Spatio-angular order in populations of self-aligning objects: Formation of oriented patches

We consider a class of models for the dynamic behaviour of ensembles of objects whose interactions depend on angular orientations as well as spatial positions. The "objects" could be particles, molecules, cells or organisms. We show how processes such as mutual alignment, pattern formation, and aggregation are describable by sets of partial differential equations containing convolution terms. Kernels of these convolutions are functions that describe the intensity of interaction of the objects at various relative angles and distances to one another. Such models appear to contain a rich diversity of possible behaviour and dynamics, depending on details of the kernels involved. They are also of great generality, with applications in the natural sciences, including physics and biology. In the latter, the examples that fall into such class include molecular, cellular, as well as social phenomena. Analysis of the equations, and predictions in several test cases are presented. This paper is related to Mogilner and Edelstein-Keshet (1995) in which the spatially-homogeneous version of these models was investigated.