Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Microscopic, kinetic and hydrodynamic hybrid models of collective motions withchemotaxis: a numerical study

Marta Menci, Roberto Natalini, Thierry Paul

TL;DR

This work numerically analyzes microscopic (P), kinetic (V), and hydrodynamic (E) descriptions of collective motion with chemotaxis in 1D, focusing on non-monokinetic initial data. It confirms that the Vlasov and Euler limits retain salient particle-scale features and that chemotaxis can reinforce alignment even beyond monokinetic regimes, while damping mitigates blow-up in the Euler system. The study also demonstrates that a nonlocal pressure term in the Euler model can improve cross-scale fidelity, identifying an approximate optimal coefficient $oldsymbol{\varepsilon}^*$ for matching with the kinetic description. Overall, the results support the viability of hybrid microscopic-continuum models as efficient, faithful digital-twin representations of chemotaxis-driven collective dynamics and guide future higher-dimensional extensions.

Abstract

A general class of hybrid models has been introduced recently, gathering the advantages multiscale descriptions. Concerning biological applications, the particular coupled structure fits to collective cell migrations and pattern formation scenarios. In this context, cells are modelled as discrete entities and their dynamics is given by ODEs, while the chemical signal influencing the motion is considered as a continuous signal which solves a diffusive equation. From the analytical point of view, this class of model has been proved to have a mean-field limit in the Wasserstein distance towards a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. Moreover, a pressureless nonlocal Euler-type system has been derived for these models, rigorously equivalent to the Vlasov one for monokinetic initial data. In the present paper, we present a numerical study of the solutions to the Vlasov and Euler systems, exploring general settings for inital data, far from the monokinetic ones.

Microscopic, kinetic and hydrodynamic hybrid models of collective motions withchemotaxis: a numerical study

TL;DR

This work numerically analyzes microscopic (P), kinetic (V), and hydrodynamic (E) descriptions of collective motion with chemotaxis in 1D, focusing on non-monokinetic initial data. It confirms that the Vlasov and Euler limits retain salient particle-scale features and that chemotaxis can reinforce alignment even beyond monokinetic regimes, while damping mitigates blow-up in the Euler system. The study also demonstrates that a nonlocal pressure term in the Euler model can improve cross-scale fidelity, identifying an approximate optimal coefficient for matching with the kinetic description. Overall, the results support the viability of hybrid microscopic-continuum models as efficient, faithful digital-twin representations of chemotaxis-driven collective dynamics and guide future higher-dimensional extensions.

Abstract

A general class of hybrid models has been introduced recently, gathering the advantages multiscale descriptions. Concerning biological applications, the particular coupled structure fits to collective cell migrations and pattern formation scenarios. In this context, cells are modelled as discrete entities and their dynamics is given by ODEs, while the chemical signal influencing the motion is considered as a continuous signal which solves a diffusive equation. From the analytical point of view, this class of model has been proved to have a mean-field limit in the Wasserstein distance towards a system given by the coupling of a Vlasov-type equation with the chemoattractant equation. Moreover, a pressureless nonlocal Euler-type system has been derived for these models, rigorously equivalent to the Vlasov one for monokinetic initial data. In the present paper, we present a numerical study of the solutions to the Vlasov and Euler systems, exploring general settings for inital data, far from the monokinetic ones.
Paper Structure (15 sections, 2 theorems, 55 equations, 17 figures, 4 tables)

This paper contains 15 sections, 2 theorems, 55 equations, 17 figures, 4 tables.

Table of Contents

  1. Introduction and rigorous results previously obtained
  2. Previous numerical results
  3. Setting and motivation of the present numerical study
  4. Main results
  5. Numerical Simulations of (P) and (V)
  6. alignment without chemotaxis
  7. alignment with chemotaxis
  8. chemotaxis without alignment
  9. Numerical Simulations of (E)
  10. without chemotaxis: finite time blow up of solution
  11. with chemotaxis and/or damping
  12. Numerical comparison between (V) and (E)
  13. Numerical comparison between (V) and (E) with pressure
  14. Conclusions
  15. Numerical schemes

Key Result

Theorem 1.1

Let $\rho^{in}$ be a compactly supported probability on $\mathbb{R}^{2dN}$, let $\Phi^t_N$ be the mapping generated by the particles system $(eq1,defG,defeqphi0,deff)$ as defined by defphitN, and let $\tau_{\rho^{in}}$ be the function defined in rt2. Then, for any $t\geq 0$, where $\rho^t$ is the solution of the Vlasov equation $(eq1V,defGV, defeqphiV,defgvlasov)$ with initial condition $\rho^{in

Figures (17)

  • Figure 1: Test 1: Numerical simulation of Cucker-Smale model with $\beta=0.05$, at particle level (first line) and kinetic (second line) level.
  • Figure 2: Test 2: Numerical simulation of Cucker-Smale model with $\beta=0.95$, at particle level (first line) and kinetic (second line) level.
  • Figure 3: Test 3: Numerical simulation of Cucker-Smale model with chemotaxis at particle level (first line) and kinetic (second line) level, with parameters as in Test 2 and $\eta=1.4$.
  • Figure 4: Test 4: Numerical simulation of \ref{['VlasovOnlyChemo']} at particle level (first line) and kinetic (second line) level, with parameters as in Test 3.
  • Figure 5: Test 5: Numerical simulation of \ref{['Euler1D_nochemo']}-\ref{['vel_0']} for $c_2=0.2$ (first line) and $c_2=0.5$ (second line).
  • ...and 12 more figures

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 1.2