General population systems. Macroscopic limit of a class of stochastic semigroups

Relationships between three possible descriptions: microscopic, mesoscopic and macroscopic that are related to a large population composed of several subpopulations, are formulated. The systems describing the evolution of a large number of individuals undergoing stochastic interactions (a description at microscopic level) in terms of stochastic semigroups are considered. The solutions of the general Lotka-Volterra-type equations without or with (weak) diffusion (descriptions at macroscopic level) are approximated by solutions of stochastic systems when the number of individuals N tends to ∞. The rate of approximation is controlled.