Numerical discovery and continuation of points of infinitesimal homeostasis

Homeostasis is the biological notion of certain outputs being stable with respect to input perturbations, at least over a relatively broad range. In general this is a result of coordination of regulatory mechanisms, and is thought to occur throughout biology. Recently the mathematical concept of infinitesimal homeostasis has been introduced, with the key idea being that the emergence of homeostasis is governed by certain geometric structures; these have previously been found in several mathematical models of biological processes. The theory of infinitesimal homeostasis has been applied to several specific classes of problems. Here we propose a more general method which allows for discovery of points of infinitesimal homeostasis, as well as their continuation, without imposing strong conditions on the underlying systems. The central idea is to construct an augmented system in which the desired point(s) of infinitesimal homeostasis occur as equilibria of the augmented system. The proposed method is developed and tested on several examples, ranging from synthetic test problems to examples drawn from the literature.