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arXiv · 1705.06785

Robust persistence and permanence of polynomial and power law dynamical systems

James D. Brunner·Gheorghe Craciun
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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.

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James D. Brunner, Gheorghe Craciun. 2017-05-18. Robust persistence and permanence of polynomial and power law dynamical systems. https://doi.org/10.1137/17m1133762

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