 $t\to+\infty$. 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.