Space and time correlations for diffusion models with prompt and delayed birth-and-death events

TL;DR

This paper analyzes the spatial and temporal correlations in diffusion models with birth-and-death events, including prompt and delayed processes, to understand fluctuations and clustering in populations relevant to physics and life sciences.

Contribution

It extends existing models by incorporating prompt and delayed birth-and-death events, providing analytical results and simulations for correlation functions and population fluctuations.

Findings

Highly non-Poissonian fluctuations can lead to clustering and critical catastrophes.

Population control mechanisms can prevent fluctuations from diverging.

Analytical results align well with Monte Carlo simulations.

Abstract

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.