Bifurcation analysis of a nonlocal two-communication mechanism model for animal aggregation with O(2) symmetry

Speaker: Alberto Alinas

Affiliation: UOIT

The study of animal aggregations has become a topic of great interest due to its theoretical significance and its practical applications. A large variety of models have been proposed and investigated over the years. I will be presenting a class of nonlocal hyperbolic models with two communication mechanisms in a domain with periodic boundary conditions. I will show that the model is symmetric with respect to the group of translations and reflections which is denoted by \textbf{O(2)}. I also show the \textbf{O(2)}-symmetry of the homogeneous steady-state solution that exists for all parameter values of the system. Using group representation theory, I perform a linear stability analysis from which curves of critical eigenvalues are obtained as a function of the wavenumber in a two-dimensional parameter space. In particular, I will show that crossing the critical curve as a parameter is varied leads to \emph{spontaneous symmetry-breaking}. That is, unique bifurcating branches of solutions that possess $\mathbf{D}_n$-symmetry emanate from the \textbf{O(2)}-symmetric equilibrium.