Ricci flow and nonlinear reaction-diffusion systems in biology, chemistry, and physics. (English) Zbl 1402.92176
Summary: This paper proposes the Ricci-flow equation from Riemannian geometry as a general geometric framework for various nonlinear reaction-diffusion systems (and related dissipative solitons) in mathematical biology. More precisely, we propose a conjecture that any kind of reaction-diffusion processes in biology, chemistry, and physics can be modeled by the combined geometric-diffusion system. In order to demonstrate the validity of this hypothesis, we review a number of popular nonlinear reaction-diffusion systems and try to show that they can all be subsumed by the presented geometric framework of the Ricci flow.
MSC:
| 92C40 | Biochemistry, molecular biology |
| 92E20 | Classical flows, reactions, etc. in chemistry |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 53C80 | Applications of global differential geometry to the sciences |
| 35K57 | Reaction-diffusion equations |
Keywords:
geometrical Ricci flow; nonlinear bio-reaction-diffusion; dissipative solitons and breathersReferences:
| [1] | Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Method for solving the sine-Gordon equation. Phys. Rev. Let. 30, 1262–1264 (1973) · doi:10.1103/PhysRevLett.30.1262 |
| [2] | Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: First-order exact solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72, 809–818 (1987) · Zbl 0656.35135 · doi:10.1007/BF01017105 |
| [3] | Amari, S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern. 27, 77–87 (1977) · Zbl 0367.92005 · doi:10.1007/BF00337259 |
| [4] | Anderson, M.T.: Geometrization of 3-manifolds via the Ricci flow. Not. Am. Math. Soc. 51(2), 184–193 (2004) · Zbl 1161.53350 |
| [5] | Barkley, D.: A model for fast computer-simulation of waves in excitable media. Physica D 49, 61–70 (1991) · doi:10.1016/0167-2789(91)90194-E |
| [6] | Bode, M.: Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions. Physica D 106, 270–286 (1997) · Zbl 0930.92002 · doi:10.1016/S0167-2789(97)00050-X |
| [7] | Cao, H.D., Chow, B.: Recent developments on the Ricci flow. Bull. Am. Math. Soc. 36, 59–74 (1999) · Zbl 0926.53016 · doi:10.1090/S0273-0979-99-00773-9 |
| [8] | Chow, B., Knopf, D.: The Ricci Flow: An Introduction. Mathematical Surveys and Monographs. AMS, Providence (2004) · Zbl 1086.53085 |
| [9] | Conway, J.M., Riecke, H.: Superlattice patterns in the complex Ginzburg-Landau equation with multi-resonant forcing. arXiv: 0803.0346 [nlin.PS] (2008) · Zbl 1183.35257 |
| [10] | Field, R.J.: Oregonator. Scholarpedia 2(5), 1386 (2007) · doi:10.4249/scholarpedia.1386 |
| [11] | Field, R.J., Körös, E., Noyes, R.M.: Oscillations in chemical systems. J. Am. Chem. Soc. 94, 8649–8664 (1972) · doi:10.1021/ja00780a001 |
| [12] | FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961) · doi:10.1016/S0006-3495(61)86902-6 |
| [13] | Friedrich, R.: Group theoretic methods in the theory of pattern formation. In: Collective Dynamics of Nonlinear and Disordered Systems. Springer, Berlin (2004) |
| [14] | Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972) · Zbl 0297.92007 · doi:10.1007/BF00289234 |
| [15] | Gurevich, S.V., Amiranashvili, S., Purwins, H.-G.: Breathing dissipative solitons in three-component reaction-diffusion system. Phys. Rev. E 74, 066201 (2006) |
| [16] | Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982) · Zbl 0504.53034 |
| [17] | Hamilton, R.S.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986) · Zbl 0628.53042 |
