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.
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