Discrimination of Domination Mode and Chaotic Mode in Species

The concepts of a population and a species play fundamental roles in biology. The existence and precise definition of higher-order hierarchies, such as division into species, is open to debate among biologists. Here we show a new metric, ‘small s’, that distinguishes the population number and various data values that are beyond the range of neutral logarithmic populations and are specific to a given species with quantization, by the data from natural environments. We modify Price equation to introduce the metric. We show prime number could be related with speciation in discontinuity in Riemann zeta function including Bose-Einstein condensation, while prime closed geodesics could represent populations in Selberg zeta function. Calculation of prime closed geodesics |N(p)| leads out non-interacting adaptive species world is in the mode |N(p)| = 2/3, while interacting neutral populations is in the mode |N(p)| = 1. Combining the calculation with phylogenetical asymmetry, this discrimination gives the investigator whether the observed hierarchy of data represents chaotic populations/non-adaptive species or adaptive species with newly defined critical temperature and Weiss field. The border of fluctuating populations and ordered species is R(s) = 2, which is proven theoretically, by calculation and by observation. mod 4 of I(s) reveals adaptive/disadaptive situations among the individuals. Furthermore, our model has partially been successful for predicting futures at a unit time after, as transitions of biologilcal phases. The time-dependent fitness function and precise ‘Hubble parameter’ of a fitness space are also predicted by utilizing Schwarz equation. Significance of biological hierarchy is also discussed. In our new model (PzDom model), calculation requires only sets of the numbers of individual densities in time course.