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Well-posedness of parabolic equations containing hysteresis with diffusive thresholds

Proceedings of the Steklov Institute of Mathematics volume 283, pages 87–109 (2013)Cite this article

Abstract

We study complex systems arising, in particular, in population dynamics, developmental biology, and bacterial metabolic processes, in which each individual element obeys a relatively simple hysteresis law (a non-ideal relay). Assuming that hysteresis thresholds fluctuate, we consider the arising reaction-diffusion system. In this case, the spatial variable corresponds to the hysteresis threshold. We describe the collective behavior of such a system in terms of the Preisach operator with time-dependent measure which is a part of the solution for the whole system. We prove the well-posedness of the system and discuss the long-term behavior of solutions.

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References

  1. H. W. Alt, “On the thermostat problem,” Control Cybern. 14, 171–193 (1985).

    MathSciNet  Google Scholar 

  2. A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations (Birkhäuser, Basel, 1994).

    Book  Google Scholar 

  3. R. J. Aumann and M. B. Maschler, Repeated Games with Incomplete Information (MIT Press, Cambridge, MA, 1995).

    MATH  Google Scholar 

  4. A. Becskei, B. Séraphin, and L. Serrano, “Positive feedback in eukaryotic gene networks: cell differentiation by graded to binary response conversion,” EMBO J. 20, 2528–2535 (2001).

    Article  Google Scholar 

  5. S. Benzer, “Induced synthesis of enzymes in bacteria analyzed at the cellular level,” Biochim. Biophys. Acta 11, 383–395 (1953).

    Article  Google Scholar 

  6. M. Cohn and K. Horbita, “Inhibition by glucose of the induced synthesis of the β-galactoside-enzyme system of Escherichia coli. Analysis of maintenance,” J. Bacteriol. 78, 601–612 (1959).

    Google Scholar 

  7. M. Cohn and K. Horbita, “Analysis of the differentiation and of the heterogeneity within a population of Eschericia coli undergoing induced β-galactosidase synthesis,” J. Bacteriol. 78, 613–623 (1959).

    Google Scholar 

  8. M. Delbrück, “Discussion,” in Unités biologiques douées de continuité génétique (CNRS, Paris, 1949), pp. 33–35.

    Google Scholar 

  9. D. Dubnau and R. Losick, “Bistability in bacteria,” Mol. Microbiol. 61, 564–572 (2006).

    Article  Google Scholar 

  10. G. Friedman, P. Gurevich, S. McCarthy, and D. Rachinskii, “Switching behaviour of two-phenotype bacteria in varying environment,” J. Phys.: Conf. Ser. (in press).

  11. T. S. Gardner, C. R. Cantor, and J. J. Collins, “Construction of a genetic toggle switch in Escherichia coli,” Nature 403, 339–342 (2000).

    Article  Google Scholar 

  12. P. Gurevich and S. Tikhomirov, “Uniqueness of transverse solutions for reaction-diffusion equations with spatially distributed hysteresis,” Nonlinear Anal., Theory Methods Appl. 75, 6610–6619 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  13. F. C. Hoppensteadt and W. Jäger, “Pattern formation by bacteria,” in Biological Growth and Spread (Springer, Berlin, 1980), Lect. Notes Biomath. 38, pp. 68–81.

    Chapter  Google Scholar 

  14. A. M. Il’in and B. A. Markov, “Nonlinear diffusion equation and Liesegang rings,” Dokl. Akad. Nauk 440(2), 164–167 (2011) [Dokl. Math. 84 (2), 730–733 (2011)].

    MathSciNet  Google Scholar 

  15. J. Kopfová, “Hysteresis in biological models,” J. Phys.: Conf. Ser. 55, 130–134 (2006).

    Google Scholar 

  16. M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis (Springer, Berlin, 1989).

    Book  MATH  Google Scholar 

  17. E. Kussell and S. Lieber, “Phenotypic diversity, population growth, and information in fluctuating environments,” Science 309, 2075–2078 (2005).

    Article  Google Scholar 

  18. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasi-linear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, RI, 1968).

    Google Scholar 

  19. V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1983) [in Russian].

    Google Scholar 

  20. J. Monod, “From enzymatic adaptation to allosteric transitions,” Science 154, 475–483 (1966).

    Article  Google Scholar 

  21. A. Novick and M. Weiner, “Enzyme induction as an all-or-none phenomenon,” Proc. Natl. Acad. Sci. USA 43, 553–566 (1957).

    Article  Google Scholar 

  22. E. M. Ozbudak, M. Thattai, H. N. Lim, B. I. Shraiman, and A. van Oudenaarden, “Multistability in the lactose utilization network of Escherichia coli,” Nature 427, 737–740 (2004).

    Article  Google Scholar 

  23. J. R. Pomerening, E. D. Sontag, and J. E. Ferrell, Jr., “Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2,” Nature Cell Biol. 5, 346–351 (2003).

    Article  Google Scholar 

  24. F. Rothe, Global Solutions of Reaction-Diffusion Systems (Springer, Berlin, 1984).

    MATH  Google Scholar 

  25. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1994).

    Book  MATH  Google Scholar 

  26. S. Spiegelman and W. F. DeLorenzo, “Substrate stabilization of enzyme-forming capacity during the segregation of a heterozygote,” Proc. Natl. Acad. Sci. USA 38, 583–592 (1952).

    Article  Google Scholar 

  27. M. Thattai and A. van Oudenaarden, “Stochastic gene expression in fluctuating environments,” Genetics 167, 523–530 (2004).

    Article  Google Scholar 

  28. A. Visintin, “Evolution problems with hysteresis in the source term,” SIAM J. Math. Anal. 17, 1113–1138 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  29. Ö. Winge and C. Roberts, “Inheritance of enzymatic characters in yeasts, and the phenomenon of long-term adaptation,” C. R. Trav. Lab. Carlsberg, Sér. Physiol. 24, 263–315 (1948).

    Google Scholar 

  30. D. M. Wolf and A. P. Arkin, “Motifs, modules and games in bacteria,” Curr. Opin. Microbiol. 6, 125–134 (2003).

    Article  Google Scholar 

  31. D. M. Wolf, V. V. Vazirani, and A. P. Arkin, “Diversity in times of adversity: probabilistic strategies in microbial survival games,” J. Theor. Biol. 234, 227–253 (2005).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Freie Universität Berlin, Kaiserswerther Str. 16-18, 14195, Berlin, Germany

    Pavel Gurevich

  2. Peoples’ Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198, Russia

    Pavel Gurevich

  3. Department of Mathematical Sciences, University of Texas at Dallas, FO 35, 800 West Campbell Road, Richardson, TX, 75080-3021, USA

    Dmitrii Rachinskii

  4. Department of Applied Mathematics, University College Cork, Western Road, Cork, Ireland

    Dmitrii Rachinskii

Authors
  1. Pavel Gurevich
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  2. Dmitrii Rachinskii
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Correspondence to Pavel Gurevich.

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Gurevich, P., Rachinskii, D. Well-posedness of parabolic equations containing hysteresis with diffusive thresholds. Proc. Steklov Inst. Math. 283, 87–109 (2013). https://doi.org/10.1134/S0081543813080075

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  • DOI: https://doi.org/10.1134/S0081543813080075

Keywords

  • Lactose
  • Parabolic Equation
  • STEKLOV Institute
  • Neumann Boundary Condition
  • Switching Threshold