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Reaction transport systems in biological modelling

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References

  1. Alt, W., Biased Random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9 (1980) 147–177

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronson, D.G., The asymptotic speed of propagation of a simple epidemic. In: W. E. Fitzgibbon, H. F. Walker (eds), Nonlinear Diffusion. Pitman Research Notes in Mathematics 14 (1977) 1–23

    Google Scholar 

  3. Aronson, D.G., Weinberger, H.F., Nonlinear diffusion in population genetics, combustion, and nerve propagation. Lect. Notes in Math. 446, p.5–49, Springer Verlag 1975

    Article  MathSciNet  MATH  Google Scholar 

  4. Bartlett, M.S., A note on random walks at constant speed. Adv. Appl. Prob. 10 (1978) 704–707

    Article  MATH  Google Scholar 

  5. Beals, R., Protopopescu, V., Abstract time-dependent transport equations. J. Math. Anal. Appl. 121, (1987) 370–405

    Article  MathSciNet  MATH  Google Scholar 

  6. Berg, H.C., Brown, D.A., Chemotaxis in Escherichia coli analyzed by three-dimensional tracking. In: Antibiotics and Chemotherapy. Vol. 19, Basel, Karger (1974) 55–78

    Google Scholar 

  7. Brayton, R., Miranker, W., A stability theory for nonlinear mixed initial boundary value problems. Arch. Rat. Mech. Anal. 17 (1964) 358–376

    Article  MathSciNet  MATH  Google Scholar 

  8. Britton, N., Reaction-diffusion equations and their application to biology. Academic Press 1986

    Google Scholar 

  9. Broadwell, J.E., Shock structure in a simple velocity gas. Phys. Fluids 7 (1964) 1243–1247

    Article  MATH  Google Scholar 

  10. Cattaneo, C., Sulla conduzione del calore. Atti del Semin. Mat. e Fis. Univ. Modena 3 (1948) 83–101

    MathSciNet  MATH  Google Scholar 

  11. Cercignani, C., The Boltzmann Equation and its Applications. Springer Verlag 1988

    Google Scholar 

  12. Cercignani, C., Illner, R., Pulvirenti, M., The Mathematical Theory of Dilute Gases. Springer Verlag 1994

    Google Scholar 

  13. Crank, J., The Mathematics of Diffusion. 2nd ed., Clarendon Press, Oxford 1975

    MATH  Google Scholar 

  14. Diekmann, O., Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6 (1978), 109–130

    Article  MathSciNet  MATH  Google Scholar 

  15. Dunbar, S., A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math. 48 (1988) 1510–1526

    Article  MathSciNet  MATH  Google Scholar 

  16. Dunbar, S., Othmer, H., On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks. In: H.G. Othmer (ed.) Nonlinear Oscillations in Biology and Chemistry. Lect. Notes in Biomath. 66, Springer Verlag 1986

    Google Scholar 

  17. Einstein, A., Zur Theorie der Brownschen Bewegung. Annalen der Physik 19 (1906) 371–381

    Article  MATH  Google Scholar 

  18. Feireisl, E., Attractors for semilinear damped wave equation on ℝ3. Nonlinear Analysis TMA 23 (1994) 187–195

    Article  MathSciNet  MATH  Google Scholar 

  19. Fife, P.C., Mathematical Aspects of Reacting and Diffusing Systems. Lect. Notes in Biomath. 28, Springer Verlag 1979

    Google Scholar 

  20. Fife, P.C., McLeod, J.B., The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65 (1977) 335–361

    Article  MathSciNet  MATH  Google Scholar 

  21. Fisher, R.A., The advance of advantageous genes. Ann. Eugenics 7 (1937) 355–361

    Article  MATH  Google Scholar 

  22. Fitzgibbon, W.E., Parrot, M.E., Convergence of singularly perturbed Hodgkin-Huxley systems. J. Nonlin. Anal. TMA 22 (1994) 363–379

