Auto-catalytic chemical reactions are open non-equilibrium systemscharacterized by feedback-driven oscillations and instabilities that arenot always transparent within classical rate-equation formalisms. Inthis work, we introduce a PT-symmetric gain–loss framework formodeling auto-catalytic reaction networks, using the Belousov–Zhabotinsky(BZ) reaction [1-3] as a motivating example. Chemicalspecies are mapped onto a three-level non-Hermitian Hamiltonian inwhich autocatalysis and decay are represented by balanced gain andloss terms. Eigenvalue analysis reveals unbroken, exceptional-point,and broken PT phases corresponding to stable oscillations, criticalthresholds, and runaway amplification in chemical dynamics. Timeevolution is computed using matrix exponentiation e −iHt , revealingnovel gain-loss induced oscillations, PT-phase transitions, andcomplex dynamical behavior in open chemical systems. This approachestablishes a dynamical-systems bridge between auto-catalyticchemistry and PT-symmetric non-Hermitian quantum physics.