Nonlinear dynamics and spatiotemporal patterns of a delayed reaction-diffusion system on a square domain

In this work, we analyze the pattern formation in a delayed reaction-diffusion system on a square domain. Using bifurcation theory and amplitude equations, we reveal symmetry-breaking dynamics near criticality, including rotating waves and quasiperiodic solutions. We apply this framework to both a Ginzburg-Landau equation and a delayed predator-prey model, explaining spatiotemporal patterns such as standing and spiral waves. The main contributions are the derivation of explicit normal forms for systems with spatial symmetry and the development of a predictive approach that links model parameters to the selection and stability of nonlinear patterns.