Codimension-Two Bifurcations in Animal Aggregation Models with Symmetry

Pattern formation in self-organised biological aggregation is a phenomenon that has been studied intensively over the past twenty years. In general, the studies on pattern formation focus mainly on identifying the biological mechanisms that generate these patterns. However, identifying the mathematical mechanisms behind these patterns is equally important, since it can offer information on the biological parameters that could contribute to the persistence of some patterns and the disappearance of other patterns. Also, it can offer information on the mechanisms that trigger transitions between different patterns (associated with different group behaviours). In this article, we focus on a class of nonlocal hyperbolic models for self-organised aggregations, and show that these models are O(2)-equivariant. We then use group-theoretic methods, linear analysis, weakly nonlinear analysis and numerical simulations to investigate the large variety of patterns that arise through O(2)-symmetric codimension-two bifurcations (i.e., Hopf/Hopf, Steady-state/Hopf and Steady-state/Steadystate mode interactions). We classify the bifurcating solutions according to their isotropy types (subgroups) and we determine the criticality and stability of primary branches of solutions. We show numerically the existence of these solutions and determine scenarios of secondary bifurcations. Also, we discuss the secondary bifurcating solutions from the biological perspective of transitions between different group behaviours.