# Periodic Solutions of Piecewise Affine Gene Network Models with Non Uniform Decay Rates: The Case of a Negative Feedback Loop

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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.

## Keywords

Piecewise linear dynamical systems Periodic trajectories Monotone, concave maps Negative feedback loop systems## Notes

### Acknowledgments

This work was partially supported by the European Commission, under project Hygeia Nest-004995. The authors would like to thank the anonymous referee for a very detailed review, which has led to substantial improvements of the paper.

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