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To what extent does the consideration of positive total flux influence the dynamics of Keller-Segel-type models?

Khadijeh Baghaei, Silvia Frassu, Yuya Tanaka, Giuseppe Viglialoro

Abstract

Since the introduction of the Keller-Segel model in 1970 to describe chemotaxis (the interactions between cell distributions u and chemical distributions v), there has been a significant proliferation of research articles exploring various extensions and modifications of this model within the scientific community. From a technical standpoint, the totality of results concerning these variants are characterized by the assumption that the total flux, involving both distributions, of the model under consideration is zero. This research aims to present a novel perspective by focusing on models with a positive total flux. Specifically, by employing Robin-type boundary conditions for u and v, we seek to gain insights into the interactions between cells and their environment, uncovering important dynamics such as how variations in boundary conditions influence chemotactic behavior. In particular, the choice of the boundary conditions is motivated by real-world phenomena and by the fact that the related analysis reveals some interesting properties of the system.

To what extent does the consideration of positive total flux influence the dynamics of Keller-Segel-type models?

Abstract

Since the introduction of the Keller-Segel model in 1970 to describe chemotaxis (the interactions between cell distributions u and chemical distributions v), there has been a significant proliferation of research articles exploring various extensions and modifications of this model within the scientific community. From a technical standpoint, the totality of results concerning these variants are characterized by the assumption that the total flux, involving both distributions, of the model under consideration is zero. This research aims to present a novel perspective by focusing on models with a positive total flux. Specifically, by employing Robin-type boundary conditions for u and v, we seek to gain insights into the interactions between cells and their environment, uncovering important dynamics such as how variations in boundary conditions influence chemotactic behavior. In particular, the choice of the boundary conditions is motivated by real-world phenomena and by the fact that the related analysis reveals some interesting properties of the system.
Paper Structure (31 sections, 24 theorems, 192 equations, 2 figures, 1 table)

This paper contains 31 sections, 24 theorems, 192 equations, 2 figures, 1 table.

Table of Contents

  1. Chemotaxis phenomena. The homogeneous Neumann boundary conditions
  2. The landmarking Keller--Segel model
  3. The Keller--Segel system with logistic and strong logistic sources
  4. Nonlinear diffusion and sensitivity and further models
  5. Modification in the boundary conditions. Requiring the total flux to vanish
  6. The uniform-in-time $L^1(\Omega)$-bound of $u$
  7. Positive total flux in chemotaxis. Real applications and mathematical formulation
  8. The Robin boundary conditions
  9. The Bacillus subtilis, Eukaryotic T-Cells, and Microbial Communities
  10. The role of the chemoattractant and positive total flux. Mathematical interpretation and formulation
  11. Boundedness of the mass by means of strong logistics in models with positive flux
  12. Logistics with damping gradient terms; applications and their role through the boundedness of the mass
  13. Aim of the investigation. Models and results
  14. Presentation of the models and of the main results
  15. Necessary tools: parameters and properties of solutions to differential equations
  16. ...and 16 more sections

Key Result

Theorem 5.1

Let the hypotheses in reglocal be fulfilled, $\tau\in\{0,1\}$, $\chi, a, h >0$ and $\alpha \in [0,1]$. Then there exists $C_{\partial \Omega}=C_{\partial \Omega}(\Omega, n)$ such that for every the following conclusion holds true: problem problem admits a unique solution such that $0\leq u,v \in L^\infty(\Omega \times (0,\infty)).$

Figures (2)

  • Figure 1: Simulations of model \ref{['KS']} with boundary conditions given in \ref{['RobinvRobinu']}. Evolution of the cells' density $u$ at six different instants of time: $t=0$ (top left), $t=0.1$ (bottom right). Some quantitative values are deductible from Figure \ref{['FigureComparison']}.
  • Figure 2: Robin conditions vs. Neumann conditions.

Theorems & Definitions (54)

  • Remark 1: On the logistic terms with strong/gradient dissipative effects
  • Remark 2: On the connection between conditions \ref{['RobinvRobinu']} and \ref{['RobinvRobinuPositiveNetFlux']}
  • Remark 3: Some numerical evidences
  • Remark 4: Quadratic-gradient-driven phenomena and their connection to positive fluxes
  • Theorem 5.1
  • Remark 5
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • proof
  • ...and 44 more