Limit theorems for the reaction-diffusion equations with the infinite number of the particles

We consider the mathematical models of the bio-populations homogeneous in space and time. The stochastic dynamics of such populations includes the birth and death processes, the migration and the immigration. The direct interaction between the species is excluded (like usually in the theory of the branching process), however the death-birth mechanism generates some kind of the mean-field attraction between the species (in the spirit of FKG property). We prove the existence of the limiting distribution, which in the dimension $d=2$ requires either a very active migration or a very active immigration. The limit theorems for the equilibrium state include CLT and the intermittency results in the case of low density populations (“patches”). The class of the models under consideration includes continuous "contact model" which was introduced and studied recently by Yu. Kondratiev, S. Pirogov, A. Skorokhod and others.