Abstract
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.
Highlights
Many applications of ODEs to physics, chemistry, biology, medicine, and life sciences give rise to non-linear non-negative compartment systems
This paper concerns the stability of the solutions of such models
Avoiding negative modelling hormone levels is necessary for a sound model and is proved in the following lemma
Summary
An n dimensional system with feedbacks from the n’th variable is introduced and some applications from bio-medicine and biochemistry are described. Analysis of a scaled version of the system is made including fixed point investigation. An easy applicable sufficient criterion for a unique, globally stable fixed point is formulated and proved. The results in this paper follow from the dimensionless form of the equations stated in (6) of Section 2. Before turning to this form we motivate and discuss the dimensional form of the equations in Section 1 as we relate the system to applications and earlier results. How to cite this paper: Andersen, M., Vinther, F. and Ottesen, J.T. (2016) Global Stability in Dynamical Systems with Multiple Feedback Mechanisms.
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