# Weakly reversible single linkage class realizations of polynomial dynamical systems: an algorithmic perspective

## TL;DR

This paper presents an algorithm to identify weakly reversible single linkage class realizations of polynomial dynamical systems, which guarantees persistent long-term behavior of the modeled species across various parameters.

## Contribution

It introduces a novel algorithmic approach to find $W ext{R}^1$ realizations, aiding the analysis of stability and persistence in polynomial dynamical systems.

## Key findings

- Algorithm successfully finds $W ext{R}^1$ realizations when they exist.
- Ensures robustness of long-term dynamics and species persistence.
- Applicable to systems in ecology, biochemistry, and epidemiology.

## Abstract

Systems of differential equations with polynomial right-hand sides are very common in applications. In particular, when restricted to the positive orthant, they appear naturally (according to the law of mass-action kinetics) in ecology, population dynamics, as models of biochemical interaction networks, and models of the spread of infectious diseases. On the other hand, their mathematical analysis is very challenging in general; in particular, it is very difficult to answer questions about the long-term dynamics of the variables (species) in the model, such as questions about persistence and extinction. Even if we restrict our attention to mass-action systems, these questions still remain challenging. On the other hand, if a polynomial dynamical system has a weakly reversible single linkage class ($W\!R^1$) realization, then its long-term dynamics is known to be remarkably robust: all the variables are persistent (i.e., no species goes extinct), irrespective of the values of the parameters in the model. Here we describe an algorithm for finding $W\!R^1$ realizations of polynomial dynamical systems, whenever such realizations exist.