Dynamical analysis of evolution equations in generalized models

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

Volumn 78, Issue 5, 2013, Pages 1051-1077

  Kuehn, Christian ^a,b   Siegmund, Stefan ^b,c   Gross, Thilo ^a,b  

^a MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

^b Center for Dynamics Dresden   (Germany)

^c DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Bifurcations; Evolution equations; Generalized models; Scaling transformation

Indexed keywords

ANALYTICAL COMPUTATIONS; ANALYTICAL INVESTIGATIONS; DYNAMICAL ANALYSIS; EVOLUTION EQUATIONS; GENERALIZED MODELS; MATHEMATICAL LITERATURE; MATHEMATICAL RESEARCH; SCALING TRANSFORMATION;

BIFURCATION (MATHEMATICS); MATHEMATICAL TECHNIQUES;

DIFFERENTIAL EQUATIONS;

EID: 84880703578     PISSN: 02724960     EISSN: 14643634     Source Type: Journal    
DOI: 10.1093/imamat/hxs008     Document Type: Article

References (68)

1. ARNOLD, L. (2003) Random Dynamical Systems. Berlin Heidelberg, Germany: Springer.
2. ARNOLD, V. I. (1983) GeometricalMethods in the Theory of Ordinary Differential Equations. New York: Springer.
3. AUFDERHEIDE, H., RUDOLF, L. & GROSS, T. (2012) Mesoscale symmetries explain dynamical equivalence of networks (in preparation).
4. BARAI, S. V. & REICH, Y. (1999) Ensemble modelling or selecting the best model: many could be better than one. Artif. Intell. Eng. Design Anal. Manufact., 13, 377-386.
5. BAURMANN, M., GROSS, T. & FEUDEL, U. (2007) Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighbourhood of Turing-Hopf bifurcations. J. Theor. Biol., 245, 220-229.
6. DE KROON, H., PLAISIER, A., VAN GROENENDAEL, J. & CASWELL, H. (1986) Elasticity: the relative contribution of demogrpahic parameters to population growth rate. Ecology, 67, 1427-1431.
7. DUMORTIER, F. (1978) Singularities of Vector Fields. Rio de Janeiro, Brazil: Instituto Nacional de Mathematica Pura e Aplicada.
8. DUMORTIER, F. (1993) Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations. Bifurcations and Periodic Orbits of Vector Fields (D. Schlomiuk ed.). Dortrecht, The Netherlands: Kluwer, pp. 19-73.
9. ENGELBORGHS, K., LUZYANINA, T. & SAMAEY, G. (2000) DDE-BIFTOOL v. 2.00: a Matlab Package for Bifurcation Analysis of Delay Differential Equations. KU Leuven.
10. ERNEUX, T. (2009) Applied Delay Differential Equations. New York: Springer.
11. FELL, D. A. (1992) Metabolic control analysis: a survey of its theoretical and experimental development. Biochem. J., 286, 313-330.
12. FRIEDMAN, M. (2007) Price Theory. New Brunswick, NJ: Transaction Publishers.
13. GARDINER, C. (2009) Stochastic Methods, 4th edn. Berlin Heidelberg, Germany: Springer.
14. GEHRMANN, E. & DROSSEL, B. (2010) Boolean versus continuous dynamics on simple two-gene modules. Phys. Rev. E, 82, 046120.
15. GOLUB, G. H. & VAN LOAN, C. (1996) Matrix Computations. Baltimore, MD: Johns Hopkins University Press.
16. GOVAERTS, W. F. (1987) Numerical Methods for Bifurcations of Dynamical Equilibria. Philadelphia, PA: SIAM.
17. GROSS, T. (2004) Population dynamics: general results from local analysis. Ph.D. Thesis, Universität Oldenburg, Oldenburg, Germany.
