Biological hierarchies emerged from natural characteristics of a number theory

Abstract: We would like to show how biological grouping, especially in the case of species formation, is emerged through a nature of interactive populations with a number theory. First, we are able to define a species as a $p$-Sylow subgroup of a particular community in a single niche, confirmed by topological analysis. We named this model the patch with zeta dominance (PzDom) model. Next, the topological nature of the system is carefully examined. We confirm the induction of hierarchy and time through a one-dimensional probability space with certain topologies. For further clarification of induced fractals including the relation to renormalization, a theoretical development is proposed based on a newly identified fact, namely that scaling parameters for magnetization analogs exactly correspond to imaginary parts of the Riemann zeta function's nontrivial zeros. In our PzDom model, calculations only require knowledge of the density of individuals over time.