On a hypercycle equation with infinitely many members

TL;DR

This paper formulates and analyzes a hypercycle model with infinitely many macromolecular types using integro-differential equations, establishing solution properties and numerical evidence for stable wave solutions.

Contribution

It introduces a novel hypercycle equation with infinitely many members and provides analytical and numerical analysis of its solutions and steady states.

Findings

Existence and uniqueness of solutions proved.

Non-uniform steady states identified.

Numerical simulations suggest stable nonlinear wave solutions.

Abstract

A hypercycle equation with infinitely many types of macromolecules is formulated and studied both analytically and numerically. The resulting model is given by an integro-differential equation of the mixed type. Sufficient conditions for the existence, uniqueness, and non-negativity of solutions are formulated and proved. Analytical evidence is provided for the existence of non-uniform (with respect to the second variable) steady states. Finally, numerical simulations strongly indicate the existence of a stable nonlinear wave in the form of the wave train.