Mathematics in Computer Science

, Volume 5, Issue 3, pp 247–262 | Cite as

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Article

Abstract

This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Gröbner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.

Keywords

Discrete dynamical system Finite dynamical system Stability Bifurcation Algebraic method Symbolic computation 

Mathematics Subject Classification (2000)

9208 92B05 

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References

  1. 1.
    Albert R., Othmer H.G.: The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. J. Theor. Biol. 223(1), 1–18 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Allen L.J.: Some discrete-time SI, SIR, and SIS epidemic models. Math. Biosci. 124(1), 83–105 (1994)CrossRefMATHGoogle Scholar
  3. 3.
    Bächler, T., Gerdt, V., Lange-Hegermann, M., Robertz, D.: Thomas decomposition of algebraic and differential systems. In: Proceedings of the 12th International Workshop, Computer Algebra in Scientific Computing (Tsakhkadzor, Armenia, September 6–12, 2010). LNCS 6244, pp. 31–54. Springer, Berlin (2010)Google Scholar
  4. 4.
    Bosma W., Cannon J., Playoust C.: The Magma algebra system I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P.-E.: Applying a rigorous quasi-steady state approximation method for proving the absence of oscillations in models of genetic circuits. In: Proceedings of the 3rd International Conference on Algebraic Biology (Hagenberg, Austria, July 31–August 2, 2008). LNCS 5147, pp. 56–64. Springer, Berlin (2008)Google Scholar
  6. 6.
    Brown, C.W., Strzeboński, A.: Black-box/white-box simplification and applications to quantifier elimination. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (Munich, Germany, July 25–28, 2010), pp. 69–76. ACM, New York (2010)Google Scholar
  7. 7.
    Buchberger B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Multidimensional Systems Theory, pp. 184–232. Reidel, Dodrecht (1985)CrossRefGoogle Scholar
  8. 8.
    Chen, C., Davenport, J.H., May, J.P., Moreno Maza, M., Xia, B., Xiao, R.: Triangular decomposition of semi-algebraic systems. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (Munich, Germany, July 25–28, 2010), pp. 187–194. ACM Press, New York (2010)Google Scholar
  9. 9.
    Chen, C., Golubitsky, O., Lemaire, F., Moreno Maza, M., Pan, W.: Comprehensive triangular decomposition. In: Proceedings of the 10th International Workshop on Computer Algebra in Scientific Computing (Bonn, Germany, September 16–20, 2007). LNCS vol. 4770, pp. 73–101. Springer, Berlin (2007)Google Scholar
  10. 10.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: The 2nd GI Conference on Automata Theory and Formal Languages (Kaiserslautern, Germany, May 20–23, 1975), pp. 134–183. Springer, Berlin (1975)Google Scholar
  11. 11.
    Collins G.E., Hong H.: Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput. 12(3), 299–328 (1991)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Dolzmann A., Sturm T.: Redlog: computer algebra meets computer logic. SIGSAM Bull. 31, 2–9 (1997)CrossRefGoogle Scholar
  13. 13.
    El Kahoui M., Otto A.: Stability of disease free equilibria in epidemiological models. Math. Comput. Sci. 2(3), 517–533 (2009)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    El Kahoui M., Weber A.: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. J. Symb. Comput. 30(2), 161–179 (2000)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Elaydi S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005)MATHGoogle Scholar
  16. 16.
    Faugère J.-C.: A new efficient algorithm for computing Gröbner bases (F 4). J. Pure Appl. Algebra 139(1–3), 61–88 (1999)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Fournier-Prunaret D., Lopez-Ruiz R., Taha A.-K.: Route to chaos in three-dimensional maps of logistic type. Grazer Math. Ber. 350, 82–95 (2006)MATHMathSciNetGoogle Scholar
  18. 18.
    Galor O.: Discrete Dynamical Systems. Springer, Berlin (2007)MATHGoogle Scholar
  19. 19.
    Gao, X.-S., Chou, S.-C.: Solving parametric algebraic systems. In: Proceedings of the 1992 International Symposium on Symbolic and Algebraic Computation (Berkeley, CA, USA, July 27–29), pp. 335–341. ACM Press, New York (1992)Google Scholar
  20. 20.
    Gao, X.-S., Huang, Z.: Efficient characteristic set algorithms for equation solving in finite fields. MM Research Preprints, KLMM, Chinese Academy of Sciences, vol. 28, pp. 1–29 (2009)Google Scholar
  21. 21.
    Hong H., Liska R., Steinberg S.: Testing stability by quantifier elimination. J. Symb. Comput. 24(2), 161–187 (1997)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Jury E.I.: Inners and Stability of Dynamic Systems. Wiley, New York (1974)MATHGoogle Scholar
  23. 23.
    Kauffman S.A.: The Origin of Order: Self-Organization and Selection in Evolution. Oxford University Press, New York (1993)Google Scholar
  24. 24.
    Kress R.: Numerical analysis. Graduate Texts in Mathematics, vol. 181. Springer, New York (1998)Google Scholar
  25. 25.
    Laubenbacher R., Stigler B.: A computational algebra approach to the reverse engineering of gene regulatory networks. J. Theor. Biol. 229, 523–537 (2004)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Laubenbacher, R., Sturmfels, B.: Computer algebra in systems biology. Am. Math. Mon. (2009, in press)Google Scholar
