Supercritical Hopf Bifurcations in Two Biochemical Reaction Systems

The characteristics of models representing biochemical phenomena exhibit complicated steady states and numerous state transitions that remain interesting in applications. Examining these states by combining the effective methods of bifurcation theory and computational algebra is profoundly appreciable to obtain bifurcation points near which the qualitative behavior of the model varies and parameter ranges that promote particular behavior. This study reveals several essential characteristics of two biochemical reaction models that have not been detected before. Utilizing the Lyapunov function, we compute the general form of the first Lyapunov coefficient to determine Hopf bifurcation for the Brusselator model. Then, for the smallest 3D biochemical reaction model, we obtain a center manifold up to third-degree to study Hopf bifurcation in this system. We demonstrate all results by numerical simulation.