A flexible 3-parameter curve for limited or unlimited somatic growth

Unlike population growth, the somatic growth of individual organisms is often well described by a curve passing through the origin, provided time (t) is measured from the moment at which the egg starts developing actively. A simple 3-parameter asymptotic growth curve, X = A/(1 + D/tc), can be derived from the logistic curve, X = A/[1 + D/exp(Ct)], by replacing time (t) by its natural logarithm. Hill (1913) used a similar curve to describe the saturation of haemoglobin by oxygen, but he considered only exponents C greater than or equal to 1. Depending upon the value of exponent C, this curve has a flexible shape which can range all the way from a rotated and translated rectangular hyperbola to a sigmoid curve. When C greater than 1, there is an inflection point of which the ordinate can assume any value between 0 and A/2, where A denotes the (upper) asymptote. When C less than 1, if the pattern of growth appears to be unlimited, within the range of the data, this curve becomes similar to allometry with respect to time, X = Btc. The asymptote A then becomes large but the ratio of parameters A/D approaches B as a limit. Unlike other 3-parameter curves, the present curve shows acceptable numerical convergence when fitted both to limited and unlimited growth data. This curve is illustrated with data on the body length of male and female elephant seals and on the body length and weight of female yellow sturgeons.