The author studied the problem of the ecology of the increase (reproduction) of vertebrates from statistics of animals, and birds of the hunting fauna. The growth of the herd of any vertebrate sp. follows the law of an exponential curve determined by the potential fertility of the sp. and by the death rate of the young before sexual maturity. A special formula for calculating potential fertility is developed. Death rate of the young, determined on a statistical basis, attains 90% in birds and 50% in herbivores. The increase of the herd is expressed as an undulating line. After a certain limit of density is attained, the period of growth is cut short by infectious disease or the elements. The species longevity of individuals determines the number of generations existing in the biocenosis. The period between 2 accumulations of the herd corresponds to the longevity of the sp. The longevity of individuals has been increasing during evolution, and the fertility diminishing. The logarithm of the rate of the species fertility varies inversely with the species longevity. Starting from these facts the author examines the theory of Volterra, concluding that this theory does not hold for vertebrates but only for cases corresponding strictly to the premises of the author. The author''s most important conclusion is the stating of a strictly mathematical regularity in the varying of the numbers of vertebrates of each sp. and of a considerable constancy of the numerical rate of the decrease in the struggle for life.