Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in²

Abstract

For the Keller–Segel model, it was conjectured by Childress and Percus (1984, Chemotactic collapse in two dimensions. InLecture Notes in Biomath. Vol. 55, Springer, Berlin-Heidelberg-New York, 1984, pp. 61–66) that in a two-dimensional domain there exists a critical numberCsuch that if the initial mass is strictly less thanC, then the solution exists globally in time and if it is strictly larger thanCblowup happens. For different versions of the Keller–Segel model, the conjecture has essentially been proved. The case of several chemotactic species introduces an additional question: What is the analogue for the critical mass obtained for the single species system? In this paper, we investigate for a two-species model for chemotaxis in2the conditions on the initial data, which determine blowup or global existence in time. Specifically, we find a curve in the plane of masses such that outside of it there is blowup and inside of it global existence in time is proved when the initial masses satisfy a threshold condition. Optimality of this condition is discussed through an analysis in the radial case. Finally, we show in the case of blowup for general data how it is possible to obtain a balance between entropies and prove what species should aggregates first.

Numerical study of two-species chemotaxis models

“…We show that different scenarios are possible: depending on the initial masses, either one or both cell densities may blow up, or a global solution may exist. In particular, our numerical results indicate answers on some open questions of possible blow up stated in [4,7]. We then introduce two regularizations of the studied models and demonstrate that their solutions are capable of developing spiky structure without blowing up.…”

“…We first numerically investigate two models proposed and analytically studied in [4,5,6,7,8,9,20,34]. The first system reads    (ρ 1 ) t + χ 1 ∇ · (ρ 1 ∇c) = µ 1 ∆ρ 1 , (ρ 2 ) t + χ 2 ∇ · (ρ 2 ∇c) = µ 2 ∆ρ 2 , c t = D∆c + α 1 ρ 1 + α 2 ρ 2 − βc,…”

Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

“…More recently, systems of two species with one chemoattractant have been studied by different research groups, the flnite-time blow-up in bounded domains for the ParabolicParabolic-Elliptic issue has been analyzed by Espejo, Stevens and Velázquez [7] and [8] for simultaneous and non-simultaneous blow-up. See also the results in Biler, Espejo and Guerra [3], Biler and Guerra for bounded domains and Conca and Espejo [5,6] for the two-dimensional case in the whole space. The Parabolic-Parabolic-Elliptic cases with competitive terms, Le., when there exists an explicit interaction between the species, have also been studied in the last years by different authors.…”

Two-species flocking particles immersed in a fluid

“…Two-species models have many applications such as pedestrian flows [29], opinion formation between two groups with different leanings [18,19], and so on. A mathematical study of existence, stability, finite-time blow up, and the large-time behavior for two competitive populations of biological species which are attracted by random diffusion and chemotaxis is another recent active research area [12,20,23,31]. We also refer to [21,25] for nonlocal interaction PDEs with two-species.…”