Darboux integrability and dynamics of the Basener-Ross population model

We deal with the Basener and Ross model for the evolution of human population in Easter island. We study the Darboux integrability of this model and characterize all its global dynamics in the Poincaré disc, obtaining 15 different topological phase portraits

The Hyperbolic Derivative in the Poincaré Ball Model of Hyperbolic Geometry

AbstractThe generic Möbius transformation of the complex open unit disc induces a binary operation in the disc, called the Möbius addition. Following its introduction, the extension of the Möbius addition to the ball of any real inner product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincaré ball model of hyperbolic geometry. The Möbius gyrovector spaces that emerge provide the setting for the Poincaré ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the Möbius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry

Dynamics of a planar Coulomb gas

We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincar{\'e} inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll -- Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.Comment: Minor revision for Annals of Applied Probabilit