The Keller-Segel Model for Chemotaxis on Metapopulation Graphs and Graphons
Abstract
Many models of spatial populations use directed motion towards environmental attractants as a mechanism for spatial pattern formation. While prior approaches typically focus on partial differential equation models for directed motion in continuous space, it is natural to consider the dynamics of biological and social systems on a metapopulation featuring well-mixed local patches and spatial motion between patches. In this paper, we formulate a version of the Keller-Segel model for chemotaxis on a graph metapopulation, modeling a chemoattractant that can diffuse across patches and a population of individuals who move through a biased random walk through comparing chemoattractant levels along edges of the metapopulation network. We also derive a continuum limit for our model describing chemotaxis in a population on patches connected by a graphon, showing that our resulting integro-differential equation model arises as the limit of our discrete patch model for families of large deterministic or random graphs. Through linear stability analysis, we derive conditions on the chemotactic sensitivity required to produce instability of a spatially uniform equilibrium for our Keller-Segel models in terms of the reaction parameters and the eigenvalues of the Laplacian operator for the metapopulation graph or graphon. Through a mix of weakly nonlinear stability analysis, numerical simulations, and analytical bounds on solutions, we can also characterize the form of pattern-forming instability in our graphon model and show how our choice of demographic dynamics for the population and the graphon adjacency function can impact the long-time total population size and chemoattractant concentration achieved in emergent spatial patterns.
Related
- Infinite time blow-up of solutions to the classical Keller-Segel model of chemotaxis in higher dimensions
- A Li-Yau and Aronson-Bénilan approach for the Keller-Segel system with critical exponent
- Hybridizable Discontinuous Galerkin methods for the Keller-Segel model
- Stability of the critical constant steady state of a Keller--Segel model
- Well-posedness of 2D stochastic parabolic-parabolic Keller-Segel system with logistic source
BibTeX
@misc{kim2026kellersegel,
title = {{The Keller-Segel Model for Chemotaxis on Metapopulation Graphs and Graphons}},
author = {Daecheol Kim and Yefei Zhang and Yuxuan Zhao and Daniel B. Cooney},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31049},
eprint = {2610.31049},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31049},
}
RIS
TY - UNPB
TI - The Keller-Segel Model for Chemotaxis on Metapopulation Graphs and Graphons
AU - Daecheol Kim
AU - Yefei Zhang
AU - Yuxuan Zhao
AU - Daniel B. Cooney
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31049
UR - https://arxiv.org/abs/2610.31049
AB - Many models of spatial populations use directed motion towards environmental attractants as a mechanism for spatial pattern formation. While prior approaches typically focus on partial differential equation models for directed motion in continuous space, it is natural to consider the dynamics of biological and social systems on a metapopulation featuring well-mixed local patches and spatial motion between patches. In this paper, we formulate a version of the Keller-Segel model for chemotaxis on a graph metapopulation, modeling a chemoattractant that can diffuse across patches and a population of individuals who move through a biased random walk through comparing chemoattractant levels along edges of the metapopulation network. We also derive a continuum limit for our model describing chemotaxis in a population on patches connected by a graphon, showing that our resulting integro-differential equation model arises as the limit of our discrete patch model for families of large deterministic or random graphs. Through linear stability analysis, we derive conditions on the chemotactic sensitivity required to produce instability of a spatially uniform equilibrium for our Keller-Segel models in terms of the reaction parameters and the eigenvalues of the Laplacian operator for the metapopulation graph or graphon. Through a mix of weakly nonlinear stability analysis, numerical simulations, and analytical bounds on solutions, we can also characterize the form of pattern-forming instability in our graphon model and show how our choice of demographic dynamics for the population and the graphon adjacency function can impact the long-time total population size and chemoattractant concentration achieved in emergent spatial patterns.
ER -