Biological Aggregations from Spatial Memory and Nonlocal Advection

TL;DR

This paper models biological aggregations using a nonlocal reaction-diffusion-advection system incorporating spatial memory and detection, providing existence, uniqueness, spectral analysis, bifurcation classification, and numerical solutions.

Contribution

It introduces a coupled PDE-ODE model with discontinuous kernels, proves well-posedness, performs spectral and bifurcation analysis, and applies numerical methods to study steady states.

Findings

Existence and uniqueness of solutions in one dimension.

Negative essential spectrum indicating stability.

Classification of bifurcations near critical points.

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.