The Langevin or Kramers approach to biological modeling

Mathematical Models and Methods in Applied Sciences

Volumn 14, Issue 10, 2004, Pages 1561-1583

  Hadeler, Karl P ^a   Hillen, Thomas ^b   Lutscher, Frithjof ^b  

^a UNIVERSITY OF TÜBINGEN   (Germany)

^b UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Biological modeling; Langevin equation; Moment closure

Indexed keywords

EID: 6044278193     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202504003726     Document Type: Article

References (52)

1. W. Alt, Correlation analysis of two-dimensional locomotion paths, Biological Motion, Lecture Notes in Biomathematics, Vol. 89, eds. W. Alt and G. Hoffmann (Springer, 1990), pp. 548-555.
2. D. Aronson, The role of diffusion in mathematical population biology: Skellam revisited, Mathematics in Biology and Medicine, Lecture Notes in Biomathematics, Vol. 57, ed. S. Levin (Springer-Verlag, 1983), pp. 2-6.
3. M. Bartlett, A note on random walks at constant speed, Adv. Appl. Probab. 10 (1978) 704-707.
4. F. Bouchut, Smoothing effect for the non-linear Vlasov-Poisson-Fokker- Planck system, J. Diff. Eqns. 122 (1995) 225-238.
5. N. Britton, Reaction-Diffusion Equations and Their Applications to Biology (Academic Press, 1986).
6. Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol. 46 (2003) 153-170.
7. R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Pop. Biol. 46 (1994) 363-394.
8. A. Einstein, Zur Theorie der Brownschen Bewegung, Ann. Phys. 19 (1906) 371-381.
9. K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, 2000).
10. U. Erdmann, W. Ebeling, L. Schimansky-Geier and F. Schweitzer, Brownian particles far from equilibrium, Euro. Phys. J. B15 (2000) 105-113.
11. P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, Vol. 28 (Springer-Verlag, 1979).
12. R. Fisher, The advance of advantageous genes, Ann. Eugenics 7 (1937) 355-369.
13. E. Geigant, K. Ladizhansky and A. Mogilner, An integro-differential model for orientational distributions of F-actin in cells, SIAM J. Appl. Math. 59 (1999) 787-809.
14. K. P. Hadeler, Travelling fronts for correlated random walks, Can. Appl. Math. Quart. 2 (1994) 27-43.
15. K. P. Hadeler, Random walk systems and reaction telegraph equations, Dynamical Systems and their Applications in Science, eds. S. v. Strien and S. Lunel (Royal Academy of the Netherlands, 1996).
16. K. P. Hadeler, Reaction transport systems in biological modelling, Mathematics Inspired by Biology, Lecture Notes Mathematics, Vol. 1714, eds. V. Capasso and O. Diekmann (Springer-Verlag, 1999), pp. 95-150.
17. T. Hillen, Transport equations with resting phases, Euro. J. Appl. Math. 14 (2003) 613-636.
18. T. Hillen and H. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math. 61 (2000) 751-775.
19. T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. 26 (2001) 280-301.
20. C. Huang, A variational principle for the Kramers equation with unbounded external forces, J. Math. Anal. Appl. 250 (2000) 333-367.
21. M. Jacobsen, Laplace and the origin of the Ornstein-Uhlenbeck process, Bernoulli 2 (1996) 271-286.
22. E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399-415.
23. A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjul. Moskovskovo Gos. Univ. 17 (1937) 1-72.
24. H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) 284-304.
25. P. Lions and B. Perthame, Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991) 415-430.
26. F. Lutscher, Modeling alignment and movement of animals and cells, J. Math. Biol. 45 (2002) 234-260.
27. N. Mantzaris, S. Webb and H. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Biol., in press.
28. H. McKean, Applications of Brownian motion to the equation of Kolmogorov-Petrovskij-Piskunov, Comm. Pure Appl. Math. 28 (1975) 323-331.
29. ∞ and Gevrey classes, in Partial Differential Equations, Lecture Notes in Mathematics, Vol. 1324, Lat. Am. Sch. Math., ELAM-8 (Springer-Verlag, 1988), pp. 197-215.
30. A. Mogilner, L. Edelstein-Keshet and G. B. Ermentrout, Selecting a common direction II, J. Math. Biol. 34 (1996) 811-842.
31. J. Moyal, Stochastic processes and statistical physics, J. Roy. Stat. Soc. (London) B11 (1949) 150-210.
32. J. Müller and T. Hillen, Modulation equations and parabolic limits of reaction random walk systems, Math. Meth. Appl. Sci. 21 (1998) 1207-1226.
33. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Springer-Verlag, Berlin, 2002).
34. A. Okubo and D. Grünbaum, Mathematical treatment of biological diffusion, in Diffusion and Ecological Problems, eds. A. Okubo and S. Levin (Springer-Verlag, 2001), pp. 127-169.
35. A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives (Springer-Verlag, New York, 2001).
36. A. Okubo, Oceanic turbulent diffusion of abiotic and biotic species, in Mathematical Ecology, Lect. Notes in Biomathematics, Vol. 54, eds. S. A. Levin and T. G. Hallam (Springer-Verlag, 1984), pp. 127-169.
37. L. Ornstein and G. Uhlenbeck, On the theory of Brownian motion, Phys. Rev. 36 (1930) 823-841.
38. H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol. 26 (1988) 263-298.
39. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953) 311-338.
40. F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Poisson-FokkerPlanck system, Math. Mod. Meth. Appl. Sci. 10 (2000) 1027-1045.
41. R. Résibois and M. De Leener, Classical Kinetic Theory of Fluids (John Wiley & Sons, 1977).
42. H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications, Springer Series in Synergetics (Springer-Verlag, 1984).
43. M. Schienbein and H. Gruler, Langevin equation, Fokker-Planck equation and cell migration, Bull. Math. Biol. 55 (1993) 585-608.
44. L. Schimansky-Geier, F. Schweitzer and M. Mieth, Iterative structure formation with Brownian particles, Self Organization of Complex Structures: From Individual to Collective Dynamics, ed. F. Schweitzer (Gordon and Breach, 1997), pp. 101-118.
45. F. Schweitzer and L. Schimansky-Geier, Clustering of active walkers in a two component system, Physica A206 (1994) 359-379.
46. H. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations, Ann. Inst. Henri Poincarè, Anal. Non Linéaire 17 (2000) 523-550
47. H. Schwetlick, Limit sets for multidimensional nonlinear transport equations, J. Diff. Eqn. 179 (2002) 356-368
48. D. Soll, Behavioral studies into the mechanism of eukaryotic chemotaxis, J. Chem. Ecol. 16 (1990) 133-150.
49. A. Stevens, The derivation of chemotaxis-equations as limit dynamics of moderately interacting stochastic many particle systems, SIAM J. Appl. Math. 61 (2000) 183-212.
50. A. Stevens and F. Schweitzer, Aggregation induced by diffusing and nondiffusing media, Dynamics of Cell and Tissue Motion, eds. W. Alt, A. Deutsch and G. Dunn (Birkhäuser, 1997), pp. 183-192.
51. H. Victory and B. O'Dwyer, On classical solutions of Vlasov-Poisson- Fokker-Planck systems, Indiana U. Math. J. 39 (1990) 105-155.
52. D. Woodward, R. Tyson, M. M. R. J. Murray and E. Budrene, Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J. 68 (1995) 2181-2189.