Limit cycles in piecewise-affine gene network models with multiple interaction loops

Etienne Farcot 1, 2 Jean-Luc Gouzé 3
1 VIRTUAL PLANTS - Modeling plant morphogenesis at different scales, from genes to phenotype
UMR AGAP - Amélioration génétique et adaptation des plantes méditerranéennes et tropicales, INRA - Institut National de la Recherche Agronomique, CRISAM - Inria Sophia Antipolis - Méditerranée
3 COMORE - Modeling and control of renewable resources
LOV - Laboratoire d'océanographie de Villefranche, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this article, we consider piecewise affine differential equations modelling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds, etc.). Our main result is an alternative theorem showing that if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions. This result extends greatly our previous work on a single negative feedback loop. We give several examples and simulations illustrating different cases. A preprint version is available as INRIA research report RR-6875 http://www-sop.inria.fr/virtualplants/Publications/2009/FG09/.
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Etienne Farcot, Jean-Luc Gouzé. Limit cycles in piecewise-affine gene network models with multiple interaction loops. International Journal of Systems Science, Taylor & Francis, 2010, 41 (1), pp.119-130. ⟨10.1080/00207720903144552⟩. ⟨hal-00831782⟩

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