    Article  MathSciNet  MATH  Google Scholar 

  23. Fitzgibbon, W.E., Parrot, M.E., Convergence of singular perturbations of strongly damped nonlinear wave equations. J. Nonl. Anal. TMA 28 (1997) 165–174

    Article  MathSciNet  MATH  Google Scholar 

  24. Friedman, A., Bei Hu, The Stefan problem for a hyperbolic heat equation. J. Math. Anal. Appl. 138 (1989) 249–279

    Article  MathSciNet  MATH  Google Scholar 

  25. Fürth, R., Die Brownsche Bewegung bei Berücksichtigung einer Persistenz der Bewegungsrichtung. Zeitschr. für Physik 2 (1920) 244–256

    Article  Google Scholar 

  26. Fusco, D., Manganaro, N., A method for finding exact solutions to hyperbolic systems of first-order PDEs. IMA J. Appl. Math. 57 (1996), 223–242

    Article  MathSciNet  MATH  Google Scholar 

  27. Gierer, A., Meinhardt, H., A theory of biological pattern formation. Kybernetik 12 (1972) 30–39

    Article  MATH  Google Scholar 

  28. Glansdorff, P., Prigogine, I., Thermodynamic Theory of Structure, Stability, and Fluctuations. Wiley, London 1971

    MATH  Google Scholar 

  29. Goldstein, S., On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951) 129–156

    Article  MathSciNet  MATH  Google Scholar 

  30. Greenberg, J.M., A hyperbolic heat transfer problem with phase change. IMA J. Appl. Math. 38 (1988) 1–21

    Article  Google Scholar 

  31. Greiner, G., Spectral properties and asymptotic behavior of the linear transport equation. Math. Zeitschr. 185 (1984) 167–177

    Article  MathSciNet  MATH  Google Scholar 

  32. Gurtin, M.E., Pipkin, A.C., A general theory of heat conduction with finite wave speeds. Arch. Rat. Mech. Anal. 31 (1968) 113–126

    Article  MathSciNet  MATH  Google Scholar 

  33. Hadeler, K.P., Hyperbolic travelling fronts. Proc. Edinburgh Math. Soc. 31 (1988) 89–97

    Article  MathSciNet  MATH  Google Scholar 

  34. Hadeler, K.P., Travelling fronts for correlated random walks. Canad. Appl. Math. Quart. 2 (1994) 27–43

    MathSciNet  MATH  Google Scholar 

  35. Hadeler, K.P., Travelling epidemic waves and correlated random walks. In: M. Martelli et al. (eds.) Differential Equations and Applications to Biology and Industry. Proc. Conf. Claremont 1994 World Scientific 1995

    Google Scholar 

  36. Hadeler, K.P., Reaction-telegraph equations with density-dependent coefficients. In: G. Lumer, S. Nicaise, B.-W. Schulze (eds) Partial Differential equations, Models in Physics and Biology, Mathematical Research 82, Akademie-Verlag, Berlin 1994, p. 152–158

    Google Scholar 

  37. Hadeler, K.P., Travelling fronts in random walk systems. Forma (Tokyo) 10 (1995) 223–233

    MathSciNet  MATH  Google Scholar 

  38. Hadeler, K.P., Reaction telegraph equations and random walk systems. In: S. van Strien, S. Verduyn Lunel (eds), Stochastic and spatial structures of dynamical systems. Roy. Acad. of the Netherlands. North Holland, Amsterdam (1996), 133–161

    Google Scholar 

  39. Hadeler, K.P., Spatial epidemic spread by correlated random walk, the case of slow infectives. In: R. A. Jarvis et al., Ordinary and partial differential equations. Proc. Conf. Dundee 1996 (1998)

    Google Scholar 

  40. Hadeler, K.P., Nonlinear propagation in reaction transport systems. In: S. Ruan, G. Wolkowicz (eds) Differential equations with Applications to Biology. Fields Institute Communications, Amer. Math. Soc. 1998

