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To understand the sophisticated control mechanisms of the human's endocrinesystem is a challenging task that is a crucial step towards precise medicaltreatment of many disfunctions and diseases. Although mathematical modelsdescribing the endocrine system as a whole are still elusive, recently somesubstantial progress has been made in analyzing theoretically its subsystems(or axes) that regulate production of specific hormones. Many of the relevantmathematical models are similar in structure to (or squarely based on) thecelebrated Goodwin's oscillator. Such models are convenient to explain stableperiodic oscillations at hormones' level by representing the correspondingendocrine regulation circuits as cyclic feedback systems. However, many realhormonal regulation mechanisms (in particular, testosterone regulation) are infact known to have non-cyclic structures and involve multiple feedbacks; aGoodwin-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 additionalnegative feedback. Local stability properties of the proposed model arestudied, and we show that the local instability of its unique equilibriumimplies oscillatory behavior of almost all solutions. Furthermore, underadditional restrictions we prove that almost all solutions converge to periodicones.

Local and global analysis of endocrine regulation as a non-cyclic feedback system