Robust persistence and permanence of polynomial and power law dynamical systems

Abstract

A persistent dynamical system in ${\mathbb{R}}_{>0}^d$ is one whose solutions have positive lower bounds for large $t$, while a permanent dynamical system in ${\mathbb{R}}_{>0}^d$ 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 two-dimensional tropically endotactic polynomial dynamical systems are permanent, irrespective of the values of (possibly time-dependent) parameters in these systems. These results generalize the permanence of two-dimensional reversible, weakly reversible, and endotactic mass action systems.

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

References in corpus (1)

• Global convergence in systems of differential equations arising from chemical reaction networks

Cited by in corpus (2)

• Permanence of Weakly Reversible Mass-Action Systems with a Single Linkage Class
• On classes of reaction networks and their associated polynomial dynamical systems