In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.

中文翻译：

---

扩散捕食者-食饵系统Hopf分支引起的点对点行波和周期行波。

在本文中，我们考虑具有Allee效应的被捕食者的扩散Leslie-Gower捕食者-被捕食者系统。首先，考虑到两种物质的扩散，我们通过上下解方法获得了连接无捕食者的恒定稳态和共存稳态的行波解的存在性。然而，由于捕食者方程式的奇异性，我们需要为食饵密度构造一个正合适的较低解。这样的行波解决方案可以对时空模型进行建模，在此过程中，捕食者入侵猎物的领土，并最终使它们共存。其次，考虑到两种情况：两个物种的扩散和仅猎物的扩散，我们使用霍普夫分岔理论证明了小振幅周期行波列车解的存在。