    Google Scholar 

  41. Hadeler, K.P., Illner, R., van den Driessche, P., A disease transport model. In preparation.

    Google Scholar 

  42. Hadeler, K.P., Rothe, F., Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2 (1975) 251–263

    Article  MathSciNet  MATH  Google Scholar 

  43. Hale, J.K., Asymptotic Behavior of Dissipative Systems. Amer. Math. Soc., Providence R.I. 1988

    MATH  Google Scholar 

  44. Hale, J.K., Diffusive coupling, dissipation, and synchronization. J. Dynamics Diff. Equ. 9 (1997) 1–52

    Article  MathSciNet  MATH  Google Scholar 

  45. Henry, D., Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math. 840 Springer Verlag 1981

    Google Scholar 

  46. Herrero, M.A., Velázquez, J.J.L., Singularity patterns in a chemotaxis model. Math. Ann. 306, (1996) 583–623

    Article  MathSciNet  MATH  Google Scholar 

  47. Hillen, T., A Turing model with correlated random walk. J. Math. Biol. 35 (1996) 49–72

    Article  MathSciNet  MATH  Google Scholar 

  48. Hillen, T., Nonlinear hyperbolic systems describing random motion and their application to the Turing model. Dissertation Summaries in Math. 1 (1996) 121–128

    MathSciNet  Google Scholar 

  49. Hillen, T., Qualitative Analysis of hyperbolic random walk systems. Preprint SFB 382, No. 43 (1996)

    Google Scholar 

  50. Hillen, T., Invariance principles for hyperbolic random walk system. J. Math. Anal. Appl. 210 (1997) 360–374

    Article  MathSciNet  MATH  Google Scholar 

  51. Hillen, T., Qualitative Analysis of semilinear Cattaneo systems. Math. Models Methods Appl. Sci. 3 (1998)

    Google Scholar 

  52. Hillen, T., Stevens, A., A random walk model with coefficients depending on an external signal as a model for chemotaxis. In preparation.

    Google Scholar 

  53. Holmes, E.E., Are diffusion models too simple? A comparison with telegraph models of invasion. Amer. Naturalist 142 (1993) 779–795

    Article  Google Scholar 

  54. Jörgens, K., An asymptotic expansion in the theory of neutron transport. Comm. Pure Appl. Math. XI (1958) 219–242

    Article  MathSciNet  MATH  Google Scholar 

  55. Joseph, D.D., Preziosi, L., Heat waves. Reviews of Modern Physics 61 (1988) 41–73

    Article  MathSciNet  MATH  Google Scholar 

  56. Kac, M., A stochastic model related to the telegrapher's equation. (1956)

    Google Scholar 

  57. reprinted in Rocky Mtn. Math. J. 4 (1974) 497–509

    Google Scholar 

  58. Källén, A., Thresholds and travelling waves in an epidemic model for rabies. Nonlinear Analysis TMA 8 (1984) 851–856

    Article  MathSciNet  MATH  Google Scholar 

  59. Kaper, H.G., Lekkerkerker, C.G., Hejtmanek, J., Spectral Methods in Linear Transport Theory. Birkhäuser Verlag 1982

    Google Scholar 

  60. Keller, E.F., Segel, L.A., Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399–415

    Article  MATH  Google Scholar 

  61. Keller, E.F., Segel, L.A., Traveling bands of chemotactic bacteria, a theoretical analysis. J. Theor. Biol. 30 (1971) 235–248

    Article  MATH  Google Scholar 

  62. Kendall, D.G., Mathematical models of the spread of infection. Mathematics and Computer Science in Biology and Medicine, p. 213–225. Medical Research Council H.M.S.O., London 1965

    Google Scholar 

  63. Kermack, W.O., McKendrick, A.G., A contribution to the mathematical theory of epidemics I. Proc. Roy. Soc. London A115 (1927) 700–721

    Article  MATH  Google Scholar 

  64. Kolmogorov, A., Petrovskij, I., Piskunov, N., Etude de l'équation de la diffusion avec croissance de la quantité de la matière et son application a une problème biologique. Bull. Univ. Moscou, Ser. Int., Sec A 1, 6 (1937) 1–25

    Google Scholar 

  65. Kuttler, C., A free boundary value problem for correlated random walk. Preprint University of Tübingen, in preparation.

