Movement-based models for abundance data

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.