Biological hierarchies emerged from natural characteristics of number theory

Abstract

Ecologists have long debated whether biological communities are fundamentally continuous or composed of discrete units. Continuum views emphasize smooth compositional change and ambiguous boundaries, whereas classification-based approaches rely on discrete community types for analysis and management. We show how biological grouping, particularly species formation, can emerge from interactions among populations governed by number-theoretic structure. In our framework, a species is identified with a $p$-Sylow subgroup of a community occupying a single niche; this identification is suported by a topological analysis. We call the resulting framework the patch with zeta dominance (PzDom) model. We then examine the system's topological properties in detail and demonstrate that both hierarchical organization and temporal ordering are induced by a one-dimensional probability space endowed with an appropriate topology. To clarify the appearance of induced fractal structure and its relation to renormalization, we develop a theoretical account based on a new observation: the scaling parameters that play the role of magnetization analogs coincide exactly with the imaginary parts of the nontrivial zeros of the Riemann zeta function. In the PzDom model, all required computations reduce to the time-dependent density of individuals. The PzDom framework reconciles these perspectives by showing that continuous community variation, represented by small $s$, can give rise to discrete species-level structure when number-theoretic constraints stabilize specific configurations. Thus, continuity and discreteness emerge as different dynamical phases of the same system, offering a unified explanation for long-standing debates in community ecology and the **species problem**.