    Google Scholar 

  66. Larsen, E.W., Zweifel, P.F., On the spectrum of the linear transport operator. J. Math. Physics 15 (1974) 1987–1997

    Article  MathSciNet  Google Scholar 

  67. Lauffenburger, D.A., Chemotaxis and cell aggregation In: W. Jäger, J. D. Murray (eds) Modelling of Patterns in Space and Time. Lect. Notes in Biomath. 55, Springer Verlag 1984

    Google Scholar 

  68. Levine, H.A., Sleeman, B.D., A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57 (1997), 683–730

    Article  MathSciNet  MATH  Google Scholar 

  69. Lieberstein, H.M., Mathematical Physiology. Blood flow and electrically active cells. Amer. Elsevier Co., New York, Amsterdam, London 1973

    Google Scholar 

  70. MacNab, R., Koshland D.E. jr., The gradient-sensing system in bacterial chemotaxis. Proc. Nat. Acad. Sci. USA 69 (1972) 2509–2512

    Article  Google Scholar 

  71. McKean, H.P., Application of Brownian motion to the equation of Kolmogorov-Petrovskij-Piskunov. Comm. Pure Appl. Math. 28 (1975) 323–331, 29 (1976) 553–554

    Article  MathSciNet  MATH  Google Scholar 

  72. Mimura, M., Kawasaki, K., Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9 (1980) 49–64

    Article  MathSciNet  MATH  Google Scholar 

  73. Mollison, D., Spatial contact models for ecological and epidemic spread. J. Roy. Statist. Soc. Ser. B 39 (1977) 283–326

    MathSciNet  MATH  Google Scholar 

  74. Müller, I., Hyperbolic equations for diffusion. Classical Mechanics and Relativity: Relationship and Consistency. Monographs and Textbooks in Physical Science, Bibliopolis, Napoli 1991, p. 121–133

    Google Scholar 

  75. Murray, J.D., Mathematical Biology. Springer Verlag 1989.

    Google Scholar 

  76. Nagai, T., Blow up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5 (1995), 581–601

    MathSciNet  MATH  Google Scholar 

  77. Neves, A.F., Ribeiro, H., Lopes, O., On the spectrum of evolution operators generated by hyperbolic systems. J. Funct. Anal. 67 (1986) 320–344

    Article  MathSciNet  MATH  Google Scholar 

  78. Okubo, A., Diffusion and Ecological Problems: Mathematical Models. Biomathematics 10, Springer Verlag 1980

    Google Scholar 

  79. Orsingher, E., A planar random motion governed by the two-dimensional telegraph equation. J. Appl. Prob. 23 (1986) 385–397

    Article  MathSciNet  MATH  Google Scholar 

  80. Othmer, H.G., Dunbar, S.R., Alt, W., Models of dispersal in biological systems. J. Math. Biol. 26 (1988) 263–298

    Article  MathSciNet  MATH  Google Scholar 

  81. Othmer, H.G., Stevens, A., Aggregation, blowup and collapse: The ABCs of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997) 1044–1081

