Exploring the error threshold through a Poincaré compactification

Abstract

[en] Continuous dynamical systems have been deeply studied since Newtonian mechanics appeared. For decades, qualitative dynamics of planar differential systems have been developed achieving a big number of results, relegating the study of infinity to a second place. On one hand, this Bachelor’s degree final project aims to examine in detail the behaviour of a vector field on a neighborhood of infinity. With this purpose, we will explain the Poincaré Compactification and use it to investigate the error threshold at infinity in the quasispecies model. On the other hand, we will focus on quasispecies theory, a biological theory widely studied in the context of the origin of life and RNA viruses. We will work on a viral quasispecies model, introducing a logistic constraint assumption that will let us analyze the error threshold in the finite plane. Under the logistic approach, the bifurcation we have characterized for the error threshold is a transcritical bifurcation. Finally, we will use numerical results to provide further insights on the dynamics and the bifurcations.