On the statistical description of large populations

A wide variety of works exists on the dynamics of large populations, ranging from simple heuristic modeling to those based on advanced computer-supported methods. However, their interconnections remain mostly vague, which, in particular, limits the effectiveness of computer methods in this domain. In this work, we propose and justify the following concept. Typically, the description of the population dynamics is based on the sole use of low-order correlations. As we demonstrate here, in important cases, where the local population structure is shaped by strong interactions, higher-order correlations become essential. To verify when one can or cannot rely on studying low-order correlations only, we suggest to explicitly use probability measures as micro-states. Among such states may be those whose adequate characterization is based on their low-order correlation functions. In particular, this is the case for sub-Poissonian states where the large $ n $ asymptotics of the probability of finding $ n $ particles in a given vessel is similar to that for non-interacting entities, which can completely be described by the density of the particles. To illustrate this concept, a general individual-based model of an infinite population of interacting entities is analyzed. The evolution of this model preserves the sub-Poissonian states, which allows one to describe it through the correlation functions of such states for which a chain of evolution equations is obtained. The corresponding kinetic equation is derived, numerically solved, and analyzed.