  Despite their noted potential in polynomial-system solving, there are few
concrete examples of Newton-Okounkov bodies arising from applications.
Accordingly, in this paper, we introduce a new application of Newton-Okounkov
body theory to the study of chemical reaction networks, and compute several
examples. An important invariant of a chemical reaction network is its maximum
number of positive steady states, which is realized as the maximum number of
positive real roots of a parametrized polynomial system. Here, we introduce a
new upper bound on this number, namely the `Newton-Okounkov body bound' of a
chemical reaction network. Through explicit examples, we show that the
Newton-Okounkov body bound of a network gives a good upper bound on its maximum
number of positive steady states. We also compare this Newton-Okounkov body
bound to a related upper bound, namely the mixed volume of a chemical reaction
network, and find that it often achieves better bounds.
