Robust persistence and permanence of polynomial and power law dynamical systems
Abstract
A persistent dynamical system in $\mathbb{R}^d_{> 0}$ is one whose solutions have positive lower bounds for large $t$, while a permanent dynamical system in $\mathbb{R}^d_{> 0}$ is one whose solutions have uniform upper and lower bounds for large $t$. These properties have important applications for the study of mathematical models in biochemistry, cell biology, and ecology. Inspired by reaction network theory, we define a class of polynomial dynamical systems called tropically endotactic. We show that twodimensional tropically endotactic polynomial dynamical systems are permanent, irrespective of the values of (possibly timedependent) parameters in these systems. These results generalize the permanence of twodimensional reversible, weakly reversible, and endotactic mass action systems.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.06785
 Bibcode:
 2017arXiv170506785B
 Keywords:

 Mathematics  Dynamical Systems
 EPrint:
 26 pages, 11 figures. Version 3 clarifies some explanations and adds a detailed calculation to an example which clarifies how the result can be applied