  We develop two statistical models for space-time abundance data based on the
modelling of an underlying continuous movement. Different from other models for
abundance in the current statistical ecology literature, our models focus
especially on an explicit connection between the movement of the individuals
and the count, and on the space-time auto-correlation thus induced. Our first
model (Snapshot) describes the count of free individuals with a false-negative
detection error. Our second model (Capture) describes the capture and retention
in traps of moving individuals, and it is constructed using an axiomatic
approach establishing three simple principles, from which it is deduced that
the density of the capture time is the solution of a Volterra integral equation
of the second kind. We make explicit the space-time mean and covariance
structures of the abundance fields thus generated, and we develop simulation
methods for both models. The joint distribution of the space-time counts is an
instance of a new multivariate distribution, here baptised the
Evolving-Categories Multinomial distribution, for which we establish some key
properties. Since a general expression of the likelihood remains intractable,
we propose an approximated MLE fitting method by replacing it by a multivariate
Gaussian one, which is justified by central limit theorem and respects mean and
covariance structures. We apply this method to experimental data of fruit flies
released in a meadow and repeatedly captured and counted in an array of traps.
We estimate spread and advection parameters, compare our models to an
Ecological Diffusion model, and conduct simulation studies to validate our
analysis. Asymptotic consistency is experimentally verified. We conclude that
we can estimate movement parameters using only abundance data, but must be
aware of the necessary conditions to avoid underestimation of spread
parameters.
