  Some ODE models treat ecological disturbance as a continuous process, even
disturbances such as fire that occur almost instantaneously on the timescale of
system recovery. Alternatively, flow-kick models resolve disturbances as
discrete impulses that change an ODE system's state periodically in time. Here
we compare the dynamics of continuously disturbed ODE models to those of
flow-kick models with the same average disturbance rate. In the case that kicks
are small and high-frequency, we find multiple similarities between continuous
and analogous discrete disturbance models. First, we prove that flow-kick maps
generate an analogous vector field in the limit as the period between kicks
approaches zero. Second, we present conditions under which equilibria,
saddle-node bifurcations, and transcritical bifurcations continue from ODE to
flow-kick systems. On the other hand, we also provide numerical evidence that
similarities between continuous and discrete disturbance models can break down
as the period between kick grows. We illustrate implications of these
differences for climate change in a nonspatial Klausmeier model of vegetation
and precipitation dynamics. We conclude that although ODEs may suffice to model
high-frequency disturbances, resolving lower-frequency disturbances in time may
be essential to effectively predicting their effects.
