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 D3-symmetric steady-state solution.
|Title of host publication||Active Particles, Volume 2|
|Editors||N. Bellomo, P. Degond, E. Tadmor|
|Place of Publication||Switzerland|
|Number of pages||26|
|Publication status||Published - 2019|
|Name||Modeling and Simulation in Science, Engineering and Technology|
Buono, P-L., Eftimie, R., Kovacic, M., & van Veen, L. (2019). Kinetic models for pattern formation in animal aggregations: a symmetry and bifurcation approach. In N. Bellomo, P. Degond, & E. Tadmor (Eds.), Active Particles, Volume 2 (Vol. 2, pp. 39-64). (Modeling and Simulation in Science, Engineering and Technology; Vol. 2). Birkhauser. https://doi.org/10.1007/978-3-030-20297-2_2