Analysis of Hopf/Hopf bifurcations in nonlocal hyperbolic models for self-organised aggregations

The modelling and investigation of complex spatial and spatio-temporal patterns exhibited by a various self-organised biological aggregations has become one of the most
rapidly-expanding research areas. Generally, the majority of the studies in this area either try to reproduce numerically the observed patterns, or use existence results to
prove analytically that the models can exhibit certain types of patterns. Here, we focus on a class of nonlocal hyperbolic models for self-organised movement and aggregations,
and investigate the bifurcation of some spatial and spatio-temporal patterns observed numerically near a codimension-2 Hopf/Hopf bifurcation point. Using weakly nonlinear
analysis and the symmetry of the model, we identify analytically all types of solutions that can exist in the neighbourhood of this bifurcation point. We also discuss the stability
of these solutions, and the implication of these stability results on the observed numerical patterns.