Skip to main content
SpringerLink
Log in

Dynamical Transitions in Chemistry and Biology

  • Chapter
  • First Online:
Phase Transition Dynamics

Abstract

Chemical reaction systems and biological models were among the first dissipative systems described by the Belgian school; see, e.g., (Prigogine and Lefever 1968; Glansdorff and Prigogine 1971). Among many other features, dissipative systems in nature demonstrate self-organized and self-assembled structures; see, for example, (Nicolis and Prigogine 1977; Kapral and Showalter 1995; Pismen 2006; Desai and Kapral 2009; Cross and Hohenberg 1993; Swinney et al. 1990; Murray 2002).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Bao, W. (2002). The random projection method for a model problem of combustion with stiff chemical reactions. Appl. Math. Comput. 130(2-3), 561–571.

    Article  MathSciNet  MATH  Google Scholar 

  • Belousov, B. P. (1959). An oscillating reaction and its mechanism. Sborn. referat. radiat. med., 145.

    Google Scholar 

  • Brenner, M. P., L. S. Levitov, and E. O. Budrene (1998). Physical mechanisms for chemotactic pattern formation by bacteria. Biophysical Journal 74, 1677–1693.

    Article  Google Scholar 

  • Budrene, E. O. and H. C. Berg (1991). Complex patterns formed by motile cells of Escherichia coli. Nature 349, 630–633.

    Article  Google Scholar 

  • Budrene, E. O. and H. C. Berg (1995). Dynamics of formation of symmetric patterns of chemotactic bacteria. Nature 376, 49–53.

    Article  Google Scholar 

  • Cross, M. and P. Hohenberg (1993). Pattern formation outside of equilibrium. Reviews of Modern Physics 65(3), 851–1112.

    Article  Google Scholar 

  • Desai, R. C. and R. Kapral (2009). Self-Organized and Self-Assembled Structures. Cambridge University Press.

    Google Scholar 

  • Field, R. J., E. Körös, and R. M. Noyes (1972). Oscillations in chemical systems, Part 2. thorough analysis of temporal oscillations in the bromate-cerium-malonic acid system. J. Am Chem. Soc. 94, 8649–8664.

    Google Scholar 

  • Field, R. J. and R. M. Noyes (1974). Oscillations in chemical systems, IV. limit cycle behavior in a model of a real chemical reaction. J. Chem. Physics. 60, 1877–1884.

    Google Scholar 

  • Glansdorff, P. and I. Prigogine (1971). Structure, stability, and fluctuations. Wiley-Interscience, New York.

    Google Scholar 

  • Guo, B. and Y. Han (2009). Attractor and spatial chaos for the Brusselator in R N. Nonlinear Anal. 70(11), 3917–3931.

    Article  MathSciNet  MATH  Google Scholar 

  • Guo, Y. and H. J. Hwang (2010). Pattern formation (I): the Keller-Segel model. J. Differential Equations 249(7), 1519–1530.

    Article  MathSciNet  MATH  Google Scholar 

  • Hastings, S. P. and J. D. Murray (1975). The existence of oscillatory solutions in the Field-Noyes model for the Belousov-Zhabotinskii reaction. SIAM J. Appl. Math. 28, 678–688.

    Article  MathSciNet  Google Scholar 

  • Kaper, H. G. and T. J. Kaper (2002). Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys. D 165(1-2), 66–93.

    Article  MathSciNet  MATH  Google Scholar 

  • Kapral, R. and K. Showalter (1995). Chemical Waves and Patterns. Kluwer, Dordrecht.

    Book  Google Scholar 

  • Keller, E. F. and L. A. Segel (1970). Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415.

    Article  MATH  Google Scholar 

  • Lapidus, I. and R. Schiller (1976). Model for the chemotactic response of a bacterial population. Biophys J. 16(7), 779–789.

    Google Scholar 

  • Liu, H., T. Sengul, and S. Wang (2012). Dynamic transitions for quasilinear systems and cahn-hilliard equation with onsager mobility. Journal of Mathematical Physics 53, 023518, doi: 10.1063/1.3687414, 1–31.

