Graphical Analysis of Ordinary Differential Equations in Biochemical Kinetics: A Pedagogical Framework for Stability and Bifurcations

Graphical analysis provides an intuitive yet rigorous framework for understanding dynamical behavior in biochemical systems. This article develops a pedagogical framework for the analysis of one-dimensional autonomous ordinary differential equation (ODE) models arising in biochemistry and biophysics. Using the production–loss dynamics of a protein P as a foundational template, we systematically introduce increasing regulatory complexity through (i) Michaelis–Menten activation, (ii) cooperative Hill kinetics, and (iii) mutual inhibition. The resulting linear and nonlinear models are analyzed using rate–balance and rate–difference representations, together with phase-line diagrams that encode the underlying one-dimensional vector field. These graphical constructions enable the systematic identification of steady states and determination of their qualitative stability. Slope-based graphical arguments are further employed to detect the onset of bifurcations through the loss of hyperbolicity at equilibrium. Although slope information alone does not fully determine the specific bifurcation type, the vanishing of the local derivative provides a reliable graphical indicator of qualitative change, thereby strengthening the connection between visual intuition and formal dynamical systems analysis. In addition, the instructional implementation of this framework is evaluated through structured classroom deployment and assessment-based analysis. The results indicate improved conceptual understanding, enhanced ability to interpret dynamical behavior graphically, and increased student confidence in linking mathematical formalism with biological intuition. These findings highlight the importance of integrating evaluation-driven pedagogical strategies to validate and refine teaching approaches in mathematical chemistry education.