SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 1-57, January 2020.
Motivated by its application in ecology, we consider an extended Klausmeier model, a singularly perturbed reaction-advection-diffusion equation with spatially varying coefficients. We rigorously establish existence of stationary pulse solutions by blending techniques from geometric singular perturbation theory with bounds derived from the theory of exponential dichotomies. Moreover, the spectral stability of these solutions is determined, using similar methods. It is found that, due to the breakdown of translation invariance, the presence of spatially varying terms can stabilize or destabilize a pulse solution. In particular, this leads to the discovery of a pitchfork bifurcation and existence of stationary multipulse solutions.

中文翻译：

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具有空间变化系数的扩展Klausmeier模型的脉冲解

SIAM应用动力系统杂志，第19卷，第1期，第1-57页，2020年1月。
由于其在生态学中的应用，我们考虑了扩展的Klausmeier模型，这是一个具有空间变化系数的奇摄动反应-对流-扩散方程。我们通过将来自几何奇异摄动理论的技术与从指数二分法理论导出的边界进行混合，来严格建立平稳脉冲解的存在性。此外，使用类似的方法确定了这些溶液的光谱稳定性。已经发现，由于平移不变性的破坏，空间变化项的存在可以使脉冲解稳定或不稳定。特别是，这导致了干草叉分叉的发现和固定多脉冲解的存在。