Abstract: This paper discusses the topic on Lyapunov stability of the population evolution proeess. It is shown that whether population systems are described in the form of partial differential equation or difference equation, there exists a problem of stability. In the paper we have theoretically proved the existence of critical fertility rate of women, which makes population systems tending to the edge of stability. Also we have derivde an cxplicit analytic expression for crictical fertility rates, thus concluding the conjecture and argument among populationistsfor decades. Critical fertility rates seem to be entirely determined by mortality functions, fertility patterns of women at their child-bearing ages and sex ratios of society population. The critical fertility at present stage in China is estimated to be 2.17. The results and conclusions Provided in this paper can be used as reference for some countries and the U. N. to tormulate their domestic or world population policies respectively.