| Abstract: |
| In this talk, I will present recent analytical and numerical results on bifurcation structure and pattern formation in nonlocal aggregation-diffusion systems on the torus. The first part, based on joint work with J. A. Carrillo, focuses on a two-species aggregation-diffusion model, in which we study long-time behaviour, stability, and local bifurcations from spatially homogeneous states. The second part concerns joint work with J. A. Carrillo and A. L. Villares on numerical stationary states for nonlocal Fokker-Planck equations. There, a semi-analytical fixed-point framework allows one to compute nontrivial stationary solutions directly and efficiently explore their bifurcation diagrams. Finally, I will discuss recent progress on an aggregation-diffusion model with nonlinear sensing, in collaboration with S. Raible, in which the local theory is extended to settings with stronger nonlinear effects and, in some regimes, higher-dimensional kernels of the linearised operator. Together, these works highlight the complementary roles of analytical bifurcation theory and semi-analytical computation in revealing rich families of stationary patterns in dissipative nonlocal systems. |
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