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Title:
Delay-Induced Hopf and Turing Instabilities in a Modified Brusselator Reaction-Diffusion Model
Authors:
Muhammad Sajjad Shabbir ORCID iD 0000-0002-7486-2493
Volume
97
Issue
3
Year
2027
Pages
1011-1056
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 residence-time transport. The analysis is developed locally near the positive equilibrium \( E^{\ast}=(1,2) \); 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.