    Google Scholar 

  • Lotka, A. J. (1925). Elements of physical biology. Baltimore: Williams & Wilkins Co.

    MATH  Google Scholar 

  • Ma, T. and S. Wang (2005b). Bifurcation theory and applications, Volume 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.

    Google Scholar 

  • Ma, T. and S. Wang (2007b). Stability and Bifurcation of Nonlinear Evolutions Equations. Science Press, Beijing.

    Google Scholar 

  • Ma, T. and S. Wang (2010b). Dynamic transition theory for thermohaline circulation. Phys. D 239(3-4), 167–189.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2011a). Dynamic transition and pattern formation for chemotactic systems.

    Google Scholar 

  • Ma, T. and S. Wang (2011c). Phase transitions for Belousov-Zhabotinsky reactions. Math. Methods Appl. Sci. 34(11), 1381–1397.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, T. and S. Wang (2011d). Phase transitions for the Brusselator model. J. Math. Phys. 52(3), 033501, 23.

    Google Scholar 

  • Murray, J. (2002). Mathematical Biology, II. 3rd Ed. Springer-Verlag.

    Google Scholar 

  • Nadin, G., B. Perthame, and L. Ryzhik (2008). Traveling waves for the Keller-Segel system with Fisher birth terms. Interfaces Free Bound. 10(4), 517–538.

    Article  MathSciNet  MATH  Google Scholar 

  • Nicolis, G. and I. Prigogine (1977). Self-organization in nonequilibrium systems. Wiley-Interscience, New York.

    Google Scholar 

  • Perthame, B. and A.-L. Dalibard (2009). Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc. 361(5), 2319–2335.

    Article  MathSciNet  MATH  Google Scholar 

  • Perthame, B., C. Schmeiser, M. Tang, and N. Vauchelet (2011). Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: existence and branching instabilities. Nonlinearity 24(4), 1253–1270.

    Article  MathSciNet  MATH  Google Scholar 

  • Pismen, L. M. (2006). Patterns and Interfaces in Dissipative Dynamics. Springer, Berlin.

    MATH  Google Scholar 

  • Prigogine, I. and R. Lefever (1968). Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695.

    Article  Google Scholar 

  • Reichl, L. E. (1998). A modern course in statistical physics (Second ed.). A Wiley-Interscience Publication. New York: John Wiley & Sons Inc.

    MATH  Google Scholar 

  • Shi, J. (2009). Bifurcation in infinite dimensional spaces and applications in spatiotemporal biological and chemical models. Front. Math. China 4(3), 407–424.

    Article  MathSciNet  MATH  Google Scholar 

  • Smoller, J. (1983). Shock waves and reaction-diffusion equations, Volume 258 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. New York: Springer-Verlag.

    Google Scholar 

  • Swinney, H. L., N. Kreisberg, W. D. McCormick, Z. Noszticzius, and G. Skinner (1990). Spatiotemporal patterns in reaction-diffusion systems. In Chaos (Woods Hole, MA, 1989), pp. 197–204. New York: Amer. Inst. Phys.

    Google Scholar 

  • Temam, R. (1997). Infinite-dimensional dynamical systems in mechanics and physics (Second ed.), Volume 68 of Applied Mathematical Sciences. New York: Springer-Verlag.

    Google Scholar 

  • Tzou, J. C., B. J. Matkowsky, and V. A. Volpert (2009). Interaction of Turing and Hopf modes in the superdiffusive Brusselator model. Appl. Math. Lett. 22(9), 1432–1437.

    Article  MathSciNet  MATH  Google Scholar 

  • Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI 2, 31–113.

    Google Scholar 

  • Zhabotinski, A. (1964). Periodic process of the oxidation of malonic acid in solution (study of the kinetics of belousov’s reaction). Biofizika 9, 306–311.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, People’s Republic of China

    Tian Ma

  2. Department of Mathematics, Indiana University, Bloomington, IN, USA

    Shouhong Wang

Authors
  1. Tian Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shouhong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Ma, T., Wang, S. (2014). Dynamical Transitions in Chemistry and Biology. In: Phase Transition Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8963-4_6

Download citation

Publish with us

Policies and ethics

Search

Navigation