Validation and calibration of models for reaction-diffusion systems

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 8, Issue 6, 1998, Pages 1163-1182

  Dilão, Rui ^a   Sainhas, Joaquim ^a  

^a INSTITUTO SUPERIOR TÉCNICO   (Portugal)

Author keywords

[No Author keywords available]

Indexed keywords

GOES;

EID: 0032093629     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127498000929     Document Type: Article

References (37)

1. Agladze, K. I. & Krinsky, V. I. [1982] "Multi-armed vortices in an active chemical medium," Nature 296, 424-426.
2. Barkley, D., Kness, M, & Tuckerman, L. S. [1990] "Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation," Phys. Rev. A42, 2489-2492.
3. Boisvert, R. F. [1981] "Families of high order accurate discretization of some elliptic problems," SIAM J. Sci. Stat. Comput. 2, 268-284.
4. Crank, J. [1975] The Mathematics of Diffusion (Oxford Univ. Press, Oxford).
5. Carslaw, H. S., & Jaeger, J. C. [1959] Conduction of Heat in Solids (Oxford Univ. Press, Oxford).
6. Castets, V., Dulos, E., Boissonade, J. & De Kepper, P. [1990] "Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern," Phys. Rev. Lett. 64, 2953-2956.
7. Chandrasekhar, S. [1943] "Stochastic problems in physics and astronomy," Rev. Mod. Phys. 15, 1-89.
8. Cross, M. C. & Hohenberg, P. C. [1993] "Pattern formation outside of equilibrium," Rev. Mod. Phys. 65, 851-1112.
9. Dejak, C., Lalatta, I. M., Messina, E. & Pecenik, G. [1987] "A two-dimensional diffusion model of the Venice lagoon and relative open boundary conditions," Ecol. Model. 37, 21-45.
10. Epstein, I. R. & Showalter, K. [1996] "Nonlinear chemical dynamics: Oscillations, patterns, and chaos," J. Phys. Chem. 100, 13132-13147.
11. Durrett, R. & Griffeath, D. [1993] "Asymptotic behavior of excitable cellular automata," Exp. Math. 2, 183-208.
12. Field, R. J., Körös, E. & Noyes, R. M. [1972] "Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system," J. Amer. Chem. Soc. 94, 3649-3665.
13. Harrison, L. G. [1993] Kinetic Theory of Living Patterns (Cambridge Univ. Press, Cambridge).
14. John, F. [1971] Partial Differential Equations (Springer, Berlin).
15. Keener, P. & Tyson, J. [1992] "The dynamics of scroll waves in excitable media," SIAM Rev. 34, 1-39.
16. Klevecz, R. R., Bolen, J. & Durán, O. [1992] "Self-organization in biological tissues: Analysis of asynchronous and synchronous periodicity, turbulence and synchronous chaos emergent in coupled chaotic arrays," Int. J. Bifurcation and Chaos 2, 941-953.
17. Kock, A. J. & Meinhardt, H. [1994] "Biological pattern formation: From basic mechanisms to complex structures," Rev. Mod. Phys. 66, 1481-1507.
18. Meinhardt, H. [1982] Models of Biological Pattern Formation (Academic Press, London).
19. Markus, M. & Hess, B. [1990] "Isotropic cellular automaton for modelling excitable media," Nature 347, 56-58.
20. Murray, J. D. [1993] Mathematical Biology (Springer, Berlin).
21. Oreskes, N., Shrader-Frechette, K. & Belitz, K. [1994] "Verification, validation, and confirmation of numerical models in the earth sciences," Science 263, 641-646.
22. Pearson, J. E. [1993] "Complex patterns in a simple system," Science 261, 189-192.
23. Press, W. H., Flannery, B. P., Teukolski, S. A. & Vetterling, W. T. [1989] Numerical Recipes (Cambridge Univ. Press, Cambridge).
24. Prigogine, I. & Lefever, R. [1968] "Symmetry breaking instabilities in dissipative systems. II," J. Chem. Phys. 48, 1695-1700.
25. Richtmeyer, R. D. & Morton, K. W. [1967] Difference Methods for Initial-Value Problems (Wiley, New York).
26. Ross, J., Müller, S. C. & Vidal, C. [1988] "Chemical waves," Science 240, 460-465.
27. Ruuth, S. J. [1995] "Implicit-explicit methods for reaction-diffusion problems in pattern formation," J. Math. Biol. 34, 148-176.
28. Sewell, G. [1988] The Numerical Solutions of Ordinary and Partial Differential Equations (Academic Press, London).
29. Smith, G. D. [1985] Numerical Solution of Partial Differential Equations (Oxford Univ. Press, Oxford).
30. Steinbock, O., Siegert, F., Müller, S. & Weijer, C. J. [1993] "Three-dimensional waves of excitation during Dictyostelium morphogenesis," Proc. Natl. Acad. Sci. USA 90, 7332-7335.
31. Tilden, J., [1974] "On the velocity of spatial wave propagation in the Belousov reaction," J. Chem. Phys. 60, 3349-3350.
32. Tyson, J. J., Alexander, K. A., Manoranjan, V. S. & Murray, J. D. [1989] "Spiral waves of cyclic AMP in a model of slime mold aggregation," Physica D34, 193-207.
33. Turing, A. M. [1952] "The chemical basis of morphogenesis," Philo. Trans. Roy. Soc. London Ser. B237, 5-72.
34. Winfree, A. T. [1990] "Stable particle-like solutions to the nonlinear wave equations of three-dimensional excitable media," SIAM Rev. 32, 1-53.
35. Weimar, J. R., Tyson, J. J. & Watson, L. T. [1992] "Diffusion and propagation in cellular automaton models of excitable media," Physica D55, 309-327.
36. Zaikin, A. N. & Zhabotinsky, A. M. [1970] "Concentration wave propagation in two-dimensional liquid-phase self-oscillating system," Nature 225, 535-537.
37. Zhabotinsky, A. M. & Zaikin, A. N. [1973] "Autowave processes in a distributed chemical system," J. Theor. Biol. 40, 45-61.