Abstract: In this paper we discuss the spectral properties of the Population operator, prove that the population operator has only one real eigenvalue γ₀, which is greater that real parts of other eigenvalues,and find the corresponding relation between γ₀ and the critical fertility rate β_cr. We also study the existence and asymptotic behaviour of the semigroup for the population system, then come to the conclusion about the stability of the population system,including the existene of the steaby population state in the critical case of the fertility rate.These are all the new results of the quantitative population theory.