Phase-translation group actions on strongly monotone skew-product semiflows
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- by Qiang Liu and Yi Wang PDF
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Abstract:
We establish a convergence property for pseudo-bounded forward orbits of strongly monotone skew-product semiflows with invariant phase-translation group actions. The results are then applied to obtain global convergence of certain chemical reaction networks whose associated systems in reaction coordinates are monotone, as well as the dynamics of certain reaction-diffusion systems in time-recurrent structure including periodicity, almost periodicity and almost automorphy.References
- Nicholas D. Alikakos and Peter Hess, On stabilization of discrete monotone dynamical systems, Israel J. Math. 59 (1987), no. 2, 185–194. MR 920081, DOI 10.1007/BF02787260
- Nicholas D. Alikakos, Peter Hess, and Hiroshi Matano, Discrete order preserving semigroups and stability for periodic parabolic differential equations, J. Differential Equations 82 (1989), no. 2, 322–341. MR 1027972, DOI 10.1016/0022-0396(89)90136-8
- David Angeli, Patrick De Leenheer, and Eduardo Sontag, Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates, J. Math. Biol. 61 (2010), no. 4, 581–616. MR 2672536, DOI 10.1007/s00285-009-0309-0
- David Angeli and Eduardo D. Sontag, Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles, Nonlinear Anal. Real World Appl. 9 (2008), no. 1, 128–140. MR 2370168, DOI 10.1016/j.nonrwa.2006.09.006
- Ovide Arino, Monotone semi-flows which have a monotone first integral, Delay differential equations and dynamical systems (Claremont, CA, 1990) Lecture Notes in Math., vol. 1475, Springer, Berlin, 1991, pp. 64–75. MR 1132019, DOI 10.1007/BFb0083480
- Anatoli V. Babin and George R. Sell, Attractors of non-autonomous parabolic equations and their symmetry properties, J. Differential Equations 160 (2000), no. 1, 1–50. MR 1734528, DOI 10.1006/jdeq.1999.3654
- J.J. Bijlsma and E.A. Groisman, Making informed decisions: Regulatory interactions between two-component systems, Trends Microbiol. 11 (2003), 359–366.
- Richard G. Casten and Charles J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations 27 (1978), no. 2, 266–273. MR 480282, DOI 10.1016/0022-0396(78)90033-5
- K. Cho, S. Shin, H. Kim, O.Wolkenhauer, B. McFerran and W. Kolch, Mathematical modeling of the influence of RKIP on the ERK signaling pathway, Computational Methods in Systems Biology, CMSB03, Lecture Notes in Computer Science, vol. 2602, Springer, Berlin, 2003.
- Klaus Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404, DOI 10.1007/978-3-662-00547-7
- S. Donovan, K.M. Shannon and G. Bollag, GTPase activating proteins: Critical regulators of intracellular signaling, Biochim, Biophys. Acta 1602 (2002), 23–45.
- Martin Feinberg, Some recent results in chemical reaction network theory, Patterns and dynamics in reactive media (Minneapolis, MN, 1989) IMA Vol. Math. Appl., vol. 37, Springer, New York, 1991, pp. 43–70. MR 1228917, DOI 10.1007/978-1-4612-3206-3_{4}
- J.E. Ferrell, Jr., Tripping the switch fantastic: How a protein kinase cascade can convert graded inputs into switch-like outputs, Trends Biochem. Sci. 21 (1996), 460–466.
- A. M. Fink, Almost periodic differential equations, Lect. Notes Math. 840 (Springer-Verlag, Berlin, 1981).
- A.D. Grossman, Genetic networks controlling the intiation of sporulation and the development of genetic competence in Bacillus subtilis, Annu. Rev. Genet. 29 (1995), 477–508.
