Department of Biomathematics, School of Medicine, Box 951766, University of California, Los Angeles, CA 90095-1766, U.S.A.
(Received December 20, 1995; Accepted January 23, 1996)
Keywords: Orientation, Tensor, Alignment, Wave, Aggregation
Abstract. Populations of interacting, oriented individuals occur on a broad range of scales in biology, from animal herds, through populations of cells, to actin filaments within the cytoplasm. The dynamics of these systems are worthy of study yet mathematical models for oriented, possibly migrating populations have typically been complicated and have necessitated a reliance on numerical simulations for insight. In this paper, we introduce a method for constructing relatively simple partial differential equations which describe how the orientation and density of individuals vary over space and time. An initial model involving integral equations is constructed based on the assumed interactions between neighbors. Formal methods are then used to generate a related model of the reaction-diffusion-advection type. The dependent variables become the components of so-called orientation tensors which approximately describe the orientation distribution at each point. We apply the method to a general class of models, construct models for orientation and aggregation in animal herds and study in more detail a model for fibroblasts in culture. Individuals reorient as a result of interactions with neighbours and waves of alignment pass through the population.