  We give a notion of branching systems on ultragraphs. From this we build
concrete representations of ultragraph C*-algebras on the bounded linear
operators of Hilbert spaces. To each branching system of an ultragraph we
describe the associated Perron-Frobenius operator in terms of the induced
representation. We show that every permutative representation of an ultragraph
C*-algebra is unitary equivalent to a representation arising from a branching
system. We give a sufficient condition on ultragraphs such that a large class
of representations of the C*-algebras of these ultragraphs is permutative. To
give a sufficient condition on branching systems so that their induced
representations are faithful we generalize Szyma{\'n}ski's version of the
Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras.
