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

by Thomas J. Jewell, Andrew L. Krause, Philip K. Maini, Eamonn A. Gaffney

We study how “chase-and-run” dynamics, where one group pursues while another escapes, are influenced by chirality, a left–right bias in movement seen in animals and cells. Using nonlocal (integro-differential) advection-diffusion models, we show that angled, chiral motion can strengthen pattern formation, prevent oscillations, and generate novel behaviours like rotating pulses of chasers and runners. We also show how these dynamics can determine whether populations mix or separate. By comparing linear stability analysis with numerical simulations, we reveal when standard theory holds and when it fails. Overall, our results highlight chirality’s potential broader role in shaping ecological and developmental patterns.

Left, a schematic showing a chiral chase-and-run interaction with a chaser cell, c, pursuing a runner cell, ρ. Their movement has a consistent clockwise or anticlockwise bias parameterised by the angles α. Right, a phase diagram of the model, showing the patterning outcome at different values of the chiral running angle and the ratio of diffusivities. Coloured regions are predicted by linear stability analysis and marked points are from numerical simulations.