  Understanding the statistical properties of a collection of individuals
subject to random displacements and birth-and-death events is key to several
applications in physics and life sciences, encompassing the diagnostic of
nuclear reactors and the analysis of epidemic patterns. Previous investigations
of the critical regime, where births and deaths balance on average, have shown
that highly non-Poissonian fluctuations might occur in the population, leading
to spontaneous spatial clustering, and eventually to a critical catastrophe,
where fluctuations can result in the extinction of the population. A milder
behaviour is observed when the population size is kept constant:
thefluctuations asymptotically level off and the critical catastrophe is
averted. In this paper, we shall extend these results by considering the
broader class of models with prompt and delayed birth-and-death events, which
mimic the presence of precursors in nuclear reactor physics or incubation
inepidemics. We shall consider models with and without population control
mechanisms. Analytical or semi-analytical results for the density, the
two-point correlation function and the mean-squared pair distance will be
derived and compared to Monte Carlo simulations, which will be used as a
reference.
