  We present a formalism for modelling parametric amplification by resonators
subject to rate-limited nonlinearity of mixed reactive/dissipative character,
with particular relevance to superconducting devices. The non-linearity is
assumed to be characterised by a single state parameter, which responds to
changes in the energy stored in the resonator with finite response time. We
show how the operating point and small signal amplification behaviour of the
pumped resonator can be calculated, characterised and optimised in terms of a
set of three dimensionless parameters. The formalism is then illustrated with a
simple, first-order, model nonlinearity and the implications for amplification
via quasiparticle generation in a superconductor discussed. Throughout we
describe how the parameters needed to characterise the device can be determined
experimentally from steady-state measurements. A key result of this paper is
that rate-limiting of a nonlinear mechanism does not preclude amplification,
although it does limit the bandwidth over which it may be achieved.
