Equilibrium and Competition in Evolutionary Dynamics
J. Medina Diaz, F. Pe\~na-Garcia, and Irbin Llanqui

TL;DR
This paper models the dynamics of polymer formation, competition, and replication, establishing conditions for equilibrium stability and the effects of resource constraints and fitness on system behavior.
Contribution
It introduces a comprehensive deterministic framework for polymer growth, competition, and replication, analyzing equilibrium existence, stability, and bifurcations under various conditions.
Findings
Existence of a unique positive equilibrium under fixed precursors.
Global exponential convergence to steady state in the polymerization model.
Resource thresholds determine the stability and number of equilibria in competition models.
Abstract
A fundamental problem in protobiological dynamics is to understand how chemically generated polymers can form persistent sequence distributions before the emergence of replication. We study deterministic polymer growth in which each finite sequence is followed along its genealogical structure. The system pictures an open polymerization cascade in which each polymer is produced from a unique precursor and lost by degradation and further extension. Setting fixed activated precursors, we show global well-posedness, positivity, uniqueness of a strictly positive equilibrium, and exponential convergence to an explicit steady state distribution. Under an additional uniform decay condition, this convergence becomes global exponential stability in a uniform norm. We then couple the polymerization to a shared environmental resource with logistic growth and depletion by two activated precursors.…
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