# Robust persistence and permanence of polynomial and power law dynamical systems

## TL;DR

This paper introduces the class of tropically endotactic polynomial dynamical systems and proves their permanence in two dimensions, extending known results for reversible and endotactic systems, with implications for biological modeling.

## Contribution

It defines tropically endotactic systems and establishes their permanence in two dimensions, generalizing previous results for specific subclasses.

## Key findings

- Two-dimensional tropically endotactic systems are permanent.
- Results extend permanence to broader classes of systems.
- Implications for biological and ecological models.

## 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 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.