Kinetic Models for Pattern Formation in Animal Aggregations: A Symmetry and Bifurcation Approach

Abstract

In this study we start by reviewing a class of 1D hyperbolic/kinetic models (with two velocities) used to investigate the collective behaviour of cells, bacteria or animals. We then focus on a restricted class of nonlocal models that incorporate various inter-individual communication mechanisms, and discuss how the symmetries of these models impact the various types of spatially-heterogeneous and spatially-homogeneous equilibria exhibited by these nonlocal models. In particular, we characterise a new type of equilibria that was not discussed before for this class of models, namely a relative equilibria. Then we simulate numerically these models and show a variety of spatio-temporal patterns (including classic equilibria and relative equilibria) exhibited by these models. We conclude by introducing a continuation algorithm (which takes into account the models symmetries) that allows us to track the solutions bifurcating from these different equilibria. Finally, we apply this algorithm to identify a D 3 -symmetric steady-state solution.