Collective motion driven by nutrient consumption

Article type: Research Article

Authors: Jabin, Pierre-Emmanuel^a | Perthame, Benoît^{b; *}

Affiliations: [a] Department of Mathematics and Huck Institutes, Pennsylvania State University, State College, PA 16801, USA | [b] Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions UMR7598, F-75005 Paris, France

Correspondence: [*] Corresponding author. E-mail: Benoit.Perthame@sorbonne-universite.fr.

Abstract: A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics.

Keywords: Asymptotic analysis, pattern formation, reaction–diffusion equations

DOI: 10.3233/ASY-221820

Journal: Asymptotic Analysis, vol. 133, no. 4, pp. 483-497, 2023