Geodesic
Class of Keller--Segel chemotactic systems based on Einstein method of Brownian motion modeling
Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 2, pp. 253-277.

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We study the movement of the living organism in a band form towards the presence of chemical substrates based on a system of partial differential evolution equations. We incorporate Einstein's method of Brownian motion to deduce the chemotactic model exhibiting a traveling band. It is the first time that Einstein's method has been used to motivate equations describing the mutual interaction of the chemotactic system. We have shown that in the presence of limited and unlimited substrate, traveling bands are achievable and it has been explained accordingly. We also study the stability of the constant steady states for the system. The linearized system about a constant steady state is obtained under the mixed Dirichlet and Neumann boundary conditions. We are able to find explicit conditions for linear instability. The linear stability is established with respect to the $L^2$-norm, $H^1$-norm, and $L^\infty$-norm under certain conditions.
Keywords: chemotactic model, Einstein's method of Brownian motion, traveling band. 
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     title = {Class of {Keller--Segel} chemotactic systems based on {Einstein} method of {Brownian} motion modeling},
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R. Islam; A. Ibragimov. Class of Keller--Segel chemotactic systems based on Einstein method of Brownian motion modeling. Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 2, pp. 253-277. http://geodesic.mathdoc.fr/item/CMFD_2024_70_2_a4/

[1] Adler J., “Effect of amino acids and oxygen on chemotaxis in escherichia coli”, J. Bacteriology, 92:1 (1966), 121–129 | DOI

[2] Belopolskaya Y. I., “Stochastic models of chemotaxis processes”, J. Math. Sci. (N. Y.), 251:1 (2020), 1–14 | DOI | MR | Zbl

[3] Carrillo J. A., Li J., Wang Zh., “Boundary spike-layer solutions of the singular Keller—Segel system: existence and stability”, Proc. Lond. Math. Soc. (3), 122:1 (2021), 42–68 | DOI | MR | Zbl

[4] Chavanis P. H., “A stochastic Keller—Segel model of chemotaxis”, Commun. Nonlinear Sci. Numer. Simul., 15:1 (2010), 60–70 | DOI | MR | Zbl

[5] Davis P. N., van Heijster P., Marangell R., “Absolute instabilities of travelling wave solutions in a Keller—Segel model”, Nonlinearity, 30:11 (2017), 4029–4061 | DOI | MR | Zbl

[6] Einstein A., “Uber die von der molekularkinetischen theorie der warme geforderte bewegung von in ruhenden flussigkeiten suspendierten teilchen”, Ann. Phys. Leipzig, 322 (1905), 549–560 | DOI

[7] Fu S., Huang G., Adam B., “Instability in a generalized multi-species Keller—Segel chemotaxis model”, Comput. Math. Appl., 72:9 (2016), 2280–2288 | DOI | MR | Zbl

[8] Gobbetti M., De Angelis M., Di Cagno R., Minervini F., Limitone A., “Cell–cell communication in food related bacteria”, Int. J. Food Microbiology, 120:1-2 (2007), 34–45 | DOI

[9] Ibragimov A., Peace A., “Light driven interactions in spatial predator–prey chemotaxis model in the presence of chemical agent”, J. Pure Appl. Math., 2:1 (2022), 222–244

[10] Keller E. F., Segel L. A., “Traveling bands of chemotactic bacteria: A theoretical analysis”, J. Theor. Biol., 30:2 (1971), 235–248 | DOI | Zbl

[11] Li Yi, Li Yong, Wu Y., Zhang H., “Spectral stability of bacteria pulses for a Keller—Segel chemotactic model”, J. Differ. Equ, 304 (2021), 229–286 | DOI | MR | Zbl

[12] Qiao Q., “Traveling waves and their spectral instability in volume-filling chemotaxis model”, J. Differ. Equ, 382 (2024), 77–96 | DOI | MR | Zbl

[13] Romanczuk P., Erdmann U., Engel H., Schimansky-Geier L., “Beyond the Keller—Segel model: Microscopic modelling of bacterial colonies”, Eur. Phys. J. Spec. Topics, 157 (2008), 61–77 | DOI

[14] Skorokhod A., Basic principles and applications of probability theory, Springer, Berlin—Heidelberg, 2005 | MR | Zbl

[15] Stevens A., “The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems”, SIAM J. Appl. Math., 61:1 (2000), 183–212 | DOI | MR | Zbl

[16] Stevens A., Othmer H. G., “Aggregation, blowup, and collapse: the abc's of taxis in reinforced random walks”, SIAM J. Appl. Math., 57:4 (1997), 1044–1081 | DOI | MR | Zbl

[17] Tindall M., Maini P., Porter S., Armitage J., “Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations”, Bull. Math. Biol., 70:6 (2008), 1570–607 | DOI | MR

[18] Tomasevic M., Talay D., “A new McKean—Vlasov stochastic interpretation of the parabolic-parabolic Keller—Segel model: The one-dimensional case”, Bernoulli, 26:2 (2020), 1323–1353 | MR | Zbl

[19] Wang Q., Yan J., Gai C., “Qualitative analysis of stationary Keller—Segel chemotaxis models with logistic growth”, Z. Angew. Math. Phys., 67:3 (2016), 51 | DOI | MR