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.