Access the full text.
Sign up today, get DeepDyve free for 14 days.
Hal Smith (1986)
Cooperative systems of differential equations with concave nonlinearitiesNonlinear Analysis-theory Methods & Applications, 10
A. Brodsky, M. Krasnoselskii, Richard Flaherty, L. Boron (1964)
Positive solutions of operator equations
Richard Casey, H. Jong, J. Gouzé (2006)
Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their StabilityJournal of Mathematical Biology, 52
R. Edwards (2000)
Analysis of continuous-time switching networksPhysica D: Nonlinear Phenomena, 146
H. Jong, J. Gouzé, Céline Hernandez, M. Page, T. Sari, J. Geiselmann (2004)
Qualitative simulation of genetic regulatory networks using piecewise-linear modelsBulletin of Mathematical Biology, 66
H. Jong (2002)
Modeling and Simulation of Genetic Regulatory Systems: A Literature ReviewJ. Comput. Biol., 9
L. Glass, J. Pasternack (1978)
Prediction of limit cycles in mathematical models of biological oscillationsBulletin of Mathematical Biology, 40
M. Elowitz, S. Leibler (2000)
A synthetic oscillatory network of transcriptional regulatorsNature, 403
Etienne Farcot (2006)
Geometric properties of a class of piecewise affine biological network modelsJournal of Mathematical Biology, 52
Tomáš Gedeon, K. Mischaikow (1995)
Structure of the global attractor of cyclic feedback systemsJournal of Dynamics and Differential Equations, 7
El Snoussi (1989)
Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approachDynamics and Stability of Systems, 4
E Farcot (2006)
Geometric properties of piecewise affine biological network modelsJ Math Biol, 52
J. Gouzé, T. Sari (2002)
A class of piecewise linear differential equations arising in biological modelsDynamical Systems, 17
L. Glass (1975)
Combinatorial and topological methods in nonlinear chemical kineticsJournal of Chemical Physics, 63
J. Mallet-Paret, Hal Smith (1990)
The Poincare-Bendixson theorem for monotone cyclic feedback systemsJournal of Dynamics and Differential Equations, 2
L. Glass, J. Pasternack (1978)
Stable oscillations in mathematical models of biological control systemsJournal of Mathematical Biology, 6
Thomas Mestl, E. Plahte, S. Omholt (1995)
Periodic solutions in systems of piecewise- linear differential equationsDynamics and Stability of Systems, 10
T Mestl, E Plahte, SW Omholt (1995)
Periodic solutions of piecewise-linear differential equationsDyn Stab Syst, 10
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.
Acta Biotheoretica – Springer Journals
Published: Oct 8, 2009
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.