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Distribution of chaos and periodic spikes in a three-cell population model of cancer

Auto-organization of oscillatory phases in parameter planes

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Abstract

We study complex oscillations generated by the de Pillis-Radunskaya model of cancer growth, a model including interactions between tumor cells, healthy cells, and activated immune system cells. We report a wide-ranging systematic numerical classification of the oscillatory states and of their relative abundance. The dynamical states of the cell populations are characterized here by two independent and complementary types of stability diagrams: Lyapunov and isospike diagrams. The model is found to display stability phases organized regularly in old and new ways: Apart from the familiar spirals of stability, it displays exceptionally long zig-zag networks and intermixed cascades of two- and three-doubling flanked stability islands previously detected only in feedback systems with delay. In addition, we also characterize the interplay between continuous spike-adding and spike-doubling mechanisms responsible for the unbounded complexification of periodic wave patterns. This article is dedicated to Prof. Hans Jürgen Herrmann on the occasion of his 60th birthday.

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References

  1. K. Araki, S. Saji, M.R. Gallas, M. Pegram, Y. Sasaki, Breast Cancer 19, 95 (2012)

    Article  Google Scholar 

  2. J. Liao, M.R. Gallas, M. Pegram, J. Slingerland, Breast Cancer (Dove Med Press) 2, 79 (2010)

  3. H. Haken, Phys. Lett. A 53, 77 (1975)

    Article  ADS  Google Scholar 

  4. S. Ayadi, O. Haeberlé, Central European J. Phys. 12, 203 (2014)

    Article  ADS  Google Scholar 

  5. M. Chaplain, J. Math. Biol. 58, 481 (2009)

    Article  MathSciNet  Google Scholar 

  6. R. Bruinsma, J.F. Joanny, J.A. Käs, Editors, Focus issue on the Physics of Cancer, New J. Phys. (2014)

  7. L.G. de Pillis, A. Radunskaya, Math. Comp. Modelling 37, 1221 (2003)

    Article  MATH  Google Scholar 

  8. V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson, Bull. Math. Bio. 56, 295 (1994)

    Article  MATH  Google Scholar 

  9. M. Itik, S.P. Banks, Int. J. Bif. Chaos 20, 71 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Duarte, C. Januario, C. Rodrigues, J. Sardanyes, Int. J. Bif. Chaos 23, 1350124 (2013)

    Article  MathSciNet  Google Scholar 

  11. C. Letellier, F. Denis, L.A. Aguirre, J. Theor. Biol. 322, 7 (2013)

    Article  MathSciNet  Google Scholar 

  12. A.G. Lopez, J. Sabuco, J.M. Seoane, J. Duarte, C. Januario, M.A.F. Sanjuan, J. Theor. Biol. 249, 74 (2014)

    Article  MathSciNet  Google Scholar 

  13. F. Ercole, S. Rinaldi, Analysis of Evolutionary Processes (Princeton University Press, Princeton, 2008)

  14. Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems (World Scientific, Singapore, 1996)

  15. J.G. Freire, J.A.C. Gallas, Phys. Chem. Chem. Phys. 13, 12191 (2011)

    Article  Google Scholar 

  16. J.G. Freire, J.A.C. Gallas, Phys. Lett. A 375, 1097 (2011)

    Article  MATH  ADS  Google Scholar 

  17. M.A. Nascimento, J.A.C. Gallas, H. Varela, Phys. Chem. Chem. Phys. 13, 441 (2011)

    Article  Google Scholar 

  18. J.G. Freire, T. Pöschel, J.A.C. Gallas, Europhys. Lett. 100, 48002 (2012)

    Article  ADS  Google Scholar 

  19. S.L.T. Souza, A.A. Lima, I.R. Caldas, R.O. Medrano-T, Z.O. Guimaães-Filho, Phys. Lett. A 376, 1290 (2012)

    Article  MATH  ADS  Google Scholar 

  20. A. Hoff, D.T. da Silva, C. Manchein, H.A. Albuquerque, Phys. Lett. A 378, 171 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  21. C. Obcemea, Chaotic Dynamics of Tumor Growth, Regeneration, Chapter 34, in Unifying Themes in Complex Systems, edited by A.A. Minai, Y. Bar-Yam (Springer, New York, 2006)

  22. Z. Bajzer, S. Vuk-Pavlovic, M. Huzak, Mathematical Modeling of Tumor Growth Kinetics, Chapter 3, in A Survey of Models for Tumor-Immune System Dynamics, edited by J.A. Adams, N. Bellomo (Birkhäuser, Boston, 1997)

  23. J.G. Freire, R.J. Field, J.A.C. Gallas, J. Chem. Phys. 131, 044105 (2009)

    Article  ADS  Google Scholar 

  24. A. Sack, J.G. Freire, E. Lindberg, T. Pöschel, J.A.C. Gallas, Nature Sci. Rep. 3, 3350 (2013)

    ADS  Google Scholar 

  25. R. Kautz, Chaos: The Science of Predictable Random Motion (Oxford University Press, Oxford, 2011)

  26. T. Tél, M. Gruiz, Chaotic Dynamics: An Introduction Based on Classical Mechanics (Cambridge University Press, Cambridge, 2006)

