Hopf bifurcation analysis of a modified Brusselator model with sub-interval distributed delay

In this paper, a modified Brusselator model incorporating a moderated autocatalytic term and a sub-interval distributed delay is investigated. The existence and uniqueness of a positive equilibrium are first established, together with explicit conditions for its local asymptotic stability. By deriving and analyzing the associated characteristic equation, two distinct Hopf bifurcation mechanisms are rigorously identified: one induced by variations in the delay center $\tau$ and the other triggered by changes in the delay distribution width $\epsilon$. Sufficient conditions for the existence of two Hopf bifurcation branches in the $(\tau ,\epsilon )$-parameter plane are obtained, and the corresponding transversality conditions are verified. Finally, numerical simulations are carried out to validate the theoretical results. The results reveal a clear destabilizing role of the delay center and a stabilizing, smoothing effect of the delay width, which highlights the important role of distributed delays in shaping the dynamical behavior of autocatalytic reaction systems.