[1] W. Alt:
Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9 (1980), 147–177.
MR 0661424 |
Zbl 0434.92001
[2] W. Alt, G. Hoffmann: Biological motion. Proceedings of a workshop held in Königswinter, Germany, March 16–19, 1989. Lecture Notes in Biomathematics, 89. Springer-Verlag, Berlin, 1990.
[3] A. R. A. Anderson, M. A. J. Chaplain: A mathematical model for capillary network formation in the absence of endothelial cell proliferation. Appl. Math. Lett. 11 (1998), 109–114.
[4] C. Bardos, R. Santos, and R. Sentis:
Diffusion approximation and computation of the critical size. Trans. Amer. Math. Soc. 284 (1984), 617–649.
MR 0743736
[5] N. Bellomo, L. Preziosi:
Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Modelling 32 (2000), 413–452.
MR 1775113
[6] M. D. Betterton, M. P. Brenner: Collapsing bacterial cylinders. Phys. Rev. E 64 (2001).
[7] P. Biler:
Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9 (1999), 347–359.
MR 1690388 |
Zbl 0941.35009
[8] P. Biler, T. Nadzieja:
A class of nonlocal parabolic problems occurring in statistical mechanics. Colloq. Math. 66 (1993), 131–145.
MR 1242651
[9] P. Biler, T. Nadzieja:
Global and exploding solutions in a model of self-gravitating systems. Rep. Math. Phys. 52 (2003), 205–225.
MR 2016216
[10] M. P. Brenner, P. Constantin, L. P. Kadanoff, A. Schenkel, and S. C. Venkataramani:
Diffusion, attraction and collapse. Nonlinearity 12 (1999), 1071–1098.
MR 1709861
[11] M. P. Brenner, L. Levitov, and E. O. Budrene: Physical mechanisms for chemotactic pattern formation by bacteria. Biophysical Journal 74 (1995), 1677–1693.
[12] C. Cercignani, R. Illner, and M. Pulvirenti:
The Mathematical Theory of Dilute Gases. Applied Math. Sciences Vol. 106, Springer-Verlag, New York, 1994.
MR 1307620
[13] F. Chalub, P. Markowich, B. Perthame, and C. Schmeiser:
Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh. Math. 142 (2004), 123–141.
MR 2065025
[14] M. A. J. Chaplain: Avascular growth, angiogenesis and vascular growth in solid tumors: the mathematical modelling of the stages of tumor development. Math. Comput. Modelling 23 (1996), 47–87.
[15] M. A. J. Chaplain, L. Preziosi: Macroscopic modelling of the growth and developement of tumor masses. Preprint No. 27, Politecnico di Torino, 2000.
[16] L. Corrias, B. Perthame, and H. Zaag:
A chemotaxis model motivated by angiogenesis. C. R. Acad. Sci. Paris, Ser. I 336 (2003), 141–146.
MR 1969568
[17] L. Corrias, B. Perthame, and H. Zaag:
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milano J. Math. 72 (2004), 1–29.
MR 2099126
[18] F. A. Davidson, A. R. A. Anderson, and M. A. J. Chaplain:
Steady-state solutions of a generic model for the formation of capillary networks. Appl. Math. Lett. 13 (2000), 127–132.
MR 1760274
[19] P. Degond, T. Goudon, and F. Poupaud:
Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math. J. 49 (2000), 1175–1198.
MR 1803225
[20] Y. Dolak, T. Hillen:
Cattaneo models for chemotaxis, numerical solution and pattern formation. J. Math. Biol. 46 (2003), 461–478.
MR 1963070
[21] J. Dolbeault, B. Perthame:
Optimal critical mass in the two dimensional Keller-Segel model in $\mathbb{R}^2$. C. R. Acad. Sci. (2004) (to appear).
MR 2103197
[22] Y. Dolak, C. Schmeiser:
Kinetic Models for Chemotaxis. ANUM preprint. (2003).
MR 2093271
[23] L. C. Evans:
Partial Differential Equations. Amer. Math. Soc., Providence, 1998.
Zbl 0902.35002
[24] F. Filbet, P. Laurençot, and B. Perthame:
Derivation of hyperbolic models for chemosensitive movement. Preprint. Ecole Normale Supérieure, 2003.
MR 2120548
[25] M. A. Fontelos, A. Friedman, and B. Hu:
Mathematical analysis of a model for the initiation of angiogenesis. SIAM J. Math. Anal. 33 (2002), 1330–1355.
MR 1920634
[26] A. Friedman, I. Tello:
Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. Appl. 272 (2002), 138–163.
MR 1930708
[27] H. Gajewski, K. Zacharias:
Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195 (1998), 77–114.
MR 1654677
[28] A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi, and F. Bussolino: Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90 (2003), .
[29] I. Gasser, P.-E. Jabin, and B. Perthame:
Regularity and propagation of moments in some nonlinear Vlasov systems. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 1259–1273.
MR 1809103
[31] M. A. Herrero, J. J. L. Velázquez:
Singularity patterns in a chemotaxis model. Math. Ann. 306 (1996), 583–623.
MR 1415081
[32] M. A. Herrero, E. Medina, and J. J. L. Velázquez:
Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10 (1997), 1739–1754.
