(Pontificia Universidad Católica del Perú, 2022-08-31) Puchuri, Liliana; Bueno, Orestes
A predator-prey model of Gause type is an extension of the classical Lotka-Volterra predator-prey model. In this work, we study a predator-prey model of Gause type, where the prey growth rate is subject to an Allee effect and the action of the predator over the prey is given by a square-root functional response, which is non-differentiable at the y-axis. This kind of functional response appropriately models systems in which the prey have a strong herd structure, as the predators mostly interact with the prey on the boundary of the herd. Because of the square root term in the functional response, studying the behavior of the solutions near the origin is more subtle and interesting than other standard models.Our study is divided into two parts: the local classification of the equilibrium points, and the behavior of the solutions in certain invariant set when the model has a strong Allee effect. In one our main results we prove, for a wide choice of parameters, that the solutions in certain invariant set approach to the y-axis. Moreover, for a certain choice of parameters, we show the existence of a separatrix curve dividing the invariant set in two regions, where in one region any solution approaches the y-axis and in the other there is a globally asymptotically stable equilibrium point. We also give conditions on the parameters to ensure the existence of a center-type equilibrium, and show the existence of a Hopf bifurcation.
(Pontificia Universidad Católica del Perú, 2022-08-31) Fernández Sánchez, Percy; García, Evelia R.; Saravia Molina, Nancy
We present a relation between the Tjurina and Milnor numbers of a holomorphic foliation of secont type and the Tjurina and Milnor numbers of its union of separatrices when this last one is holomorphic.
(Pontificia Universidad Católica del Perú, 2022-08-31) Vargas Pallete, Franco
In this paper we are going to develop an unpublished argument of William Thurston that shows the existence of hyperbolic groups with no non-trivial representations to linear groups over any ring, for a given dimension. His approach studies the space of representations of any non-elementary hyperbolic group G to conclude that there is a hyperbolic quotient with such property.
