  Many spatio-temporal data record the time of birth and death of individuals,
along with their spatial trajectories during their lifetime, whether through
continuous-time observations or discrete-time observations. Natural
applications include epidemiology, individual-based modelling in ecology,
spatio-temporal dynamics observed in bio-imaging, and computer vision. The aim
of this article is to estimate in this context the birth and death intensity
functions, that depend in full generality on the current spatial configuration
of all alive individuals. While the temporal evolution of the population size
is a simple birth-death process, observing the lifetime and trajectories of all
individuals calls for a new paradigm. To formalise this framework, we introduce
spatial birth-death-move processes, where the birth and death dynamics depends
on the current spatial configuration of the population and where individuals
can move during their lifetime according to a continuous Markov process with
possible interactions.We consider non-parametric kernel estimators of their
birth and death intensity functions. The setting is original because each
observation in time belongs to a non-vectorial, infinite dimensional space and
the dependence between observations is barely tractable. We prove the
consistency of the estimators in presence of continuous-time and discrete-time
observations, under fairly simple conditions. We moreover discuss how we can
take advantage in practice of structural assumptions made on the intensity
functions and we explain how data-driven bandwidth selection can be conducted,
despite the unknown (and sometimes undefined) second order moments of the
estimators. We finally apply our statistical method to the analysis of the
spatio-temporal dynamics of proteins involved in exocytosis in cells, providing
new insights on this complex mechanism.
