  The relationships between urban area and population size have been
empirically demonstrated to follow the scaling law of allometric growth. This
allometric scaling is based on exponential growth of city size and can be
termed "exponential allometry", which is associated with the concepts of
fractals. However, both city population and urban area comply with the course
of logistic growth rather than exponential growth. In this paper, I will
present a new allometric scaling based on logistic growth to solve the
abovementioned problem. The logistic growth is a process of replacement
dynamics. Defining a pair of replacement quotients as new measurements, which
are functions of urban area and population, we can derive an allometric scaling
relation from the logistic processes of urban growth, which can be termed
"logistic allometry". The exponential allometric relation between urban area
and population is the approximate expression of the logistic allometric
equation when the city size is not large enough. The proper range of the
allometric scaling exponent value is reconsidered through the logistic process.
Then, a medium-sized city of Henan Province, China, is employed as an example
to validate the new allometric relation. The logistic allometry is helpful for
further understanding the fractal property and self-organized process of urban
evolution in the right perspective.
