A Small Parameter Method for the Chemotaxis Problem

Authors

  • O.M. Vasyliev Taras Shevchenko National University of Kyiv

DOI:

https://doi.org/10.15407/ujpe71.11.861

Keywords:

chemotaxis, attractant, small parameter, distribution, convergence, iteration

Abstract

The paper proposes an iterative method for constructing an approximate analytical solution for the chemotaxis problem in a spatially constrained one-dimensional system. Within the proposed approach, an approximate analytical solution is obtained, sufficient conditions for its applicability are identified, and the approximate and numerical solutions are compared.

References

1. E. Keller, L. Segel. Model for chemotaxis. J. Theor. Biol. 30, 225 (1971).

https://doi.org/10.1016/0022-5193(71)90050-6

2. J. Adler, G.L. Hazelbauer, M.M. Dahl. Chemotaxis toward sugars in escherichia coli. J. Bacteriology 115, 824 (1973).

https://doi.org/10.1128/jb.115.3.824-847.1973

3. R. Lapidus, R. Schiller. Bacterial chemotaxis in a fixed attractant gradient. J. Theor. Biol. 53, 215 (1975).

https://doi.org/10.1016/0022-5193(75)90112-5

4. M. Eisenbach. Chemotaxis (World Scientific Publishing, 2004).

https://doi.org/10.1142/p303

5. H.C. Berg. E. coli in Motion (Springer, 2004).

https://doi.org/10.1007/b97370

6. P. Van Haastert, P. Devreotes. Chemotaxis: Signalling the way forward. Nat. Rev. Mol. Cell. Biol. 5, 626 (2004).

https://doi.org/10.1038/nrm1435

7. R. Kay, P. Langridge, D. Traynor et al. Changing directions in the study of chemotaxis. Nat. Rev. Mol. Cell. Biol. 9, 455 (2008).

https://doi.org/10.1038/nrm2419

8. J. Adler. Chemotaxis in bacteria. Science 153, 708 (1966).

https://doi.org/10.1126/science.153.3737.708

9. J. Adler. Chemoreceptors in bacteria. Science 166, 1588 (1969).

https://doi.org/10.1126/science.166.3913.1588

10. E.F. Keller, L.A. Segel. Travelling bands of chemotactic bacteria: A theoretical analysis. J. Theor. Biology 30, 235 (1971).

https://doi.org/10.1016/0022-5193(71)90051-8

11. H.C. Berg, D.A. Brown. Chemotaxis in escherichia coli analysed by three-dimensional tracking. Nature 239, 500 (1972).

https://doi.org/10.1038/239500a0

12. R. Lapidus, R. Schiller. Model for the chemotactic response of a bacterial population. Biophys. J. 16, 779 (1976).

https://doi.org/10.1016/S0006-3495(76)85728-1

13. J. Zhuang, G. Wei, R.W. Carlsen, M.R. Edwards, R. Marculescu, P. Bogdan, M. Sitti. Analytical modeling and experimental characterization of chemotaxis in serratia marcescens. Phys. Rev. E 89, 052704 (2014).

https://doi.org/10.1103/PhysRevE.89.052704

14. L. Yang. Global solutions to the Cauchy problem in a higher-dimensional chemotaxis model with indirect signal production mechanism. Appl. Anal. 1 (2026).

https://doi.org/10.1080/00036811.2026.2660789

15. A. Paiva. Interaction of Dirac δ-waves in the inviscid levine and sleeman chemotaxis model. Math. Methods Appl. Sci. 49 (7), 6492 (2026).

https://doi.org/10.1002/mma.70400

16. X. Pan, C. Mu. Large time behavior in a doubly degenerate nutrient taxis system with logistic source. Nonlinear Anal. 271, 114142 (2026).

https://doi.org/10.1016/j.na.2026.114142

17. A. Agosti, A. Signori. Analysis of a multi-species Cahn-Hilliard-Keller-Segel tumor growth model with chemotaxis and angiogenesis. J. Differ. Equ. 403, 308 (2024).