- J. R. Haddock, M. N. Nkashama, and J. Wu, Asymptotic constancy for pseudomonotone dynamical systems on function spaces, J. Differential Equations 100 (1992), no. 2, 292–311. MR 1194812, DOI 10.1016/0022-0396(92)90116-5
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- Morris W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 1–64. MR 741723, DOI 10.1090/S0273-0979-1984-15236-4
- Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1–53. MR 921986, DOI 10.1515/crll.1988.383.1
- M. W. Hirsch and Hal Smith, Monotone dynamical systems, Handbook of differential equations: ordinary differential equations. Vol. II, Elsevier B. V., Amsterdam, 2005, pp. 239–357. MR 2182759
- Hongxiao Hu and Jifa Jiang, Translation-invariant monotone systems, I: autonomous/periodic case, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 3211–3217. MR 2661981, DOI 10.1016/j.nonrwa.2009.11.015
- Hongxiao Hu and Jifa Jiang, Translation-invariant monotone systems II: Almost periodic/automorphic case, Proc. Amer. Math. Soc. 138 (2010), no. 11, 3997–4007. MR 2679621, DOI 10.1090/S0002-9939-2010-10389-1
- J. F. Jiang, Sublinear discrete-time order-preserving dynamical systems, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 561–574. MR 1357065, DOI 10.1017/S0305004100074417
- Ji-Fa Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal. 27 (1996), no. 6, 1738–1744. MR 1416516, DOI 10.1137/S003614109326063X
- Jifa Jiang and Xiao-Qiang Zhao, Convergence in monotone and uniformly stable skew-product semiflows with applications, J. Reine Angew. Math. 589 (2005), 21–55. MR 2194677, DOI 10.1515/crll.2005.2005.589.21
- James Keener and James Sneyd, Mathematical physiology, Interdisciplinary Applied Mathematics, vol. 8, Springer-Verlag, New York, 1998. MR 1673204
- B.N. Kholodenko, A. Kiyatkin, F. Bruggenman, E.D. Sontag, H. Westerhoff and J. Hoek, Untangling the wires: A novel strategy to trace gunctional interactions in signaling and gene networks, Proc. Nat. Acad. Sci. USA 99 (2002), 12841–12846.
- D.J. Lew and D.J. Burke, The spindle assembly and spindle position checkpoings, Annu. Rev. Genet. 37 (2003), 251–282.
- Hiroshi Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci. 15 (1979), no. 2, 401–454. MR 555661, DOI 10.2977/prims/1195188180
- Janusz Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal. 18 (1987), no. 3, 642–646. MR 883558, DOI 10.1137/0518049
- Janusz Mierczyński and Peter Poláčik, Group actions on strongly monotone dynamical systems, Math. Ann. 283 (1989), no. 1, 1–11. MR 973801, DOI 10.1007/BF01457499
- Janusz Mierczyński and Wenxian Shen, Spectral theory for random and nonautonomous parabolic equations and applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 139, CRC Press, Boca Raton, FL, 2008. MR 2464792, DOI 10.1201/9781584888963
- Sylvia Novo and Rafael Obaya, Strictly ordered minimal subsets of a class of convex monotone skew-product semiflows, J. Differential Equations 196 (2004), no. 1, 249–288. MR 2025194, DOI 10.1016/S0022-0396(03)00152-9
- Sylvia Novo, Rafael Obaya, and Ana M. Sanz, Stability and extensibility results for abstract skew-product semiflows, J. Differential Equations 235 (2007), no. 2, 623–646. MR 2317498, DOI 10.1016/j.jde.2006.12.009
- Toshiko Ogiwara and Hiroshi Matano, Stability analysis in order-preserving systems in the presence of symmetry, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 2, 395–438. MR 1686708, DOI 10.1017/S0308210500021429
- Toshiko Ogiwara and Hiroshi Matano, Monotonicity and convergence results in order-preserving systems in the presence of symmetry, Discrete Contin. Dynam. Systems 5 (1999), no. 1, 1–34. MR 1664441, DOI 10.3934/dcds.1999.5.1
- H.G. Othmer, Analysis of complex reaction networks, (2003) Available on-line at http://www.math. leidenuniv.nl/ verduyn.
