Title: Stability, convergence, and limit cycles in some human physiological processes,
Abstract: Mathematical models for physiological processes aid qualitative understanding
of the impact of various parameters on the underlying process. We analyse two
such models for human physiological processes: the Mackey-Glass and the Lasota
equations, which model the change in the concentration of blood cells in the
human body. We first study the local stability of these models, and derive
bounds on various model parameters and the feedback delay for the concentration
to equilibrate. We then deduce conditions for non-oscillatory convergence of
the solutions, which could ensure that the blood cell concentration does not
oscillate. Further, we define the convergence characteristics of the solutions
which govern the rate at which the concentration equilibrates when the system
is stable. Owing to the possibility that physiological parameters can seldom be
estimated precisely, we also derive bounds for robust stability\textemdash
which enable one to ensure that the blood cell concentration equilibrates
despite parametric uncertainty. We also highlight that when the necessary and
sufficient condition for local stability is violated, the system transits into
instability via a Hopf bifurcation, leading to limit cycles in the blood cell
concentration. We then outline a framework to characterise the type of the Hopf
bifurcation and determine the asymptotic orbital stability of limit cycles. The
analysis is complemented with numerical examples, stability charts and
bifurcation diagrams. The insights into the dynamical properties of the
mathematical models may serve to guide the study of dynamical diseases.