Competing effects of attraction vs. repulsion in chemotaxis

Mathematical Models and Methods in Applied Sciences

Volumn 23, Issue 1, 2013, Pages 1-36

  Tao, Youshan ^a,b   Wang, Zhi An ^b  

^a DONGHUA UNIVERSITY   (China)

^b HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

Author keywords

Attraction repulsion; Boundedness; Chemotaxis; Convergence; Entropy inequality; Stationary solutions

Indexed keywords

ATTRACTION-REPULSION; BOUNDEDNESS; CHEMOTAXIS; CONVERGENCE; ENTROPY INEQUALITY; STATIONARY SOLUTIONS;

BOUNDARY CONDITIONS;

BIOCHEMISTRY;

EID: 84872393159     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202512500443     Document Type: Article

References (57)

1. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math. 12 (1959) 623-727.
2. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Commun. Pure Appl. Math. 17 (1964) 35-92.
3. N. D. Alikakos, Lp bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations 4 (1979) 827-868.
4. N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems, Math. Models Methods Appl. Sci. 20 (2010) 1179-1207.
5. P. Biler, Local and global solvability of some parabolic system modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998) 715-743.
6. J. P. Boon and B. Herpigny, Model for chemotactic bacterial bands, Bull. Math. Biol. 48 (1986) 1-19.
7. J. A. Carrillo, A. Jüngle, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001) 1-82. (Pubitemid 33585637)
8. M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. A. Glazier and C. J. Weijer, Cell movement during chick primitive streak formation, Dev. Biol. 296 (2006) 137-149.
9. T. Ciéslak and P. Laurečot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H. Poincaŕe Anal. Non Lińeaire 27 (2010) 437-446.
10. T. Ciéslak, P. Laurečot and C. Morales-Rodrigo, Global Existence and Convergence to Steady-States in a Chemorepulsion System, Banach Center Publications, Vol. 81 (Polish Acad. Sci., 2008), pp. 105-117.
11. T. Ciéslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity 21 (2008) 1057-1076.
12. M. Eisenbach, Chemotaxis (Imperial College Press, 2004).
13. E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis 29 (2009) 317-338.
14. A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subjected to chemotaxis, Math. Models Methods Appl. Sci. 14 (2004) 503-533.
15. A. Friedman, Partial Differential Equations (Holt, Rinehart & Winston, 1969).
16. M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-sriatal circuit, Euro. J. Neurosci. 19 (2004) 831-844. (Pubitemid 38297020)
17. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer-Verlag, 1983).
18. M. A. Herrero and J. L. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Super. Pisa Cl. Sci. 24 (1997) 633-683.
19. T. Hillen and K. Painter, A users' guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183-217.
20. T. Hillen and J. Renclawowicz, Analysis of an attraction-repulsion chemotaxis model, Unpublished manuscript (2007).
21. D. Horstmann, The nonsymmetric case of the Keller-Segel model in chemotaxis: Some recent results, NoDEA: Nonlinear Differential Equations Appl. 8 (2001) 399-423.
22. D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verien. 105 (2003) 103-106.
23. D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci. 21 (2011) 231-270.
24. D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, Eur. J. Appl. Math. 12 (2001) 159-177.
25. D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005) 52-107. (Pubitemid 40621900)
26. W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992) 819-824.
27. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399-415.
28. R. Kowalczyk and Z. Szymánska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl. 343 (2008) 379-398.
29. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type (Amer. Math. Soc., 1968).
30. D. Le, Coexistence with chemotaxis, SIAM J. Math. Anal. 32 (2000) 504-521.
31. D. Le and H. L. Smoth, Steady states of models of microbial growth and competition with chemotaxis, J. Math. Anal. Appl. 229 (1999) 295-318.
32. J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction- repulsion chemotaxis in one dimension, J. Biol. Dynam. 6 (2012) 31-41, doi:10.1080/ 17513758.2011.571722.
33. M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimer's disease senile plague: Is there a connection? Bull. Math. Biol. 65 (2003) 673-730.
34. M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, J. Math. Anal. Appl. 49 (1975) 215-225.
35. T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. 5 (1995) 581-601.
36. T. Nagai, Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl. 6 (2001) 37-55.
37. L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20 (1966) 733-737.
38. K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac. 44 (2001) 441-469.
39. K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart. 10 (2002) 501-543.
40. B. Perthame, Transport Equations in Biology (Birkhäuser, 2007).
41. B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Traveling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion-existence and branching instabilities, Nonlinearity 24 (2011) 1253-1270.
42. J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulse, PLoS Comput. Biol. 6 (2010) e1000890.
43. T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl. 10 (2000) 191-224.
44. T. Suzuki, Free Energy and Self-Interacting Particles (Birkhäuser, 2005).
45. Y. Tao, Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl. 381 (2011) 521-529.
46. Y. Tao and M. Wang, A combined chemotaxis-haptotaxis system: The role of logistic source, SIAM J. Math. Anal. 41 (2009) 1533-1558.
47. Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal. 43 (2011) 685-704.
48. Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller- Segel system with subcritical sensitivity, J. Differential Equations 252 (2012) 692-715.
49. Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations 252 (2012) 2520-2534.
50. G. Wang and J. Wei, Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr. 233-234 (2002) 221-236.
51. Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos 17 (2007) 037108.
52. M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci. 25 (2002) 911-925.
53. M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2010) 2889-2905.
54. M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differential Equations 37 (2012) 319-352.
55. M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, arXiv:1112.4156v1.
56. G. Wolansky, Multi-components chemotaxis system in absence of conflict, Eur. J. Appl. Math. 13 (2002) 641-661.
57. D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volumefilling effect, Proc. Roy. Soc. Edinburgh A 136 (2006) 431-444. (Pubitemid 43657667)