Global Stability in Dynamical Systems with Multiple Feedback Mechanisms

A class of n-dimensional ODEs with up to n feedbacks from the n’th variable is analysed. The feedbacks are represented by non-specific, bounded, non-negative C1 functions. The main result is the formulation and proof of an easily applicable criterion for existence of a globally stable fixed point of the system. The proof relies on the contraction mapping theorem. Applications of this type of systems are numerous in biology, e.g., models of the hypothalamic-pituitary-adrenal axis and testosterone secretion. Some results important for modelling are: 1) Existence of an attractive trapping region. This is a bounded set with non-negative elements where solutions cannot escape. All
solutions are shown to converge to a “minimal” trapping region. 2) At least one fixed point exists. 3) Sufficient criteria for a unique fixed point are formulated. One case where this is fulfilled is when the feedbacks are negative.