Curvature in chemotaxis: A model for ant trail pattern formation

TL;DR

This paper introduces a novel chemotaxis model inspired by ant trail formation, featuring a unique transport term based on second derivatives of the chemical field, with proven existence, uniqueness, and numerical simulations.

Contribution

It presents a new PDE model for chemotaxis with a second-derivative-dependent transport term, and establishes its mathematical well-posedness and numerical trail formation evidence.

Findings

Global existence and uniqueness of solutions

Propagation of regularity from initial data

Numerical simulations showing trail pattern formation

Abstract

In this paper, we propose a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic local PDE system, for the population density and the chemical field. The main novelty lies in the transport term of the population density, which depends on the second-order derivatives of the chemical field. This term is derived as an anticipation-reaction steering mechanism of an infinitesimally small ant as its size approaches zero. We establish global-in-time existence and uniqueness for the model, and the propagation of regularity from the initial data. Then, we build a numerical scheme and present various examples that provide hints of trail formation.