18. GROSS, T., BAURMANN, M., FEUDEL, U. & BLASIUS, B. (2006) Generalized models-a new tool for the investigation of ecological systems. Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics (B. Blasius, J. Kurths, & L. Stone, eds), Singapore: World Scientific, pp. 21-48.
19. GROSS, T., EBENHÖH, W. & FEUDEL, U. (2004) Enrichment and foodchain stability: the impact of different functional forms. J. Theor. Biol., 227, 349-358.
20. GROSS, T., EBENHÖH, W. & FEUDEL, U. (2005) Long food chains are in general chaotic. Oikos, 109, 133-155.
21. GROSS, T. & FEUDEL, U. (2004) Analytical search for bifurcation surfaces in parameter space. Physica D, 195, 292-302.
22. GROSS, T. & FEUDEL, U. (2006) Generalized models as an universal approach to the analysis of nonlinear dynamical systems. Phys. Rev. E, 73, 016205-14.
23. GROSS, T. & FEUDEL, U. (2009) Local dynamical equivalence of certain food webs. Ocean Dyn., 59, 417-427.
24. GROSS, T., RUDOLF, L., LEVIN, S. A. & DIECKMANN, U. (2009) Generalized models reveal stabilizing factors in food webs. Science, 325, 747-750.
25. GUCKENHEIMER, J. (1998) Computer simulation and beyond-for the 21st century. Notices Am. Math. Soc., 45, 1120-1123.
26. GUCKENHEIMER, J. & HOLMES, P. (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer.
27. GUCKENHEIMER, J. & MYERS, M. (1996) Computing Hopf Bifurcations II. SIAM J. Sci. Comput., 17, 1275-1301.
28. GUCKENHEIMER, J., MYERS, M. & STURMFELS, B. (1997) Computing Hopf Bifurcations I. SIAM J. Numer. Anal., 34, 1-21.
29. HALE, J. K. (2009) Ordinary Differential Equations. New York: Dover.
30. HALE, J. K. & VERDUYN LUNEL, S. M. (1993) Introduction to Functional Differential Equations. New York: Springer.
31. HARTMAN, P. (2002) Ordinary Differential Equations, 2nd edn. Philadelphia, PA: SIAM.
32. HEINRICH, R., RAPOPORT, S. M. & RAPOPORT, T. A. (1977) Metabolic regulation and mathematical models. Prog. Biophys. Mol. Biol., 32, 1-82.
33. HEINRICH, R. & SCHUSTER, S. (1996) The Regulation of Cellular Systems. New York: Chapman & Hall.
34. HENRY, D. (1981) Geometric Theory of Semilinear Parabolic Equations. Berlin Heidelberg, Germany: Springer.
35. HEPPELL, S. S., CASWELL, H. & CROWDER, L. B. (2000) Life histories and elasticity patterns: perturbation analysis for species with minimal demographic data. Ecology, 81, 654-665.
36. ḦOFENER, J. M., SETHIA, G. C. & GROSS, T. (2011) Stability of networks of delay-coupled delay oscillators. Europhys. Lett., 95, 40002.
37. HORSTHEMKE, W. & LEFEVER, R. (2006) Noise-Induced Transitions. Berlin Heidelberg, Germany: Springer.
38. JOST, J. (2006) Partial Differential Equations. Berlin Heidelberg, Germany: Springer.
39. KACSER, H. & BURNS, J. A. (1973) The control of flux. Rate Control of Biological Processes. London: CUP, pp. 65-104.
40. KALLENBERG, O. (2002) Foundations of Modern Probability, 2nd edn. New York: Springer.
41. KARRASCH, D. (2009) Ü ber Verallgemeinerte Modelle. Diplomarbeit, Dresden, Germany: TU Dresden.
42. KUEHN, C. (2009) Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes. J. Phys. A Math. Theor., 42, 045101-10.
43. KUEHN, C. (2010) From first Lyapunov coefficients to maximal canards. Int. J. Bif. Chaos, 20, 1467-1475.
