Delay-Induced Hopf and Turing Instabilities in a Modified Brusselator Reaction-Diffusion Model
Abstract
A delayed reaction–diffusion extension of a restrained modified Brusselator model is studied under homogeneous Neumann boundary conditions. The delay is inserted into the restrained autocatalytic channel and is interpreted as an effective memory generated by intermediate-complex formation, catalytic activation, or residencetime transport. The analysis is developed locally near the positive equilibrium E ∗ = (1, 2); no global invariance of the nonnegative cone is claimed for the delayed inhibitor equation. After linearization, the modal characteristic equation reduces to a single-exponential form, which gives explicit formulas for homogeneous delay-induced Hopf-crossing thresholds and for linear stationary Turing thresholds. Their intersection is identified as a linear Turing–Hopf threshold rather than as a fully classified codimension-two nonlinear bifurcation. Numerical diagrams and direct simulations illustrate stable relaxation, stationary pattern onset, and mixed spatiotemporal modulation. A proportional feedback law is also examined. The activator feedback gain raises the stationary threshold, whereas the inhibitor feedback gain can lower part of that threshold; therefore positive feedback gains should not be interpreted as uniformly stabilizing the full reaction–diffusion system. The manuscript is thus framed as a corrected local threshold analysis and numerical exploration of delayed restrained Brusselator dynamics.