Stability properties of the Goodwin-Smith oscillator model with additional feedback

The Goodwin oscillator is a simple yet instructive mathematical model, describing a wide range of self-controlled biological and biochemical processes, among them are self-inhibitory metabolic pathways and genetic circadian clocks. One of its most important applications is concerned with the hormonal cycles modeling. One of the first model of hormonal rhythm, squarely based on the conventional Goodwin oscillator, was suggested by W.R. Smith to describe the testosterone regulation in male and is often referred to as the Goodwin-Smith model. The Goodwin-Smith model describes an endocrine regulatory circuit, consisting of three hormones. Gonadotropin-releasing hormone controls the secretion of the luteinizing hormone, which influences the secretion of testosterone. In its turn, testosterone inhibits the secretion of gonadotropin, which effect is represented by the nonlinear feedback. Further theoretical and experimental works on hormonal regulation revealed that the structure of this control circuit is in fact more complicated, since the testosterone directly influences the secretion of both precursor hormones. This motivates us to consider the modified Goodwin-Smith model with additional negative feedback loop. The potential applications of such an extended model are not limited to hormonal regulation; similar models, where extra feedbacks can be either positive or negative, arise in some metabolic reactions. We show that many basic properties of the Goodwin-Smith model retain their validity, introducing an additional feedback, this holds e.g. for the existence and uniqueness of equilibria, Hopf bifurcation, and the existence of periodic solutions.