Mass-Conservative Numerical Analysis of an Attraction–Repulsion Keller–Segel System for Self-Organized Biological Pattern Dynamics

Abstract

Collective migration driven by attractive and repulsive chemical cues is a basic mechanism behind self-organized biological pattern formation. In this paper, we study a one-dimensional attraction–repulsion Keller–Segel (ARKS) system as a minimal mathematical model for such competing interactions. Our emphasis is not on a purely analytical characterization of the model, but on a numerical investigation of biologically relevant spatiotemporal dynamics. To this end, we construct a conservative finite-difference scheme by adapting the structure-preserving framework of Saito for the classical Keller–Segel system to the ARKS setting through a single effective potential that combines the attractive and repulsive signals into one discrete drift, to which Saito’s conservative upwind treatment can be directly applied. The proposed scheme preserves the total mass of the cell density at the discrete level, and the discrete density is shown to remain nonnegative under an adaptive time-step restriction. Numerical experiments systematically reveal, within this single conservative framework, three characteristic dynamical regimes—traveling patterns, multi-pulse interactions including bouncing behavior, and breathing pulses—and quantitatively characterize their dependence on the attraction–repulsion balance. These results illustrate how the balance between attraction and repulsion controls self-organized information processing in living systems, and demonstrate that structure-preserving computation is an effective tool for interpreting such dynamics from a biological viewpoint.