Statistical description of closed biocenoses described by a Volterra chain subject to periodic boundary conditions

Statistical properties of closed biocenoses described by a Volterra chain subject to periodic boundary conditions are studied. The periodicity of the boundary conditions makes the interaction matrix singular. This causes certain difficulties in statistical description. The system dynamics is described by a class of Hamiltonians. The change of Hamiltonians within the class preserves the system dynamics, but it changes its statistical properties. Hence, the problem of selecting a unique Hamiltonian that correctly describes the statistical properties of the system. This problem is solved, and the statistical properties of the system are described.