| [18] | Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988) · Zbl 0663.53031 · doi:10.1090/conm/071/954419 |
| [19] | Hamilton, R.S.: The Harnack estimate for the Ricci flow. J. Differ. Geom. 37, 225–243 (1993) · Zbl 0804.53023 |
| [20] | Hamilton, R.S.: Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7(4), 695–729 (1999) · Zbl 0939.53024 |
| [21] | Hawking, S., Penrose, R.: The Nature of Space and Time. Princeton University Press, Princeton (1996) · Zbl 0962.83500 |
| [22] | Hsu, S.Y.: Some results for the Perelman LYH-type inequality. arXiv: 0801.3506 [math.DG] (2008) |
| [23] | Ivancevic, V.: Symplectic rotational geometry in human biomechanics. SIAM Rev. 46(3), 455–474 (2004) · Zbl 1063.92003 · doi:10.1137/S003614450341313X |
| [24] | Ivancevic, V., Ivancevic, T.: Natural Biodynamics. World Scientific, Singapore (2006) · Zbl 1120.92002 |
| [25] | Ivancevic, V., Ivancevic, T.: Geometrical Dynamics of Complex Systems. Springer, Berlin (2006) · Zbl 1092.53001 |
| [26] | Ivancevic, V., Ivancevic, T.: High-Dimensional Chaotic and Attractor Systems. Springer, Berlin (2006) · Zbl 1092.53001 |
| [27] | Ivancevic, V., Ivancevic, T.: Complex Dynamics: Advanced System Dynamics in Complex Variables. Springer, Berlin (2007) · Zbl 1134.81002 |
| [28] | Ivancevic, V., Ivancevic, T.: Applied Differential Geometry: A Modern Introduction. World Scientific, Singapore (2007) · Zbl 1126.53001 |
| [29] | Ivancevic, V., Ivancevic, T.: Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling. Springer, Berlin (2007) · Zbl 1193.91131 |
| [30] | Ivancevic, V., Ivancevic, T.: Computational Mind: A Complex Dynamics Perspective. Springer, Berlin (2007) · Zbl 1131.68497 |
| [31] | Ivancevic, V., Ivancevic, T.: Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals. Springer, Berlin (2008) · Zbl 1152.37002 |
| [32] | Ivancevic, V., Ivancevic, T.: Quantum Leap: From Dirac and Feynman, Across the Universe, to Human Body and Mind. World Scientific, Singapore (2008) · Zbl 1417.81005 |
| [33] | Ivancevic, T., Jain, L., Pattison, J., Hariz, A.: Nonlinear dynamics and chaos methods in neurodynamics and complex data analysis. Nonlinear Dyn. 56(1–2), 23–44 (2009) · Zbl 1170.92005 · doi:10.1007/s11071-008-9376-9 |
| [34] | Kaminaga, A., Vanag, V.K., Epstein, I.R.: ”Black spots” in a surfactant-rich Belousov-Zhabotinsky reaction dispersed in a water-in-oil microemulsion system. J. Chem. Phys. 122, 174706 (2005) |
| [35] | Kaminaga, A., Vanag, V.K., Epstein, I.R.: A reaction-diffusion memory device. Angew. Chem. 45, 3087 (2006) · doi:10.1002/anie.200600400 |
| [36] | Kohler, G., Milstein, C.: Continuous cultures of fused cells secreting antibody of predefined specificity. Nature 256, 495 (1975) · doi:10.1038/256495a0 |
| [37] | Kolokolnikov, T., Tlidi, M.: Spot deformation and replication in the two-dimensional Belousov-Zhabotinsky reaction in water-in-oil microemulsion. Phys. Rev. Lett. 98, 188303 (2007) |
| [38] | Kunz-Schughart, L.A.: Multicellular tumor spheroids: intermediates between monolayer culture and in vivo tumor. Cell Biol. Int. 23(3), 157–161 (1999) · doi:10.1006/cbir.1999.0384 |
| [39] | Li, J.: First variation of the Log Entropy functional along the Ricci flow. arXiv: 0712.0832 [math.DG] (2007) |
| [40] | Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986) · Zbl 0611.58045 · doi:10.1007/BF02399203 |
| [41] | Mackenzie, D.: Perelman declines math’s top prize; three others honored in Madrid. Science 313, 1027 (2006) |
| [42] | Meinhardt, H.: Gierer-Meinhardt model. Scholarpedia 1(12), 1418 (2006) · doi:10.4249/scholarpedia.1418 |
