Dynamics of a discrete hypercycle

Abstract

[en] The concept of the Hypercycle was introduced in 1977 by Manfred Eigen and Peter Schuster within the framework of origins of life and prebiotic evolution. Hypercycle are catalytic sets of macromolecules, where each replicator catalyzes the replication of the next species of the set. This system was proposed as a possible solution to the information crisis in prebiotic evolution. Hypercycles are cooperative systems that allow replicators to increase their information content beyond the error threshold. This project studies the dynamics of a discrete-time model of the hypercycle considering heterocatalytic interactions. To date, hypercycles’ dynamics has been mainly studied using continuous-time dynamical systems. We follow the Hofbauer’s discrete model [13]. First, we introduce some important and necessary mathematical notions. Then, we also review the biological concept of the hypercycle and some criticisms that it has received. We present a complete proof of the fact that the hypercycle is a cooperative system. Also, we present an analytic study of the fixed point in any dimension and its stability. In particular, in dimension three we prove that fixed point is globally asymptotically stable. In dimension four we have obtained a stable invariant curve for all values of the discreteness parameter.