Selecting a common direction
We present results and analysis of models for contact-induced turning responses and alignment in populations of interacting individuals. Such models describe distributions of orientation, and how these evolve under different assumptions about the turning behaviour of individuals. One of these models was first used to describe interactions between mammalian cells called fibroblasts in Edelstein-Keshet and Ermentrout (1990) J. Math. Biol. 29: 33–58 (henceforth abbreviated EKE 1990). Here the original model is generalized to encompass motion in both 2 and 3 dimensions. Two modifications of this model are introduced: in one, the turning is described by a gradual direction shift (rather than abrupt transition). In another variant, the interactions between individuals changes as the density of the population increases to include the effects of crowding. Using linear stability analysis and synergetics analysis of interacting modes we describe the nature and stability properties of the steady state solutions. We investigate how nonhomogeneous pattern evolves close to the bifurcation point. We find that individuals tend to cluster together in one direction of alignment.