J. Math. Biol. (1998) 36: 321—348 A methodological basis for description and analysis of systems with complex switch-like interactions Erik Plahte, Thomas Mestl, Stig W. Omholt  Department of Mathematical Sciences, Agricultural University of Norway, N-1432 A s s, Norway  Department of Animal Science, Agricultural University of Norway, N-1432 A s s, Norway Received 25 January 1996; received in revised form 29 June 1997 Abstract. A wide range of complex systems appear to have switch-like inter- actions, i.e. below (or above) a certain threshold x has no or little influence on y, while above (or below) this threshold the effect of x on y saturates rapidly to a constant level. Switching functions are frequently described by sigmoid functions or combinations of these. Within the context of ordinary differential equations we present a very general methodological basis for designing and analysing models involving complicated switching functions together with any other non-linearities. A procedure to determine position and stability properties of all stationary points lying close to a threshold for one or several variables, so-called singular stationary points, is developed. Such points may represent homeostatic states in models, and are therefore of considerable interest. The analysis provides a profound insight into the generic effects of steep sigmoid interactions on the dynamics around homeostatic points. It leads to qualitative as well as quantitative predictions without using advanced mathematical methods. Thus, it may have an important heuristic function in connection with numerical simulations aimed at unfolding the predictive potential of realistic models. Key words: Complex systems — Nonlinear — Sigmoid — Threshold — Homeo- stasis — Switch-like 1 Introduction A predominant feature of biological and other higher level systems is the presence of threshold dominated cause and effect relationships between the system variables. Below (or above) a certain level, variable x has little or no influence on the behavior of variable y, while above (or below) this level the effect of x on y saturates rapidly to a constant level. Such interactions are found in a wide range of biological phenomena (Holling, 1959; Goodwin, 1965; Walter and Parker, 1967; Murdoch, 1969; Murdoch and Oaten, 1975;