Global stability for monotone tridiagonal systems with negative feedback

This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincare-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique equilibrium is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Different approaches are discussed to rule out period orbits. One is based on direct linearization, while the other uses the theory of second additive compound matrices. Among the examples that will illustrate our main theoretical results is the classical Goldbeter model of the circadian rhythm.

[1]  L. Sanchez,et al.  Dynamics of the modified Michaelis–Menten system , 2006 .

[2]  J. Mallet-Paret,et al.  The Poincare-Bendixson theorem for monotone cyclic feedback systems , 1990 .

[3]  James S. Muldowney,et al.  Compound matrices and ordinary differential equations , 1990 .

[4]  Michael Y. Li,et al.  Global stability for the SEIR model in epidemiology. , 1995, Mathematical biosciences.

[5]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[6]  James S. Muldowney,et al.  Dynamics of Differential Equations on Invariant Manifolds , 2000 .

[7]  Michael Y. Li,et al.  A Criterion for Stability of Matrices , 1998 .

[8]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[9]  A. Goldbeter,et al.  Biochemical Oscillations And Cellular Rhythms: Contents , 1996 .

[10]  Michael Y. Li,et al.  Global Stability in Some Seir Epidemic Models , 2002 .

[11]  A. Goldbeter A model for circadian oscillations in the Drosophila period protein (PER) , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[12]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[13]  L. Sanchez,et al.  Global asymptotic stability of the Goodwin system with repression , 2009 .

[14]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[15]  G. Sell,et al.  THE POINCARE-BENDIXSON THEOREM FOR MONOTONE CYCLIC FEEDBACK SYSTEMS WITH DELAY , 1996 .

[16]  David Angeli,et al.  Oscillations in I/O Monotone Systems Under Negative Feedback , 2007, IEEE Transactions on Automatic Control.

[17]  V. Boichenko,et al.  Dimension theory for ordinary differential equations , 2005 .