  Ordinary differential equations have been used to model dynamical systems in
a broad range. Model checking for parametric ordinary differential equations is
a necessary step to check whether the assumed models are plausible. In this
paper we introduce three test statistics for their different purposes. We first
give a trajectory matching-based test for the whole system. To further identify
which component function(s) would be wrongly modelled, we introduce two test
statistics that are based on integral matching and gradient matching
respectively. We investigate the asymptotic properties of the three test
statistics under the null, global and local alternative hypothesis. To achieve
these purposes, we also investigate the asymptotic properties of nonlinear
least squares estimation and two-step collocation estimation under both the
null and alternatives. The results about the estimations are also new in the
literature. To examine the performances of the tests, we conduct several
numerical simulations. A real data example about immune cell kinetics and
trafficking for influenza infection is analyzed for illustration.