44. KUEHN, C. & GROSS, T. (2012) Nonlocal generalized models of predator-prey systems. preprint, 1-30, arXiv:1105.3662.
45. KUZNETSOV, YU. A. (2004) Elements of Applied Bifurcation Theory, 3rd edn. New York: Springer.
46. LADE, S. J. & GROSS, T. (2012) Early warning signals for critical transitions: a generalized modeling approach. PLoS Comput. Biol., 8, e1002360-6.
47. LU, Y.-C. (1976) Singularity Theory and an Introduction to Catastrophe Theory. Berlin Heidelberg, Germany: Springer.
48. ØKSENDAL, B. (2003) Stochastic Differential Equations. Springer.
49. ORTH, J. D., THIELE, I. & PALSSON, B. Ø. (2010) What is flux balance analysis? Nat. Biotechnol., 28, 245-248.
50. REDER, C. (1988) Metabolic control theory: a structural approach. J. Theor. Biol., 135, 175-201.
51. REZNIK, E. & SEGŔE, D. (2010) On the stability of metabolic cycles. J. Theor. Biol., 266, 536-549.
52. SALTELLI, A., CHAN, K. & SCOTT, E. M. (2000) Sensitivity Analysis. Chichester: Wiley.
53. SALTELLI, A., TARANTOLA, S., CAMPOLONGO, F. & RATTO, M. (2004) Sensitivity Analysis in Practice. Chichester: Wiley.
54. SAVAGEAU, M. A. & VOIT, E. O. (1987) Recasting nonlinear differential equations as S-systems: a canonical nonlinear form. Math. Biosci., 87, 83-115.
55. SMITH, E. D., SZIDAROVSZKY, F., KARNAVAS, W. J. & BAHIL, A. T. (2008) Sensitivity analysis, a powerful system validation technique. Open Cybernet. Syst. J., 2, 39-56.
56. STEUER, R., GROSS, T., SELBIG, J. & BLASIUS, B. (2006) Structural kinetic modeling of metabolic networks. Proc. Natl. Acad. Sci. USA, 103, 11868-11873.
57. STEUER, R., NUNES NESI, A., FERNIE, A. R., GROSS, T., BLASIUS, B. & SELBIG, J. (2007) From structure to dynamics of metabolic pathways. Bioinformatics, 23, 1378-1385.
58. STIEFS, D. (2009) Relating generalized and specific modeling in population dynamical systems. Ph.D. Thesis, Universität Oldenburg, Oldenburg, Germany.
59. STIEFS, D., GROSS, T., STEUER, R. & FEUDEL, U. (2008) Computation and visualization of bifurcation surfaces. Int. J. Bif. Chaos, 18, 2191-2206.
60. STIEFS, D., VAN VOORN, G. A. K., KOOI, B. W., FEUDEL, U. & GROSS, T. (2010) Food quality in producergrazer models. Am. Nat., 176, 367-380.
61. STIEFS, D., VENTURINO, E. & FEUDEL, U. (2009) Evidence of chaos in eco-epidemic models. Math. Biosci. Eng., 6, 855-871.
62. VAN VOORN, G. A. K., STIEFS, D., GROSS, T., KOOI, B. W., FEUDEL, U. & KOOIJMAN, S. A. L. M. (2008) Stabilization due to predator interference. Math. Biosci. Eng., 5, 567-583.
63. VOIT, E. O. & SAVAGEAU, M. A. (1986) Equivalence between S-systems and Volterra systems. Math. Biosci., 78, 47-55.
64. WIGGINS, S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. New York: Springer.
65. YEAKEL, J. D., STIEFS, D., NOVAK, M. & GROSS, T. (2011) Generalized modeling of ecological population dynamics. Theor. Ecol., 4, 179-194.
66. ZUMSANDE, M. & GROSS, T. (2010) Bifurcations and chaos in the MAPK signaling cascade. J. Theor. Biol., 265, 481-491.
67. ZUMSANDE, M. (2012) Extension of generalized modeling and applications to problems from cell biology, Ph.D. Thesis, TU Dresden, Dresden, Germany.
68. ZUMSANDE, M., STIEFS, D., SIEGMUND, S. & GROSS, T. (2011) General analysis of mathematical models for bone remodeling. Bone, 48, 910-917.