The characteristic features of the fossil record of the planet Earth are an equilibrium state (stasis) of ecosystems during the Precambrian period (over 550 million years BC) and a periodicity of changes in species diversity during the past 500 million years. None explanation for this phenomena have not yet been found. In the paper the minimal logistic model of the biota development is presented, which predicts stasis for ecosystems that have predominantly asexual type of reproduction and cyclical development for ecosystems with sexual reproduction. According to results of theoretical studies presented in this paper, documented cases of global extinctions observed in the Earth's fossil record during the past 500 million years could be a reflection of the internal global properties of living systems. Such the properties are birth, death, and evolutionary selection processes of the organisms that are the best adapted ones to the environmental conditions.