  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.
