Speed and stability of segregated waves in a pressure-based model of heterogeneous cell populations
Abstract
We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction--cross--diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous--Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.
Related
- Medial septum glutamatergic neurons support speed coding and grid cell spatial stability in the medial entorhinal cortex
- Individual-Specific Gaussian Graphical Models for Heterogeneous Populations with Application to Epigenetic Gene Regulation in Lung Adenocarcinoma
- Bioimaging With Fluorescent Nucleic‐Acid Aptamers for the Specific Detection and Quantification of Pseudomonas aeruginosa Alone and in Heterogeneous Bacterial Populations
- A Unified Unsupervised Framework for Genome-Wide Association Studies in Heterogeneous Populations
- Regularized logistic regression-based gene selection for functional analysis of heterogeneous scRNA-seq macrophage populations in muscle aging and regeneration.
BibTeX
@misc{falco2026speed,
title = {{Speed and stability of segregated waves in a pressure-based model of heterogeneous cell populations}},
author = {Carles Falcó and Rebecca M. Crossley and Martina Conte and Tommaso Lorenzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09043},
eprint = {2609.09043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09043},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Speed and stability of segregated waves in a pressure-based model of heterogeneous cell populations
AU - Carles Falcó
AU - Rebecca M. Crossley
AU - Martina Conte
AU - Tommaso Lorenzi
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.09043
UR - https://arxiv.org/abs/2609.09043
AB - We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction--cross--diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous--Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.
ER -