https://doi.org/10.1016/j.jde.2024.05.025

18. H. Garcke, K.F. Lam. Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete Contin. Dyn. Syst. 37 (8), 4277 (2017).

https://doi.org/10.3934/dcds.2017183

19. H. Garcke, K.F. Lam. Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport. Europ. J. Appl. Math. 28, 2, 284 (2017).

https://doi.org/10.1017/S0956792516000292

20. C. Beta, C. Martinez-Torres. Cell Motility and Chemotaxis (Humana New York, 2024).

https://doi.org/10.1007/978-1-0716-4023-4

21. T. Sagawa, Y. Kikuchi, Y. Inoue, H. Takahashi, T. Muraoka, K. Kinbara, A. Ishijima, H. Fukuoka. Single-cell E. coli Response to an instantaneously applied chemotactic signal. Biophys. J. 10, 730 (2014).

https://doi.org/10.1016/j.bpj.2014.06.017

22. T. Namba, M. Nishikawa, T. Shibata. The relation of signal transduction to the sensitivity and dynamic range of bacterial chemotaxis. Biophys. J. 103, 1390 (2012).

https://doi.org/10.1016/j.bpj.2012.08.034

23. A.N. Vasilev. Analytical approach for calculating the chemotaxis sensitivity function. Ukr. J. Phys. 63 (3), 255 (2018).

https://doi.org/10.15407/ujpe63.3.255

24. A.N. Vasilev, D.E. Sakovich. The Modeling of bacterial chemotaxis in a one-dimensional system. J. Phys. Studies 19 (1/2), 1801 (2015).

https://doi.org/10.30970/jps.19.1801

25. A.N. Vasilev, B.Y. Serhushev. Peculiarities of bacterial chemotaxis in a cylindrical pore. Ukr. J. Phys. 64 (2), 137 (2019).

https://doi.org/10.15407/ujpe64.2.137

26. O.M. Vasyliev. Model of chemotaxis in a combined environment. Ukr. J. Phys. 70 (12), 852 (2025).

https://doi.org/10.15407/ujpe70.12.852

27. O.M. Vasilev, V.O. Karpenko. Modeling of bacterial chemotaxis in a medium with a repellent. Ukr. J. Phys. 63, 802, (2018).

https://doi.org/10.15407/ujpe63.9.802

28. O.M. Vasyliev, A.O. Slobodianiuk. The chemotaxis sensitivity function for a system with a spherical geometry. Ukr. J. Phys. 68 (7), 458 (2023).

https://doi.org/10.15407/ujpe68.7.456

29. O.M. Vasyliev. A Model of chemotaxis in an aggressive environment. J. Phys. Studies 30 (2), 2801 (2026).

https://doi.org/10.30970/jps.30.2801

30. D.V. Bohdanov, A.N. Vasilev. Chemotaxis sensitivity function for two-dimensional system with radial symmetry. J. Phys. Studies. 21 (3), 3801 (2017).

https://doi.org/10.30970/jps.21.3801

31. R. Lapidus, R. Schiller. A mathematical model for bacterial chemotaxis. Biophys. J. 14, 825 (1974).

https://doi.org/10.1016/S0006-3495(74)85952-7

32. J.D. Murray. Mathematical Biology: I. An Introduction (Springer, 2007).

33. M.J. Tindall, S.K. Porter, P.K. Maini, G. Gaglia, J.P. Armitage. Overview of mathematical approaches used to model bacterial chemotaxis. II: Bacterial populations. Bull. Math. Biol. 70, 1570 (2008).

https://doi.org/10.1007/s11538-008-9322-5

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Published

2026-10-29

Issue

Vol. 71 No. 11 (2026)

Section

Physics of liquids and liquid systems, biophysics and medical physics

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How to Cite

A Small Parameter Method for the Chemotaxis Problem. (2026). Ukrainian Journal of Physics, 71(11), 861. https://doi.org/10.15407/ujpe71.11.861

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