Pattern Formation Induced by Diffusion, Taxis, and Delay on Sector Domains

Diffusion processes in ecology, biology, chemistry, and physics have been extensively studied on one-dimensional line segments, as well as on regular two-dimensional domains such as squares and disks. In contrast to such typical geometric regions, we investigate diffusive behavior on sector domains to model environments with both straight and curved boundaries, where taxis behavior is also considered. We first analyze the eigenvalue problem for the Laplace operator on sector domains, elucidating the relationship between the sector angle and nonzero eigenvalues. Building on this foundation, we perform linearization and phase space decomposition, and apply center manifold reduction to derive normal forms, thereby rigorously determining the direction and stability of the Hopf bifurcation. Subsequently, we introduce two ecological models with time delay to investigate the rich dynamical behaviors incorporating nonlocal effects and taxis, respectively, including spatially homogeneous and inhomogeneous periodic solutions, as well as spatially inhomogeneous steady states. Comparative studies on limiting cases are also presented to highlight spectral discontinuities. Finally, we present numerical simulations to demonstrate the effects of sector angles and boundary conditions on pattern formation.