%0 Journal Article
%T Global Stability in Dynamical Systems with Multiple Feedback Mechanisms
%A Morten Andersen
%A Frank Vinther
%A Johnny T. Ottesen
%J Advances in Pure Mathematics
%P 393-407
%@ 2160-0384
%D 2016
%I Scientific Research Publishing
%R 10.4236/apm.2016.65027
%X A class of <i>n</i>-dimensional ODEs with up to <i>n</i> feedbacks from the <i>n</i>¡¯th variable is analysed. The feedbacks
are represented by non-specific, bounded, non-negative <i>C</i><sup>1</sup> 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.
%K Odes
%K Multiple Feedbacks
%K Stability
%K Global Stability
%K Attracting Trapping Region
%K Nonlinear Dynamics
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=65885