  27. C. Bonatto, J.A.C. Gallas, Phys. Rev. Lett. 101, 054101 (2008)

    Article  ADS  Google Scholar 

  28. J.G. Freire, J.A.C. Gallas, Phys. Rev. E 82, 037202 (2010)

    Article  ADS  Google Scholar 

  29. J.A.C. Gallas, Int. J. Bifurc. Chaos 20, 197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Vitolo, P. Glendinning, J.A.C. Gallas, Phys. Rev. E 84, 016216 (2011)

    Article  ADS  Google Scholar 

  31. R. Barrio, F. Blesa, S. Serrano, A. Shilnikov, Phys. Rev. E 84, 035201 (2011)

    Article  ADS  Google Scholar 

  32. R. Barrio, A. Shilnikov, L.P. Shilnikov, Int. J. Bif. Chaos 22, 1230016 (2011)

    Article  MathSciNet  Google Scholar 

  33. R. Stoop, P. Banner, Y. Uwate, Phys. Rev. Lett. 105, 074102 (2010)

    Article  ADS  Google Scholar 

  34. C. Bonatto, J.C. Garreau, J.A.C. Gallas, Phys. Rev. Lett. 95, 143905 (2005)

    Article  ADS  Google Scholar 

  35. C. Bonatto, J.A.C. Gallas, Y. Ueda, Phys. Rev. E 77, 026217 (2008)

    Article  ADS  Google Scholar 

  36. L. Junges, J.A.C. Gallas, Phys. Lett. A 376, 2109 (2012)

    Article  MATH  ADS  Google Scholar 

  37. L. Junges, J.A.C. Gallas, Opt. Comm. 285, 4500 (2012)

    Article  ADS  Google Scholar 

  38. L. Junges, T. Pöschel, J.A.C. Gallas, Eur. Phys. J. D 67, 149 (2013)

    Article  ADS  Google Scholar 

  39. H.A. Albuquerque, P.C. Rech, Int. J. Circuit Theory Appl. 40, 189 (2012)

    Article  Google Scholar 

  40. C. Cabeza, C.A. Briozzo, R. Garcia, J.G. Freire, A. Marti, J.A.C. Gallas, Chaos Sol. Frac. 52, 59 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  41. R.E. Francke, T. Pöschel, J.A.C. Gallas, Phys. Rev. E 87, 042907 (2013)

    Article  ADS  Google Scholar 

  42. V. Kovanis, A. Gavrielides, J.A.C. Gallas, Eur. Phys. J. D 58, 181 (2010)

    Article  ADS  Google Scholar 

  43. J.G. Freire, C. Cabeza, A. Marti, T. Pöschel, J.A.C. Gallas, Nature Sci. Rep. 3, 1958 (2013)

    ADS  Google Scholar 

  44. C. Stegemann, P.C. Rech, Int. J. Bif. Chaos 24, 1450023 (2014)

    Article  Google Scholar 

  45. E.N. Lorenz, Physica D 237, 1689 (2008)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  46. W. Façanha, B. Oldeman, L. Glass, Phys. Lett. A 377, 1264 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  47. J.A.C. Gallas, Appl. Phys. B 60, S-203 (1995)

    Article  Google Scholar 

  48. J.A.C. Gallas, Physica A 202, 196 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  49. J.A.C. Gallas, Phys. Rev. Lett. 70, 2714 (1993)

    Article  ADS  Google Scholar 

  50. Handbook of Chaos Control, edited by E. Schöll, H.G. Schuster (Wiley-VCH, Weinheim, 2007)

  51. Introduction to Control of Oscillations and Chaos, edited by A.L. Fradkov, A.Yu. Pogromsky (World Scientific, Singapore, 1999)

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Authors and Affiliations

  1. Center for Research & Grants, Baptist Health South Florida, Miami, FL, 33143, USA

    Michelle R. Gallas

  2. Departamento de Física, Universidade Federal da Paraíba, 58051-970, João Pessoa, Brazil

    Marcia R. Gallas & Jason A.C. Gallas

  3. Instituto de Altos Estudos da Paraíba, Rua Infante Dom Henrique 100–1801, 58039-150, João Pessoa, Brazil

    Marcia R. Gallas & Jason A.C. Gallas

  4. Institute for Multiscale Simulations, Friedrich-Alexander-Universität, 91052, Erlangen, Germany

    Marcia R. Gallas & Jason A.C. Gallas

  5. Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187, Dresden, Germany

    Marcia R. Gallas & Jason A.C. Gallas

  6. Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London, SW7 2AZ, UK

    Marcia R. Gallas & Jason A.C. Gallas

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  1. Michelle R. Gallas
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  2. Marcia R. Gallas
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  3. Jason A.C. Gallas
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R. Gallas, M., R. Gallas, M. & A.C. Gallas, J. Distribution of chaos and periodic spikes in a three-cell population model of cancer. Eur. Phys. J. Spec. Top. 223, 2131–2144 (2014). https://doi.org/10.1140/epjst/e2014-02254-3

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  • DOI: https://doi.org/10.1140/epjst/e2014-02254-3

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