Assess graduate students' ability to analyze nonlinear reaction–diffusion systems by (a) performing linear stability analysis of homogeneous steady states and deriving dispersion relations to identify Turing instabilities, (b) classifying and interpreting local bifurcations (e.g., transcritical, pitchfork, Hopf) and secondary bifurcations relevant to pattern onset, (c) carrying out weakly nonlinear (amplitude) expansions to determine pattern selection, stability and sub/supercritical behavior, (d) predicting selected wavelengths and pattern types under parameter changes and boundary conditions, and (e) interpreting numerical continuation and simulation results to connect theory with observed spatial patterns. Scope: two-species reaction–diffusion models, analytic derivations, and comparison with numerical examples; assumes familiarity with PDEs, perturbation methods, and dynamical systems.