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The collective behavior of active agents, whether herds of wildebeest ormicroscopic actin filaments propelled by molecular motors, is an excitingfrontier in biological and soft matter physics. Almost three decades ago, Tonerand Tu developed a continuum theory of the collective action of flocks, orherds, that helped launch the modern field of active matter. One challengefaced when applying continuum active matter theories to living phenomena is thecomplex geometric structure of biological environments. Both macroscopic andmicroscopic herds move on asymmetric curved surfaces, like undulating grassplains or the surface layers of cells or embryos, which can render problemsanalytically intractable. In this work, we present a formulation of theToner-Tu flocking theory that uses the finite element method to solve thegoverning equations on arbitrary curved surfaces. First, we test the developedformalism and its numerical implementation in channel flow with scatteringobstacles and on cylindrical and spherical surfaces, comparing our results toanalytical solutions. We then progress to surfaces with arbitrary curvature,moving beyond previously accessible problems to explore herding behavior on avariety of landscapes. Our approach allows the investigation of transients anddynamic solutions not revealed by analytic methods. It also enables versatileincorporation of new geometries and boundary conditions and efficient sweeps ofparameter space. Looking forward, the work presented here lays the groundworkfor a dialogue between Toner-Tu theory and data on collective motion inbiologically-relevant geometries, from drone footage of migrating animal herdsto movies of microscopic cytoskeletal flows within cells.

Wildebeest Herds on Rolling Hills: Flocking on Arbitrary Curved Surfaces