论文标题
扩散的Lotka-Volterra系统的整个解决方案
Entire Solutions of Diffusive Lotka-Volterra System
论文作者
论文摘要
这项工作涉及存在扩散性Lotka-volterra竞争系统的整个解决方案\ BEGIN {equation} \ label {eq：acmpact} \ begin {cases} u_ {t} = u_ {xx} = u_ {xx} + u（1- u-av），＆\ qquad v_ {xx}+ rv（1-v-bu），＆\ qquad \ x \ in \ mathbb {r} \ end {cases} \ quad（case} \ quad（1）\ end {equation {qore}，其中$ d，r，a $，a $和$ b $是$ a \ a \ a \ neq 1 $和$ b \ b $ $ b \ b \ neq $ b \ neq $ b \ neq $ b \ neq 1 $ by。 We prove the existence of some entire solutions $(u(t,x),v(t,x))$ of $(1)$ corresponding to $(Φ_{c}(ξ),0)$ at $t=-\infty$ (where $ξ=x-ct$ and $Φ_c$ is a traveling wave solution of the scalar Fisher-KPP defined by the first equation of $(1)$ when $a=0$).此外，我们还将这些解决方案的渐近行为描述为$ t \ to+\ infty $。我们证明了针对弱势竞争和强大竞争案件的新解决方案的存在。在弱势竞争案例中，我们证明了形成4维流形的整个解决方案的存在。
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 $(Φ_{c}(ξ),0)$ at $t=-\infty$ (where $ξ=x-ct$ and $Φ_c$ 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.