  27. 27.
    Lazard D.: A new method for solving algebraic systems of positive dimension. Discrete Appl. Math. 33(1–3), 147–160 (1991)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Lazard D., Rouillier F.: Solving parametric polynomial systems. J. Symb. Comput. 42(6), 636–667 (2007)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Li X., Mou C., Wang D.: Decomposing polynomial sets into simple sets over finite fields: the zero-dimensional case. Comput. Math. Appl. 60(11), 2983–2997 (2010)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Maynard Smith J.: Mathematical Ideas in Biology. Cambridge University Press, Cambridge (1968)CrossRefGoogle Scholar
  31. 31.
    Montes A.: A new algorithm for discussing Gröbner bases with parameters. J. Symb. Comput. 33(2), 183–208 (2002)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Murakami K.: Stability and bifurcation in a discrete-time predator-prey model. J. Differ. Equ. Appl. 13(10), 911–925 (2007)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Niu, W., Wang, D.: Algebraic analysis of bifurcation and limit cycles for biological systems. In: Proceedings of the 3rd International Conference on Algebraic Biology (Hagenberg, Austria, July 31–August 2, 2008). LNCS 5147, pp. 156–171. Springer, Berlin (2008)Google Scholar
  34. 34.
    Niu W., Wang D.: Algebraic approaches to stability analysis of biological systems. Math. Comput. Sci. 1(3), 507–539 (2008)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Oldenbourg R.C., Sartorius H.: The Dynamics of Automatic Controls. The American Society of Mechanical Engineers, New York (1948)Google Scholar
  36. 36.
    Pomerening J., Sontag E., Ferrell J.: Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2. Nat. Cell Biol. 5(4), 346–351 (2003)CrossRefGoogle Scholar
  37. 37.
    Stigler, B., Veliz-Cuba, A.: Network topology as a driver of bistability in the lac operon. arXiv:0807.3995 (2008)Google Scholar
  38. 38.
    Strzeboński, A.: Computation with semialgebraic sets represented by cylindrical algebraic formulas. In: Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (Munich, Germany, July 25–28, 2010), pp. 61–68. ACM Press, New York (2010)Google Scholar
  39. 39.
    Sturm T., Weber A., Abdel-Rahman E.O., El Kahoui M.: Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Math. Comput. Sci. 2(3), 493–515 (2009)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Suzuki, A., Sato, Y.: A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In: Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation (Genova, Italy, July 9–12, 2006), pp. 326–331. ACM Press, New York (2006)Google Scholar
  41. 41.
    Tamura T., Akutsu T.: Algorithms for singleton attractor detection in planar and nonplanar AND/OR Boolean networks. Math. Comput. Sci. 2(3), 401–420 (2009)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    van Kooten T., de Roos A.M., Persson L.: Bistability and an Allee effect as emergent consequences of stage-specific predation. J. Theor. Biol. 237(1), 67–74 (2005)CrossRefGoogle Scholar
  43. 43.
    Wang D.: Computing triangular systems and regular systems. J. Symb. Comput. 30(2), 221–236 (2000)CrossRefMATHGoogle Scholar
  44. 44.
    Wang D.: Elimination Methods. Springer, Wien (2001)CrossRefMATHGoogle Scholar
  45. 45.
    Wang, D., Xia, B.: Stability analysis of biological systems with real solution classification. In: Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation (Beijing, China, July 24–27, 2005), pp. 354–361. ACM Press, New York (2005)Google Scholar
  46. 46.
    Wang J., Feng G.: Bifurcation and chaos in discrete-time BVP oscillator. Int. J. Non-Linear Mech. 45(6), 608–620 (2010)CrossRefGoogle Scholar
  47. 47.
    Weispfenning V.: Comprehensive Gröbner bases. J. Symb. Comput. 14(1), 1–29 (1992)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Weispfenning, V.: Quantifier elimination for real algebra—the cubic case. In: Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation (Oxford, UK, July 20–22, 1994), pp. 258–263. ACM Press, New York (1994)Google Scholar
  49. 49.
    Weispfenning V.: Quantifier elimination for real algebra—the quadratic case and beyond. Appl. Algebra Eng. Commun. Comput. 8, 85–101 (1997)CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Wen G.: Criterion to identify Hopf bifurcations in maps of arbitrary dimension. Phys. Rev. E 72(2), 026201 (2005)CrossRefMathSciNetGoogle Scholar
  51. 51.
    Wu W.-T.: Mathematics Mechanization. Science Press/Kluwer, Beijing (2000)MATHGoogle Scholar
  52. 52.
    Xia B.: DISCOVERER: a tool for solving semi-algebraic systems. ACM Commun. Comput. Algebra 41(3), 102–103 (2007)CrossRefGoogle Scholar
  53. 53.
    Yang, L., Xia, B.: Real solution classifications of parametric semi-algebraic systems. In: Algorithmic Algebra and Logic—Proceedings of the A3L 2005 (Passau, Germany, April 3–6, 2005), pp. 281–289. Herstellung und Verlag, Norderstedt (2005)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Xiaoliang Li
    • 1
  • Chenqi Mou
    • 1
    • 2
  • Wei Niu
    • 2
  • Dongming Wang
    • 2
  1. 1.LMIB – SKLSDE – School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.Laboratoire d’Informatique de Paris 6Université Pierre et Marie Curie – CNRSParis Cedex 05France

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