[Paper Review] Existence and diffusive limit of a two-species kinetic model of chemotaxis

Luís Almeida, Casimir Emako+1 more|arXiv (Cornell University)|Apr 18, 2014

Mathematical Biology Tumor Growth18 references6 citations

TL;DR

This paper proposes a two-species kinetic model of chemotaxis where cells emit a common chemoattractant and respond to its gradient via a run-and-tumble mechanism. It establishes the existence of weak solutions and proves the diffusive limit converges to a two-species Keller-Segel-type macroscopic model, validating the hydrodynamic scaling in a multi-species context.

ABSTRACT

In this paper, we propose a kinetic model describing the collective motion by chemotaxis of two species in interaction emitting the same chemoattractant. Such model can be seen as a generalisation to several species of the Othmer-Dunbar-Alt model which takes into account the run-and-tumble process of bacteria. Existence of weak solutions for this two-species kinetic model is studied and the convergence of its diffusive limit towards a macroscopic model of Keller-Segel type is analysed.

Motivation & Objective

• To develop a kinetic model describing collective motion via chemotaxis for two interacting species emitting the same chemoattractant.
• To establish the existence of weak solutions for the proposed two-species kinetic model.
• To rigorously analyze the diffusive limit of the kinetic model and derive the corresponding macroscopic two-species Keller-Segel system.
• To validate the hydrodynamic scaling limit by showing convergence of the kinetic solution to the macroscopic model under diffusive scaling.
• To ensure global well-posedness of the limiting macroscopic system under bounded chemosensitivities and positive definite diffusivities.

Proposed method

• Formulate a two-species kinetic model using velocity distribution functions $f_i(x,v,t)$ for each species, coupled to a reaction-diffusion equation for the chemoattractant $S(x,t)$.
• Model cell reorientation via a tumbling kernel $T[S](x,v,v',t) = \phi(\partial_t S + v' \cdot \nabla_x S)$, with $\phi$ decreasing to reflect preference for favorable chemical gradients.
• Apply a diffusive scaling $\varepsilon \to 0$ to the kinetic equations, introducing a small parameter $\varepsilon$ to describe the limit from microscopic to macroscopic dynamics.
• Use weak compactness and moment methods to pass to the limit in the kinetic equations, extracting weak limits of the distribution functions and fluxes.
• Derive the macroscopic system by identifying the limit of the flux $J_i^\varepsilon = \frac{1}{\varepsilon} \int_V v f_i^\varepsilon \, dv$ as $\varepsilon \to 0$, leading to drift-diffusion equations.
• Establish a priori estimates for the macroscopic system using energy methods and Gronwall's inequality, ensuring global existence of weak solutions under bounded chemosensitivities and positive definite diffusivities.

Experimental results

Research questions

• RQ1 Does a two-species kinetic model of chemotaxis with a common chemoattractant admit weak solutions for initial data in $L^1 \cap L^q$?
• RQ2 Can the diffusive limit of this two-species kinetic model be rigorously derived to yield a macroscopic Keller-Segel-type system?
• RQ3 What are the limiting expressions for the effective diffusion and chemotactic drift in the macroscopic model in terms of microscopic parameters?
• RQ4 How do the chemosensitivities $\chi_i[S]$ and diffusivities $D_i$ in the macroscopic model emerge from the kinetic model's tumbling kernel and velocity space?
• RQ5 Is the macroscopic two-species Keller-Segel system globally well-posed under physically reasonable assumptions on the parameters?

Key findings

• Weak solutions exist for the two-species kinetic model under standard assumptions on initial data and the tumbling kernel $T[S]$, with $f_i^{ini} \in L^1 \cap L^q$.
• The diffusive limit of the kinetic model converges weakly to a macroscopic two-species Keller-Segel system, with the flux $J_i^\varepsilon$ converging to $\int_V v r_i^0 \, dv$ in $L^2((0,\tau], L^2_{\text{loc}})$.
• The limiting macroscopic system features drift-diffusion equations with effective diffusivity $D_i = \frac{1}{|V|^2 \psi_i} \int_V v \otimes v \, dv$ and chemotactic sensitivity $\chi_i[S] = -\int_V v \theta_i(v \cdot \nabla S) \frac{dv}{|V|}$.
• The macroscopic system is globally well-posed in the sense of weak solutions, with uniform bounds in $L^q$ and $H^1$ norms derived via energy estimates and Gronwall's inequality.
• The a priori estimates ensure no finite-time blow-up occurs, even in the two-dimensional case, under bounded chemosensitivities and positive definite diffusivities.
• The derivation confirms that the macroscopic model arises as a hydrodynamic limit of the kinetic model, justifying its use in modeling collective chemotactic behavior.