The Evolution of States in a Spatial Population Model

Abstract

The evolution of states in a spatial population model is studied. The model describes an infinite system of point entities in Rd which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system’s states are probability measures on the space of configurations, and their evolution is obtained from a hierarchical chain of differential equations for the corresponding correlation functions derived from the Fokker–Planck equation for the states. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that at each moment of time the correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are described.