Comparative dynamics and normal form of nonlocal chemotaxis models with spatial average and top-hat kernels

This study conducts a comparative analysis of two nonlocal chemotaxis models: one incorporating a spatial average kernel and the other a top-hat kernel with a perception radius M. We investigate how taxis and kernel selection impact model dynamics, employing linear stability and bifurcation theory to characterize the conditions for pattern formation. Through numerical simulations, we illustrate how the chemotactic coefficient χ modulates the critical perception radius MT and the spatial wave number. The key contributions of this work are the proof of local existence and uniqueness for a classical solution to model (1.1), as well as the systematic derivation of an explicit normal form for the Turing bifurcation under periodic boundary conditions with a spatial average kernel. This normal form provides analytical insight into the stability and direction of the resulting inhomogeneous patterns. Our results demonstrate that the top-hat kernel supports richer spatiotemporal dynamics than the spatially averaged kernel, and that stronger chemotaxis tends to suppress pattern formation unless compensated by a larger sensing range.