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

Abstract

The study of animal aggregations has become a topic of great interest due to its practical applications and theoretical significance. The common occurrence of these aggregations in nature lead to a large variety of models that have been proposed and investigated over the years. In this thesis, we present a class of models that considers communication among the individuals to be the basis of the social interaction within a group. In particular, we present the nonlocal hyperbolic model with two-communication mechanisms in a domain with periodic boundary conditions introduced by Eftimie (J. Theoretical Biol., 337, 42-53, 2013). We show that the system is symmetric with respect to the group O(2) of spatial translations and reflection. Using symmetry techniques from the text by Golubitsky et al. (Singularities and Groups in Bifurcation Theory, Vol II. Springer, 1988), we focus on studying a spatially homogeneous equilibrium and its linear stability as a function of parameters for attraction and repulsion. Given this symmetry perspective, we obtain a decomposition of the linearization at the equilibrium in terms of 4 x 4 matrices. We obtain a diagram in the attraction and repulsion parameter space for which the critical curves forming the boundary of the asymptotic stability region for the equilibrium are shown. Given the obtained boundaries, we then describe the patterns obtained via steady-state and Hopf bifurcations with symmetry as critical curves are crossed.