Title: Biological Aggregations from Spatial Memory and Nonlocal Advection Show Chinese title

Title: 从空间记忆和非局部对流产生的生物聚集

Abstract: We investigate a nonlocal single-species reaction-diffusion-advection model that integrates the spatial memory of previously visited locations and nonlocal detection in space, resulting in a coupled PDE-ODE system reflective of several existing models found in spatial ecology. We prove the existence and uniqueness of a H\"older continuous weak solution in one spatial dimension under some general conditions, allowing for discontinuous kernels such as the top-hat detection kernel. A robust spectral and bifurcation analysis is also performed, providing the rigorous analytical study not yet found in the existing literature. In particular, the essential spectrum is shown to be entirely negative, and we classify the nature of the bifurcation near the critical values obtained via a linear stability analysis. A pseudo-spectral method is used to solve and plot the steady states near and far away from these critical values, complementing the analytical insights. Show Chinese abstract

Abstract: 我们研究了一个非局部的单物种反应-扩散-对流模型，该模型结合了对先前访问过的位置的空间记忆和空间中的非局部检测，导致了一个反映空间生态学中几种现有模型的耦合PDE-ODE系统。 我们在一些一般条件下证明了一维空间中Hölder连续弱解的存在性和唯一性，允许使用不连续的核，例如顶帽检测核。 还进行了稳健的谱分析和分岔分析，提供了现有文献中尚未找到的严格分析研究。 特别是，显示本质谱完全是负的，并通过线性稳定性分析获得的关键值附近的分岔性质进行了分类。 使用伪谱方法求解并绘制这些关键值附近和远离的稳态，补充了分析见解。