January  2017, 37(1): 485-504. doi: 10.3934/dcds.2017020

Monotone dynamical systems: Reflections on new advances & applications

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

Received  April 2016 Revised  July 2016 Published  November 2016

Fund Project: The author is supported by a Simons Foundation Grant 355819.

The article contains the author's reflections on recent developments in a very select portion of the now vast subject of monotone dynamical systems. Continuous timesystems generated by cooperative systems of ordinary differential equations, delay differential equations, parabolic partial differential equations, and controlsystems are the main focus and results are included which the author feels have had a major impact in the applications. These include the theory of competition betweentwo species or two teams and the theory of monotone control systems.

Citation: Hal L. Smith. Monotone dynamical systems: Reflections on new advances & applications. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 485-504. doi: 10.3934/dcds.2017020
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D. AngeliP. De Leenheer and E. Sontag, Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates, J. Math. Biol., 61 (2010), 581-616.  doi: 10.1007/s00285-009-0309-0.

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D. Angeli and E. D. Sontag, Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles, Nonlinear Analysis Series B: Real World Applications, 9 (2008), 128-140.  doi: 10.1016/j.nonrwa.2006.09.006.

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show all references

References:
[1]

D. AngeliG. Enciso and E. Sontag, A small-gain result for orthant-monotone systems under mixed feedback, Systems & Control Letters, 68 (2014), 9-19.  doi: 10.1016/j.sysconle.2014.03.002.

[2]

D. Angeli and E. D. Sontag, Monotone control systems, IEEE Trans. Automat. Control, 48 (2003), 1684-1698.  doi: 10.1109/TAC.2003.817920.

[3]

D. Angeli and E. D. Sontag, Multi-stability in monotone input/output systems, Systems Control Lett., 51 (2004), 185-202.  doi: 10.1016/j.sysconle.2003.08.003.

[4]

D. AngeliP. De Leenheer and E. Sontag, Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates, J. Math. Biol., 61 (2010), 581-616.  doi: 10.1007/s00285-009-0309-0.

[5]

D. Angeli and E. D. Sontag, Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles, Nonlinear Analysis Series B: Real World Applications, 9 (2008), 128-140.  doi: 10.1016/j.nonrwa.2006.09.006.

[6]

D. AngeliM. W. Hirsch and E. D. Sontag, Attractors in coherent systems of differential equations, J. Differential Equations, 246 (2009), 3058-3076.  doi: 10.1016/j.jde.2009.01.025.

[7]

L. BourouibaS. GourleyR. Liu and J. Wu, The interaction of migratory birds and domestic poultry and its role in sustaining avian influenza, SIAM Journal on Applied Mathematics, 71 (2011), 487-516.  doi: 10.1137/100803110.

[8]

C. BowmanA. GumelP. van den DriesscheJ. Wu and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133.  doi: 10.1016/j.bulm.2005.01.002.

[9]

R. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, W. Sussex, U. K. , 2003. doi: 10.1002/0470871296.

[10]

R. CantrellC. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214.  doi: 10.1016/j.mbs.2006.09.003.

[11]

R. CantrellC. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.

[12]

R. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal in heterogeneous landscapes, Chapter 11 in "Spatial Ecology" Ed. by S. Cantrell, C. Cosner, S. Ruan. Chapman & Hall/CRC Press,

[13]

F. Cao and J. Jiang, On the global attractivity of monotone random dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 891-898.  doi: 10.1090/S0002-9939-09-09912-2.

[14]

A. Carvalho, J. Langa and J. Robinson, Attractors for infinite-dimensional non-autonomous dynamical systems, Applied Math. Sciences, 182, Springer, NY, 2013. doi: 10.1007/978-1-4614-4581-4.

[15]

X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658.  doi: 10.1512/iumj.2008.57.3204.

[16]

I. Chueshov, Monotone Random Systems-Theory and Applications, Springer Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[17]

C. CosnerJ. C. BeierR. S. CantrellD. ImpoinvilL. KapitanskiM. D. PottsA. Troyoe and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, Journal of Theoretical Biology, 258 (2009), 550-560.  doi: 10.1016/j.jtbi.2009.02.016.

[18]

M. Di MarcoM. FortiM. Grazzini and L. Pancioni, Limit set dichotomy and convergence of cooperative piecewise linear neural networks, IEEE Trans. Circuits Syst. Ⅰ. Regul. Pap., 58 (2011), 1052-1062.  doi: 10.1109/TCSI.2010.2091194.

[19]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

[20]

G. EncisoH. L. Smith and E. Sontag, Non-monotone systems decomposable into monotone systems with negative feedback, J. Diff. Eqns., 224 (2006), 205-227.  doi: 10.1016/j.jde.2005.05.007.

[21]

G. EncisoM. W. Hirsch and H. L. Smith, Prevalent behavior of strongly order preserving semiflows, J.Dynamics and Diff. Eqns., 20 (2008), 115-132.  doi: 10.1007/s10884-007-9084-z.

[22]

F. Forni and R. Sepulchre, Differentially Positive Systems, IEEE Transactions on Automatic Control, 61 (2016), 346-359. 

[23]

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