# Monotone systems theory applied to reaction networks

From Mathematics of Reaction Networks

**Monotone systems theory** was originally pioneered by Morris Hirsch in the 1980s ^{[1]} ^{[2]} ^{[3]} ^{[4]} ^{[5]} ^{[6]} ^{[7]} ^{[8]} ^{[9]} ^{[10]} ^{[11]} ^{[12]} ^{[13]} ^{[14]}. The theory was further developed by a number of other mathematicians, including Stephen Smale and Hal Smith.

A dynamical system is *monotone* if its mapping is order preserving, i.e. given any two initial states that are ordered with respect to some partial order, , their images are also ordered: .

## References

- ↑ M. W. Hirsch, Convergence in ordinary and partial differential equations,
*Notes for Colloquium Lectures at University of Toronto*, August 23-26, 1982. Providence, R. I. : Amer. Math. Soc. (1982). - ↑ M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I: limit sets.
*SIAM J. Appl. Math.*13 (1982), 167-179. - ↑ M. W. Hirsch, Differential equations and convergence almost everywhere in strongly monotone semiflows,
*Contemp. Math.*17 (1983), 267-285. - ↑ M. W. Hirsch, The dynamical systems approach to differential equations.
*Bull. A. M. S.*11 (1984), 1-64. - ↑ M. W. Hirsch, Systems of differential equations which are competitive or cooperative. II: convergence almost everywhere.
*SIAM J. Math. Anal.*16 (1985), 423-439. - ↑ M. W. Hirsch, Attractors for discrete-time monotone dynamical systems in strongly ordered spaces,
*Geometry and Topology, Lecture Notes in Mathematics*vol. 1167, J. Alexander and J. Harer eds., 141-153. New York: Springer-Verlag (1985). - ↑ M. W. Hirsch, Systems of differential equations which are competitive or cooperative. III: Competing species.
*Nonlinearity*1 (1988), 51-71. - ↑ M. W. Hirsch, Stability and Convergence in Strongly Monotone dynamical systems,
*J. reine angew. Math.*383 (1988), 1-53. - ↑ M. W. Hirsch, Convergent activation dynamics in continuous time neural networks,
*Neural Networks*2 (1989) - ↑ M. W. Hirsch, Systems of differential equations that are competitive or cooperative. IV: Structural stability in three dimensional systems.
*SIAM J. Math. Anal.*21 (1990), 1225-1234. - ↑ M. W. Hirsch, (1989) Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems.
*J. Diff. Eqns.*80 (1989), 94-106. - ↑ M. W. Hirsch, Systems of differential equations that are competitive or cooperative. VI: A local closing lemma for 3-dimensional systems.
*Ergod. Th. Dynamical Sys.*11 (1991), 443-454. - ↑ M. W. Hirsch, Fixed points of monotone maps,
*J. Differential Equations*123(1995), 171-179. - ↑ M. W. Hirsch, Chain transitive sets for smooth strongly monotone dynamical systems,
*Diff. Eqns. and Dynamical Systems 5*(1999) 529-543.