Stability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations

Time delays are an integral part of various physiological processes. In this paper, we analyse two models for physiological systems: the Mackey-Glass and Lasota equations. We first exhibit a sufficient condition to ensure local stability, and then outline the associated necessary and sufficient condition for stability. Using a non-dimensional bifurcation parameter, we then highlight that stability will be lost via a Hopf bifurcation. We also explicitly characterise the type of the Hopf bifurcation using Poincaré normal forms and the center manifold theory. The theoretical analysis is complemented with some numerical examples, stability charts and bifurcation diagrams.