  This paper develops low-dimensional moment methods for advective problems on
networks of domains. The evolution of a density function is described by a
linear advection-diffusion-reaction equation on each domain, combined via
advective flux coupling across domains in the network graph. The PDEs'
coefficients vary in time and across domains but they are fixed along each
domain. As a result, the solution on each domain is frequently close to a
Gaussian that moves, decays, and widens. For that reason, this work studies
moment methods that track only three degrees of freedom per domain -- in
contrast to traditional PDE discretization methods that tend to require many
more variables per domain. A simple ODE-based moment method is developed, as
well as an asymptotic-preserving scheme. We apply the methodology to an
application that models the life cycle of forest pests that undergo different
life stages and developmental pathways. The model is calibrated for the spotted
lanternfly, an invasive species present in the Eastern USA. We showcase that
the moment method, despite its significant low-dimensionality, can successfully
reproduce the prediction of the pest's establishment potential, compared to
much higher-dimensional computational approaches.
