Aggregation under local reinforcement: From lattice to continuum

European Journal of Applied Mathematics

Volumn 15, Issue 5, 2004, Pages 545-576

  Horstmann, Dirk ^a   Painter, Kevin J ^b   Othmer, Hans G ^c  

^a UNIVERSITY OF COLOGNE   (Germany)

^b HERIOT WATT UNIVERSITY   (United Kingdom)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 16644373361     PISSN: 09567925     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0956792504005571     Document Type: Article

References (36)

1. ADAMS, R. A. (1975) Sobolev Spaces. Academic Press.
2. p Bounds of solutions of reaction-diffusion equations. Comm. Partial Diff. Eqns. 4, 827-868.
3. ALT, W. (1980) Biased Random Walk models for chemotaxis and related diffusion approximations. J. Math Biol. 9, 147-177.
4. ALT, H. W. (1992) Lineare Funktionalanalysis. Springer Verlag.
5. ARONSON, D. G. (1985) The role of diffusion in mathematical population biology: Skellam revisited. In: V. Capasso, E. Grosso and S. L. Paveri-Fontana (eds.), Mathematics in Biology and Medicine, pp. 2-6. Springer-Verlag.
6. BILLEN, J. & GOBIN, B. (1996) Trail following in army ants (Hymenoptera, Formicidae). Neth. J. Zool. 46, 272-280.
7. CARR, J., GURTIN, M. E. & SLEMROD, M. (1984) Structured phase transitions on a finite interval. Arch. Rational Mech. Anal. 86, 317-351.
8. CONWAY, E., HOFF, D. & SMOLLER, J. (1978) Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 35, 1-16.
9. ELLIOTT, C. M. (1989) The Cahn-Hilliard model for the kinetics of phase separation. Mathematical Models for Phase Change Problems, International Series of Numerical Mathematics, Vol. 88, Birkhäuser Verlag.
10. EVANS, L. C. (1998) Partial Differential Equations, American Mathematical Society, Graduate Studies in Mathematics, 19, Providence, RI.
11. HEUSER, H. (1975) Funktionalanalysis, Teubner.
12. HILLEN, T. & OTHMER, H. G. (2000) The diffusion limit of transport equation derived from velocity-jump processes. Siam J. Appl. Math. 61, 751-775.
13. HÖLLDOBLER, B. & WILSON, E. O. (1990) Ants. Harvard University Press.
14. HORSTMANN, D. (2003) From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I. Jahresberichte der DMV, 105, 103-165.
15. HORSTMANN, D. On some cross-diffusion models in population dynamics and their connections to well-posed filters in signal enhancement processes. IMA J. Appl. Math. (to appear).
16. KAWOHL, B. & KUTEV, N. (1998) Maximum principle and comparison theorems for anisotropic diffusion. Mathematische Annalen 311, 107-123.
17. KICHENASSAMY, S. (1997) The Perona-Malik paradox. SIAM J. Appl. Math. 57, 1328-1342.
18. LADYZENSKAYA, O. A. & URAL'CEVA, N. N. (1968) Linear and Quasilinear Elliptic Equations, Academic Press.
19. LADYZENSKAYA, O. A., SOLONNIKOV, V. A. & URAL'CEVA, N. N. (1969) Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Volume 23, AMS.
20. LUTSCHER, F. & STEVENS, A. (2002) Emerging patterns in a hyperbolic model for locally interacting cell systems. J. Nonlinear Sci. 12, 619-640.
21. LEVINE, H. A. & SLEEMAN, B. D. (1997) A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683-730.
22. OTHMER, H. G. (1969) Interactions of reaction and diffusion in open systems. PhD Thesis, University of Minnesota.
23. OTHMER, H. G., DUNBAR, S. R. & ALT, W. (1988) Models of dispersal in biological systems. J. Math. Biol. 26, 263-298.
24. OTHMER, H. G. & SCHAAP, P. (1998) Oscillatory cAMP Signaling in the Development of Dictyostelium discoideum. Comments on Theor. Biol. 5, 175-282.
25. OTHMER, H. G. & STEVENS, A. (1997) Aggregation, Blowup and Collapse: The ABC's of generalized taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044-1081.
26. OTHMER, H. G. & KILLEN, T. (2002) The diffusion limit of transport equations, Part II: chemotaxis equations. SIAM J. Appl. Math. 62, 1222-1260.
27. PAINTER, K. J., HORSTMANN, D. & OTHMER, H. G. (2003) Localization in Lattice and Continuum Models of Reinforced Random Walks. Appl. Math. Lett. 16, 375-381.
28. Painter, K. J., Horstmann, D. & Othmer, H. G. Applications of models for localization of populations. In preparation.
29. PATLAK, C. S. (1953) Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311-338.
30. PERONA, P. & MALIK, J. (1990) Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629-639.
31. ROSENAU, P. (1987) Dynamics of dense lattice. Physical Rev. B, 36, 5868-5876.
32. SMOLLER, J. (1994) Shock waves and reaction-diffusion equations, Springer-Verlag, New York, Heidelberg, second edition.
33. SPIRO, P. A., PARKINSON, J. S. & OTHMER, H. G. (1997) A model of excitation and adaptation in bacterial chemotaxis. Proc. Nat. Acad. Sci. 94, 7263-7268.
34. STRUWE, M. (1996) Variational Methods. Springer-Verlag, second ed.
35. YANG, Y., CHEN, H. & LIU, W. (2001) On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling chemotaxis. SIAM J. Math. Anal. 33, 763-785 (electronic).
36. YANG, Y., CHEN, H. & LIU, W. (2002) Different behaviour for the solutions of 1-dimensional chemotaxis model with exponential growth. Geometry and nonlinear partial differential equations (Hangzhou, 2001), pp. 181-189, AMS/IP Stud. Adv. Math., 29, Amer. Math. Soc., Providence, RI.