Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop |
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Authors: | Etienne Farcot Jean-Luc Gouzé |
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Institution: | (1) INRIA, Virtual Plants Project-Team, UMR DAP, CIRAD, TA A96/02, 34398 Montpellier Cedex 5, France;(2) INRIA, Comore Project-Team, UR Sophia Antipolis, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France |
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Abstract: | This paper concerns periodic solutions of a class of equations that model gene regulatory networks. Unlike the vast majority
of previous studies, it is not assumed that all decay rates are identical. To handle this more general situation, we rely
on monotonicity properties of these systems. Under an alternative assumption, it is shown that a classical fixed point theorem
for monotone, concave operators can be applied to these systems. The required assumption is expressed in geometrical terms
as an alignment condition on so-called focal points. As an application, we show the existence and uniqueness of a stable periodic orbit for negative feedback loop systems in
dimension 3 or more, and of a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed
the existence of these orbits only. |
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Keywords: | |
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