MR 1483563
[33] T. Hillen, H. Othmer:
The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61 (2000), 751–775.
MR 1788017
[34] D. Horstmann:
Lyapunov functions and $L^p$ estimates for a class of reaction-diffusion systems. Colloq. Math. 87 (2001), 113–127.
MR 1812147
[35] D. Horstmann:
From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. Jahresber. Dtsch. Math.-Ver. Vol. 105, 2003, pp. 103–165.
MR 2013508 |
Zbl 1071.35001
[36] H. J. Hwang, K. Kang, and A. Stevens:
Global solutions of nonlinear transport equations for chemosensitive movement. SIAM J. Math. Anal (to appear).
MR 2139206
[37] W. Jäger, S. Luckhaus:
On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329 (1992), 819–824.
MR 1046835
[38] E. F. Keller:
Assessing the Keller-Segel model: How has it fared? Biological growth and spread. Proc. Conf., Heidelberg, 1979. Lecture Notes in Biomath. Vol. 38, Springer-Verlag, Berlin-New York, 1980, pp. 379–387.
MR 0609374
[39] E. F. Keller, L. A. Segel: Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 (1970), 399–415.
[40] E. F. Keller, L. A. Segel: Model for chemotaxis. J. Theoret. Biol. 30 (1971), 225–234.
[41] E. F. Keller, L. A. Segel: Travelling bands of chemotactic bacteria: a theoretical analysis. J. Theoret. Biol. 30 (1971), 235–248.
[42] H. A. Levine, B. D. Sleeman:
A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57 (1997), 683–730.
MR 1450846
[43] H. A. Levine, B. D. Sleeman:
Partial differential equations of chemotaxis and angiogenesis. Math. Methods Appl. Sci. 24 (2001), 405–426.
MR 1821934
[44] H. A. Levine, M. Nilsen-Hamilton, and B. D. Sleeman:
Mathematical modelling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42 (2001), 195–238.
MR 1828815
[45] P. K. Maini:
Applications of mathematical modelling to biological pattern formation. Coherent Structures in Complex Systems (Sitges, 2000). Lecture Notes in Phys. Vol. 567, Springer-Verlag, Berlin, 2001, pp. 205–217.
MR 1995108
[46] D. Manoussaki: Modeling and simulation of the formation of vascular networks. ESAIM Proc. 12 (2002 (electronic)), 108–114.
[47] A. Marrocco:
2D simulation of chemotactic bacteria agreggation. ESAIM: Math. Model. Numer. Anal. 37 (2003), 617–630.
MR 2018433
[48] P. Michel, S. Mischler, and B. Perthame:
General entropy equations for structured population models and scattering. C. R. Acad. Sci. Paris (to appear).
MR 2065377
[49] J. D. Murray:
Mathematical Biology, Vol. 2, third revised edition. Spatial Models and Biomedical Applications. Springer-Verlag, , 2003.
MR 1952568
[50] T. Nagai:
Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5 (1995), 581–601.
MR 1361006 |
Zbl 0843.92007
[51] T. Nagai, T. Senba:
Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis. Adv. Math. Sci. Appl. 8 (1998), 145–156.
MR 1623326
[52] J. Nieto, F. Poupaud, and J. Soler:
High field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Rational. Mech. Anal. 158 (2001), 29–59.
MR 1834113
[53] H. G. Othmer, A. Stevens:
Aggregation, blowup and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997), 1044–1081.
MR 1462051
[54] H. G. Othmer, S. R. Dunbar, and W. Alt:
Models of dispersal in biological systems. J. Math. Biol. 26 (1988), 263–298.
MR 0949094
[55] C. S. Patlak:
Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311–338.
MR 0081586
[56] B. Perthame:
Mathematical tools for kinetic equations. Bull. Amer. Math. Soc. (NS) 41 (2004), 205–244.
MR 2043752 |
Zbl 1151.82351
[57] M. Rascle: On a system of non-linear strongly coupled partial differential equations arising in biology. Proc. Conf. on Ordinary and Partial Differential Equation. Lectures Notes in Math. Vol. 846, Everitt and Sleeman (eds.), Springer-Verlag, New-York, 1981, pp. 290–298.
[58] M. Rascle, C. Ziti:
Finite time blow-up in some models of chemotaxis. J. Math. Biol. 33 (1995), 388–414.
MR 1320430
[59] G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi, and F. Bussolino: Modeling the early stages of vascular network assembly. The EMBO Journal 22 (2003), 1771–1779.
[60] T. Sanba and T. Suzuki:
Weak solutions to a parabolic-elliptic system of chemotaxis. J. Functional. Analysis 47 (2001), 17–51.
MR 1909263
[61] H. R. Schwetlick:
Travelling fronts for multidimensional nonlinear transport equations. Ann. Inst. H. Poincaré. Anal. non Linéaire 17 (2000), 523–550.
MR 1782743
[62] 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.
MR 1776393 |
Zbl 0963.60093
[63] A. Stevens, M. Schwelick: Work in preparation.
[64] M. I. Weinstein:
Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983), 567–576.
MR 0691044 |
Zbl 0527.35023
[65] Y. Yang, H. Chen, and W. Liu:
On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis. SIAM J. Math. Anal. 33 (2001), 763–785.
MR 1884721