| [43] | Milnor, J.: Towards the Poincaré conjecture and the classification of 3-manifolds. Not. Am. Math. Soc. 50(10), 1226–1233 (2003) · Zbl 1168.57303 |
| [44] | Misner, C., Thorne, K., Wheeler, J.A.: Gravitation. Freeman, New York (1973) |
| [45] | Missel, A.R., Dahmen, K.A.: Hopping conduction and bacteria: transport in disordered reaction-diffusion systems. Phys. Rev. Let. 100, 058301 (2007) |
| [46] | Morgan, S.W., Biktasheva, I.V., Biktashev, V.N.: Control of scroll wave turbulence using resonant perturbations. arXiv: 0806.2262 [nlinPS] (2008) |
| [47] | Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962) · doi:10.1109/JRPROC.1962.288235 |
| [48] | Nelson, D.R., Shnerb, N.M.: Non-hermitian localization and population biology. Phys. Rev. E 58, 1383–1403 (1998) · doi:10.1103/PhysRevE.58.1383 |
| [49] | Nielsen, K., Hynne, F., Sorensen, P.G.: Hopf bifurcation in chemical kinetics. J. Chem. Phys. 94, 1020–1029 (1991) · doi:10.1063/1.460057 |
| [50] | Penrose, R.: Singularities and time-asymmetry. In: Hawking, S., Israel, W. (eds.) General Relativity: An Einstein Centenary Survey, pp. 581–638. Cambridge University Press, Cambridge (1979) |
| [51] | Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv: math/0211159 [math.DG] (2002) · Zbl 1130.53001 |
| [52] | Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv: math/0303109 [math.DG] (2003) · Zbl 1130.53002 |
| [53] | Preziosi, L.: Cancer Modeling and Simulation. CRC Press, Boca Raton (2003) · Zbl 1039.92022 |
| [54] | Prigogine, I.: From Being to Becoming: Time and Complexity in the Physical Sciences. Freeman, New York (1980) |
| [55] | Purwins, H.-G., Bodeker, H.U., Liehr, A.W.: Dissipative Solitons in Reaction-Diffusion Systems. In: Akhmediev, N., Ankiewicz, A. (eds.) Dissipative Solitons. Lecture Notes in Physics. Springer, Berlin (2005) · Zbl 1080.35112 |
| [56] | Rabinovich, M.I., Ezersky, A.B., Weidman, P.D.: The Dynamics of Patterns. World Scientific, Singapore (2000) · Zbl 1043.37513 |
| [57] | Roose, T., Chapman, S.J., Maini, P.K.: Mathematical models of avascular tumor growth. SIAM Rev. 49(2), 179–208 (2007) · Zbl 1117.93011 · doi:10.1137/S0036144504446291 |
| [58] | Schöner, G.: Dynamical systems approaches to cognition. In: Sun, R. (ed.) Cambridge Handbook of Computational Cognitive Modeling. Cambridge University Press, Cambridge (2007) |
| [59] | Sutherland, R.M.: Cell and environment interactions in tumor microregions: The multicell spheroid model. Science 240, 177–184 (1988) · doi:10.1126/science.2451290 |
| [60] | Thurston, W.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. 6, 357–381 (1982) · Zbl 0496.57005 · doi:10.1090/S0273-0979-1982-15003-0 |
| [61] | Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952) · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012 |
| [62] | Tyson, J.J.: A quantitative account of oscillations, bistability and travelling waves in the Belousov-Zhabotinsky reaction. In: Field, R.J., Burger, M. (eds.) Oscillation and Travelling Waves in Chemical Systems. Wiley, New York (1985) |
| [63] | Yau, S.T.: Structure of three-manifolds–Poincaré and geometrization conjectures. Talk given at the Morningside Center of Mathematics on 20 June 2006 |
| [64] | Ye, R.: The log entropy functional along the Ricci flow. arXiv: 0708.2008v3 [math.DG] (2007) |
| [65] | Zhabotinsky, A.M.: Belousov–Zhabotinsky reaction. Scholarpedia 2(9), 1435 (2007) · doi:10.4249/scholarpedia.1435 |
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