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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2017
- DOI:
- 10.48550/arXiv.1705.06785
- arXiv:
- arXiv:1705.06785
- Bibcode:
- 2017arXiv170506785B
- Keywords:
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- Mathematics - Dynamical Systems
- E-Print:
- 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