Codimension-two bifurcations in animal aggregation models with symmetry

Pattern formation in self-organized biological aggregation is a phenomenon that has been studied intensively over the past 20 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 behaviors). In this article, we focus on a class of nonlocal hyperbolic models for self-organized aggregations and show that these models are ${{\bf O(2)}}$-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 ${{\bf O(2)}}$-symmetric codimension-two bifurcations (i.e., Hopf/Hopf, steady-state/Hopf, and steady-state/steady-state 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 numerically show 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 behaviors.
© 2014, Society for Industrial and Applied Mathematics