Homeostasis with Multiple Inputs

Abstract

Homeostasis is a regulatory mechanism whereby some output variables of a system are kept approximately constant as input parameters vary over some region. Important applications include biological and chemical systems. In [M. Golubitsky and I. Stewart, J. Math. Biol., 74 (2017), pp. 387--407], we reformulated homeostasis in the context of singularity theory by replacing “approximately constant over an interval” by “zero derivative with respect to the input at a point” and discussed coordinate changes that put the singularity in normal form. We call this form of homeostasis infinitesimal homeostasis. Reed et al. [Bull. Math. Biol., 79 (2017), pp. 1--24] show by example that biologically relevant biochemical mechanisms can exhibit homeostasis even though infinitesimal homeostasis does not occur for realistic model parameters.) The main focus in [M. Golubitsky and I. Stewart, J. Math. Biol., 74 (2017), pp. 387--407] was on systems with one input and one output variable, classified by a subfamily of the elementary catastrophes of Thom [Structural Stability and Morphogenesis, Benjamin, Reading MA, 1975] and Zeeman [Catastrophe Theory: Selected Papers 1972--1977, Addison-Wesley, London, 1977]. In this paper, we use the singularity theory approach to study simultaneous homeostasis for two input parameters. As before, coordinate changes that preserve homeostasis play a prominent role because such coordinate changes provide the basis for singularity theory. We classify the singularities concerned and discuss their effect on homeostasis. We also speculate on why homeostasis in models may look as though it comes from infinitesimal homeostasis. This speculation uses the classification of elementary catastrophes and discusses why certain singularities (chairs and hyperbolic umbilics) may be expected to occur as a system evolves towards homeostasis.