- Peter Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), no. 1, 89–110. MR 997611, DOI 10.1016/0022-0396(89)90115-0
- P. Poláčik and I. Tereščák, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 339–360. MR 1132766, DOI 10.1007/BF00375672
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- M. Samoilov, S. Plyasunov and A.P. Arkin, Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations, Proc. Nat. Acad. Sci. USA 102 (2005), 2310–2315.
- George R. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc. 127 (1967), 241–262. MR 212313, DOI 10.1090/S0002-9947-1967-0212313-2
- George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
- Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
- Wenxian Shen and Xiao-Qiang Zhao, Convergence in almost periodic cooperative systems with a first integral, Proc. Amer. Math. Soc. 133 (2005), no. 1, 203–212. MR 2085171, DOI 10.1090/S0002-9939-04-07556-2
- H. L. Smith, Cooperative systems of differential equations with concave nonlinearities, Nonlinear Anal. 10 (1986), no. 10, 1037–1052. MR 857738, DOI 10.1016/0362-546X(86)90087-8
- Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
- E.D. Sontag, Some new directions in control theory inspired by systems biology, Systems Biol. 1 (2004), 9–18.
- L. Stryer, Biochemistry, Freeman, New York, 1995.
- M.L. Sulis and R. Parsons, PTEN: from pathology to biology, Trends Cell Biol. 13 (2003), 478–483.
- Peter Takáč, Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry, J. Differential Equations 100 (1992), no. 2, 355–378. MR 1194815, DOI 10.1016/0022-0396(92)90119-8
- Peter Takáč, Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology, Nonlinear Anal. 14 (1990), no. 1, 35–42. MR 1028245, DOI 10.1016/0362-546X(90)90133-2
- Bao Rong Tang, Yang Kuang, and Hal Smith, Strictly nonautonomous cooperative system with a first integral, SIAM J. Math. Anal. 24 (1993), no. 5, 1331–1339. MR 1234019, DOI 10.1137/0524076
- W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87 (1965), 719–751. MR 187014, DOI 10.2307/2373071
- Yi Wang, Asymptotic symmetry in strongly monotone skew-product semiflows with applications, Nonlinearity 22 (2009), no. 4, 765–782. MR 2486356, DOI 10.1088/0951-7715/22/4/005
- Yi Wang and Xiao-Qiang Zhao, Convergence in monotone and subhomogeneous discrete dynamical systems on product Banach spaces, Bull. London Math. Soc. 35 (2003), no. 5, 681–688. MR 1989498, DOI 10.1112/S0024609303002273
- Jian Hong Wu, Convergence of monotone dynamical systems with minimal equilibria, Proc. Amer. Math. Soc. 106 (1989), no. 4, 907–911. MR 1004632, DOI 10.1090/S0002-9939-1989-1004632-7
- Xiao-Qiang Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations 187 (2003), no. 2, 494–509. MR 1949451, DOI 10.1016/S0022-0396(02)00054-2
Additional Information
- Qiang Liu
- Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
- Yi Wang
- Affiliation: Wu Wen-Tsun Key Laboratory of Mathematics, Department of Mathematics, University of Science and Technology of China Hefei, Anhui, 230026, People’s Republic of China – and – Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014, Finland
- Received by editor(s): September 10, 2009
- Received by editor(s) in revised form: January 22, 2011
- Published electronically: February 27, 2012
- Additional Notes: The second author was the corresponding author and was partially supported by NSF of China No. 10971208 and by the Finnish Center of Excellence in Analysis and Dynamics and the FRF for the Central Universities
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 3781-3804
- MSC (2010): Primary 37B55, 37C65, 37L15, 37N25
- DOI: https://doi.org/10.1090/S0002-9947-2012-05555-3
- MathSciNet review: 2901234