  In this paper we investigate the homeostatic mechanism in two biologically
motivated models: intracellular copper regulation and self immune recognition.
The analysis is based on the notions of infinitesimal homeostasis and
near-perfect homeostasis. We introduce a theoretical background that makes it
possible to consider points of infinitesimal homeostasis that lie at the
boundary of the domain of definition of the input-output function. We show that
the two models display near-perfect homeostasis. Moreover, we show that, unlike
the examples of [M. Reed, J. Best, M. Golubitsky, I. Stewart, and H. F.
Nijhout. Analysis of homeostatic mechanisms in biochemical networks. Bull.
Math. Biol., 79(11):2534-2557, 2017], the obstruction of occurrence of
infinitesimal homeostasis in both of them is caused by the modeling assumptions
that force the point of infinitesimal homeostasis to lie at the boundary of
domain of definition of the respective input-output functions.
