Geodesic
Chemotaxis stochastic model for two populations
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 37-63 
Ya. I. Belopolskaya; E. I. Nemchenko. Chemotaxis stochastic model for two populations. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 37-63. http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a2/
@article{ZNSL_2020_495_a2,
     author = {Ya. I. Belopolskaya and E. I. Nemchenko},
     title = {Chemotaxis stochastic model for two populations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {37--63},
     year = {2020},
     volume = {495},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a2/}
}
TY  - JOUR
AU  - Ya. I. Belopolskaya
AU  - E. I. Nemchenko
TI  - Chemotaxis stochastic model for two populations
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2020
SP  - 37
EP  - 63
VL  - 495
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a2/
LA  - ru
ID  - ZNSL_2020_495_a2
ER  - 
%0 Journal Article
%A Ya. I. Belopolskaya
%A E. I. Nemchenko
%T Chemotaxis stochastic model for two populations
%J Zapiski Nauchnykh Seminarov POMI
%D 2020
%P 37-63
%V 495
%U http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a2/
%G ru
%F ZNSL_2020_495_a2

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We construct a probabilistic representation of the Cauchy problem weak solution for a system of parabolic equations describing a chemotaxis process in a system of two interacting populations. We derive a stochastic system describing the Keller–Segel type chemotaxis process and the Lotka–Voltera type interatction between two populations and prove existence and uniqueness theorem for its solution. Finally, we show connections between solutions of the stochastic system and the Cauchy problem weak solution of the original PDE system.

[1] T. Black, J. Lankeit, M. Mizukami, “On the weakly competitive case in a two-species chemotaxis model”, IMA J. Appl. Math., 81 (2016), 860–876 | DOI | MR | Zbl

[2] H. Qiu, S. Guo, “Global existence and stability in a two-species chemotaxis system”, Discrete Continuous Dynamical Systems, B, 24:4 (2019), 1569–1587 | DOI | MR | Zbl

[3] Ya. I. Belopolskaya, “Stokhasticheskie modeli protsessov khemotaksisa”, Zapiski nauchn. sem. POMI, 484, 2018, 7–27

[4] D. Talay, M. Tomašević, “A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case”, Bernoulli Soc. Math. Stat. and Prob., 26:2 (2020), 1323–1353 | MR | Zbl

[5] Le Cavil, N. Oudjane, F. Russo, “Probabilistic representation of a class of for nonconseravtive nonlinear partial differential equations”, ALEA, Lat Am. J. Prob Math. Stat., 13 (2016), 1189–1233 | DOI | MR | Zbl

[6] N. Danford, Dzh. Shvarts, Lineinye operatory, v. 1, Obschaya teoriya, IL, M., 1962