# Chase-and-Run and Chirality in Nonlocal Models of Pattern Formation

## TL;DR

This paper studies how angled, asymmetric movement in chase-and-run dynamics influences pattern formation in natural systems like zebrafish skin and animal motion.

## Contribution

The paper introduces a nonlocal advection-diffusion model that incorporates arbitrary angles and chirality to reveal new dynamical structures in pattern formation.

## Key findings

- Chirality enhances pattern formation and suppresses oscillations in chase-and-run systems.
- Chirality leads to new structures like rotating pulses of chasers and runners.
- Chase-and-run dynamics can cause populations to either mix or separate.

## Abstract

Chase-and-run dynamics, in which one population pursues another that flees from it, are found throughout nature, from predator-prey interactions in ecosystems to the collective motion of cells during development. Intriguingly, in many of these systems, the movement is not straight; instead, ‘runners’ veer off at an angle from their pursuers. This angled movement often exhibits a consistent left–right asymmetry, known as lateralisation or chirality. Inspired by such phenomena in zebrafish skin patterns and evasive animal motion, we explore how chirality shapes the emergence of patterns in nonlocal (integro-differential) advection-diffusion models. We extend such models to allow movement at arbitrary angles, uncovering a rich landscape of behaviours. We find that chirality can enhance pattern formation, suppress oscillations, and give rise to entirely new dynamical structures, such as rotating pulses of chasers and runners. We also uncover how chase-and-run dynamics can cause populations to mix or separate. Through linear stability analysis, we identify physical mechanisms that drive some of these effects, whilst also exposing striking limitations of this theory in capturing more complex dynamics. Our findings suggest that chirality could have roles in ecological and cellular patterning beyond simply breaking left-right symmetry.