School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, China.
J Math Biol. 2024 Jul 15;89(3):28. doi: 10.1007/s00285-024-02125-5.
This paper aims to establish the existence of traveling wave solutions connecting different equilibria for a spatial eco-epidemiological predator-prey system in advective environments. After applying the traveling wave coordinates, these solutions correspond to heteroclinic orbits in phase space. We investigate the existence of the traveling wave solution connecting from a boundary equilibrium to a co-existence equilibrium by using a shooting method. Different from the techniques introduced by Huang, we directly prove the convergence of the solution to a co-existence equilibrium by constructing a special bounded set. Furthermore, the Lyapunov-type function we constructed does not need the condition of bounded below. Our approach provides a different way to study the existence of traveling wave solutions about the co-existence equilibrium. The existence of traveling wave solutions between co-existence equilibria are proved by utilizing the qualitative theory and the geometric singular perturbation theory. Some other open questions of interest are also discussed in the paper.
本文旨在为具有扩散环境的空间生态传染病模型建立连接不同平衡点的行波解的存在性。在行波坐标下，这些解对应于相空间中的异宿轨道。我们通过打靶法研究了从边界平衡点到共存平衡点的行波解的存在性。与 Huang 等人引入的技术不同，我们通过构造一个特殊的有界集直接证明了解的收敛性。此外，我们构造的 Lyapunov 型函数不需要有下界的条件。我们的方法为研究共存平衡点的行波解的存在性提供了一种不同的途径。通过利用定性理论和几何奇异摄动理论，证明了共存平衡点之间的行波解的存在性。本文还讨论了一些其他感兴趣的开放性问题。
