Pattern Formation in a Nonlocal Model for Cell Attraction and Repulsion in 2 or More Dimensions #

Thomas Jun Jewell, Andrew L Krause, Philip K Maini, Eamonn A Gaffney

12:10 Monday in 2Q49.

Part of the Dynamics of reaction-transport systems session.

Abstract #

Nonlocal interactions are common in biological and ecological systems, such as the extension of filopodia from one cell to another across many cell lengths. In (Painter et al. Bulletin of Mathematical Biology, 2015, vol 77, issue 6), a nonlocal model of coupled integro-differential equations is proposed for contact attraction and repulsion in cell populations, with a focus on pattern forming potential in 1D. Here, we generalise their stability analysis to any number of spatial dimensions, showing that the dispersion relation takes the same form, but with more generalised integral transforms. We determine some key behaviours common to the system in any number of dimensions, such as the presence of Turing bifurcations. However, we also note some fundamental differences from the 1D case, such as the ability to form Turing patterns for repulsive interactions in two or more dimensions. From simulations in the 2 dimensional case, which is closer to real biological systems such as zebrafish pigmentation patterning, we show that the system can exhibit a rich range of behaviours. These include spatio-temporal oscillations, spots and stripes that depend on attractive vs repulsive interactions, and a robust mechanism for fine-tuning pattern wavelength. If nothing else, come along if you want to look at videos of cool shifting patterns!