Biological aggregations from spatial memory and nonlocal advection

Abstract

We investigate a nonlocal reaction–diffusion–advection model of a population of organisms that integrates spatial memory of previously visited locations and nonlocal detection in space, resulting in a coupled PDE–ODE system reflective of several models found in spatial ecology. Our study advances the mathematical understanding of such models by proving the existence and uniqueness of a global weak solution in one spatial dimension using an iterative approach. This result includes potentially discontinuous detection kernels, explicitly emphasizing the so-called ‘top-hat’ detection function, and does not place any restriction on the rate of advection, thereby addressing some analytical voids in the mathematical discourse on such models. A comprehensive spectral and stability analysis is also performed, providing analytical expressions for bifurcation values contingent on various model parameters, such as species advection rate, diffusion rate, memory uptake and decay rates. Unlike classical reaction–diffusion systems, the point spectrum may now include elements that have an infinite-dimensional kernel. We show the existence of such a point and that it remains negative, ensuring that it does not influence the stability of the constant steady state. Linear stability analysis then provides critical values for destabilizing the constant steady state. We explicitly describe the form of the non-constant steady state near these critical values and classify the nature of the pitchfork bifurcation as forward/backward and stable/unstable. To complement our analytical insights, we explore a targeted case study of three particular instances with the top-hat detection function. Using a pseudo-spectral method, we depict a numerical bifurcation diagram showing cases with sub or supercritical behaviour.