    Article  MathSciNet  MATH  Google Scholar 

  82. Papanicolaou, G.C., Asymptotic analysis of transport processes. Bull. AMS 81 (1975) 330–392

    Article  MathSciNet  MATH  Google Scholar 

  83. Patlak, C., Random walk with persistence and external bias. Bull. Math. Biophysics 15 (1953) 311–318

    Article  MathSciNet  MATH  Google Scholar 

  84. Pearson, K., Nature 72 (1905) 294

    Article  Google Scholar 

  85. Poincaré, H., Sur la propagation de l'électricité. Compt. Rend. Ac. Sci. 107, 1027–1032. ∄uvres IX, 278–283

    Google Scholar 

  86. Rascle, M., Ziti, C., Finite time blow up in some model of chemotaxis. J. Math. Biol. 33 (1995), 388–414

    Article  MathSciNet  MATH  Google Scholar 

  87. Rivero, M.A., Tranquillo, R.T., Buettner, H.M., Lauffenburger, D.A.

    Google Scholar 

  88. Transport models for chemotactic cell populations based on individual cell behavior. Chemical Engin. Sci. 44 (1989)

    Google Scholar 

  89. Rothe, F., Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics 1072, Springer Verlag 1984

    Google Scholar 

  90. Ruggieri, T., Cattaneo equation and relativistic extended thermodynamics. Classical Mechanics and Relativity: Relationship and Consistency. Monographs and Textbooks in Physical Science, Bibliopolis, Napoli 1991, p. 135–150

    Google Scholar 

  91. Schaaf, R., Global behaviour of solution branches for some Neumann problems depending on one or several parameters. J. Reine Angew. Math. 346 (1984) 1–31

    MathSciNet  MATH  Google Scholar 

  92. Schwetlick, H., On the minimal speed of travelling waves in reaction transport equations. Preprint University of Tübingen. SFB 382, No. (1997)

    Google Scholar 

  93. Senba, T., Blow-up of radially symmetric solutions to some systems of partial differential equations modelling chemotaxis. Adv. Math. Sciences Appl. (Tokyo) 7 (1997), 79–92

    MathSciNet  MATH  Google Scholar 

  94. Sharov, O.I., Random walks in the euclidean space R n associated with the telegraph equation. Theor. Prob. Math. Statist. 49 (1994) 165–171

    MathSciNet  Google Scholar 

  95. Skellam, J.G., The formulation and interpretation of mathematical models of diffusional processes in population biology. In: M. S. Bartlett, R. W. Hiorns (eds) The Mathematical Theory of the Dynamics of Biological Populations. Academic Press 1973, p. 63–85

    Google Scholar 

  96. Smoller, J., Shock Waves and Reaction-Diffusion Equations. Springer Verlag 1982

    Google Scholar 

  97. Stadje, W., Exact probability distributions for noncorrelated random walk models. J. Stat. Physics 56 (1989), 415–435

    Article  MathSciNet  MATH  Google Scholar 

  98. Stevens, A., Trail following and aggregation of myxobacteria. J. Biol. Systems 3 (1995), 1059–1068

    Article  Google Scholar 

  99. Tang, Y., Othemer, H.G., Excitation, oscillations and wave propagation in a G-protein based model of signal transduction in Dictyostelium discoideum. Phil. Trans. R. Soc. London B 349 (1995), 179–195

    Article  Google Scholar 

  100. Taylor, G.I., Diffusion by continuous movements. Proc. London Math. Soc. 20 (1920) 196–212

    MathSciNet  MATH  Google Scholar 

  101. Temam, R., Infinite-dimensional systems in Mechanics and Physics.

    Google Scholar 

  102. Springer Verlag 1988

    Google Scholar 

  103. Turing, A.M., The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. London B 237 (1952) 37–72

    Article  Google Scholar 

  104. Voigt, J. Spectral properties of the neutron transport equation. J. Math. Anal. Appl. 106 (1985) 140–153

    Article  MathSciNet  MATH  Google Scholar 

  105. Webb, G., Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Canad. J. Math. 32 (1980) 631–643

    Article  MathSciNet  MATH  Google Scholar 

  106. Weiss, G.H., Aspects and applications of the random walk. North Holland Publ., Amsterdam 1994

    MATH  Google Scholar 

  107. Witt, I., Existence and continuity of the attractor for a singularly perturbed hyperbolic equation. J. Dynamics Diff. Equ. 7 (1995) 591–639

    Article  MathSciNet  MATH  Google Scholar 

  108. Zauderer, E., Partial Differential Equations of Applied Mathematics. Wiley, New York 1983

    MATH  Google Scholar 

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    K. P. Hadeler

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Hadeler, K.P. (1999). Reaction transport systems in biological modelling. In: Capasso, V. (eds) Mathematics Inspired by Biology. Lecture Notes in Mathematics, vol 1714. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092376

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