Delay-differential extension of the logistic two-gene oscillator undergoes a Hopf bifurcation at explicit critical total delay τ_c with closed-form ω_c, transversality bound, and supercritical amplitude laws, matching p53 data within 3%.
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Logistic reformulations of delay-coupled gene regulatory networks are globally smooth and positive at zero, with matched parameters, unique equilibria, Hopf bifurcation at critical delays, and substantially smaller Lipschitz constants than Hill-based versions.