On Chemotaxis Models with Cell Population Interactions

This paper extends the volume filling chemotaxis model (18, 26) by taking into account the cell population interactions. The extended chemotaxis models have nonlinear diffusion and chemotactic sensitivity depending on cell population density, which is a modification of the clas- sical Keller-Segel model in which the diffusion and chemotactic sensitivity are constants (linear). The existence and boundedness of global solutions of these models are discussed and the numerical pattern formations are shown. The further improvement is proposed in the end.

[1]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[2]  Thomas Hillen,et al.  Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding , 2001, Adv. Appl. Math..

[3]  Yung-Sze Choi,et al.  Prevention of blow-up by fast diffusion in chemotaxis , 2010 .

[4]  Dariusz Wrzosek,et al.  Long-time behaviour of solutions to a chemotaxis model with volume-filling effect , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Thomas Hillen,et al.  Classical solutions and pattern formation for a volume filling chemotaxis model. , 2007, Chaos.

[6]  T. Iwasaki,et al.  Formal analysis of resonance entrainment by central pattern generator , 2008, Journal of mathematical biology.

[7]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[8]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[9]  P. Chavanis.  A stochastic Keller-Segel model of chemotaxis , 2008, 0804.4425.

[10]  Ronald W. Shonkwiler.  Mathematical Biology: An Introduction with Maple and Matlab , 2009 .

[11]  Kevin J Painter,et al.  Modelling the movement of interacting cell populations. , 2003, Journal of theoretical biology.

[12]  Cell-Cell Interactions , 2006 .

[13]  P. Devreotes,et al.  Signaling pathways mediating chemotaxis in the social amoeba, Dictyostelium discoideum. , 2006, European journal of cell biology.

[14]  Dariusz Wrzosek,et al.  Global attractor for a chemotaxis model with prevention of overcrowding , 2004 .

[15]  B. Perthame.  Transport Equations in Biology , 2006 .

[16]  Philip K. Maini,et al.  Dictyostelium discoideum: Cellular self-organisation in an excitable medium , 1995 .

[17]  J. V. Hurley,et al.  Chemotaxis , 2005, Infection.

[18]  R. Kowalczyk,et al.  Preventing blow-up in a chemotaxis model , 2005 .

[19]  Philippe Laurençot,et al.  A Chemotaxis Model with Threshold Density and Degenerate Diffusion , 2005 .

[20]  N. Shigesada,et al.  Biological Invasions: Theory and Practice , 1997 .

[21]  W. I. Newman.  The long-time behavior of the solution to a non-linear diffusion problem in population genetics and combustion , 1983 .

[22]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[23]  Mark Alber,et al.  Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Hendrik Kuiper,et al.  Global attractors for cross diffusion systems on domains of arbitrary dimension , 2007 .

[25]  D. Wrzosek.  Model of chemotaxis with threshold density and singular diffusion , 2010 .

[26]  A. Ōkubo.  Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. , 1986, Advances in biophysics.

[27]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[28]  Hans G. Othmer,et al.  Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[29]  明 大久保,et al.  Diffusion and ecological problems : mathematical models , 1980 .