Deriving five growth functions from bertalanffy function based on symmetry and complexity

This study was designed to derive five growth functions from Bertalanffy function using symmetry and complexity of them by relating each function to its first and second derivatives. The results obtained were as follows. In equations constructed by relating functions to their derivatives, the left-hand sides were the same in form, but the right-hand sides showed differences from the symmetry essential to exponential function. These differences were related to complexity of functions approaching asymptotes. The asymptotic properties of five functions were shown by phenomena that the right-hand sides tended to zero as í tended to infinity. Six growth functions arranged by the complexity were Bertalanffy > Richards > Mitscherlich = logistic = Gompertz > basic growth. Despite different function forms, Mitscherlich, logistic and Gompertz functions were not distinguished each other from the viewpoint of complexity, a kind of symmetry existing at the base of them. Based on symmetry and complexity, five growth functions were derived from Bertalanffy function, a hierarchic structure of growth functions from Bertalanffy function on down.