Phenotype-structured nonlocal aggregation models I: signal production and response

Abstract

Individuals within a cellular or ecological population can exhibit substantial variation in their traits or phenotype. Such heterogeneity can fundamentally alter population-level behaviour, including the ability to aggregate and form spatial patterns. We investigate these effects by extending nonlocal aggregation models, widely used to model cell--cell adhesion and collective animal movement, to incorporate phenotype structure. We characterise how heterogeneity and switching between phenotypes affect whether patterns form, how phenotypes are distributed within them, and whether they are stationary or exhibit spatio-temporal oscillations. We show that pattern formation is promoted when phenotypes that produce stronger interaction signals also respond more strongly to them, whereas specialisation towards either signal production or response inhibits pattern formation. Increasing rates of switching between phenotypes dampen these effects. Heterogeneity can also enable weakly interacting, and even repulsive, individuals to pattern as part of the collective. Furthermore, movement in phenotype and physical space can interact to restructure patterns and generate spatio-temporal oscillations. A further key message of this study is the value of considering both discrete $N$-phenotype and continuous phenotype formulations. We develop both in parallel, demonstrating formal correspondences, but also advantages to each framework in terms of mathematical analyses and interpretability.