Goodwin's oscillators with an additional negative feedback for modeling hormonal regulation systems

To understand the sophisticated control mechanisms of the human's endocrine system is a challenging task that is a crucial step towards precise medical treatment of many disfunctions and diseases. Although mathematical models describing the endocrine system as a whole are still elusive, recently some substantial progress has been made in analyzing theoretically its subsystems (or \emph{axes}) that regulate production of specific hormones. Many of the relevant mathematical models are similar in structure to (or squarely based on) the celebrated \emph{Goodwin's oscillator}. Such models are convenient to explain stable periodic oscillations at hormones' level by representing the corresponding endocrine regulation circuits as \emph{cyclic} feedback systems. However, many real hormonal regulation mechanisms (in particular, testosterone regulation) are in fact known to have non-cyclic structures and involve multiple feedbacks; a Goodwin-type model thus represents only a part of such a complicated mechanism. In this paper, we examine a new mathematical model of hormonal regulation, obtained from the classical Goodwin's oscillator by introducing an additional negative feedback. Local stability properties of the proposed model are studied, and we show that the local instability of its unique equilibrium implies oscillatory behavior of almost all solutions. Furthermore, under additional restrictions we prove that almost all solutions converge to periodic ones.