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?:abstract
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The concepts of population and species play a fundamental role in biology. The existence and precise definition of higher-order hierarchies, such as division into species, is open to debate among biologists. First, we seek to show a fractal structure of species. 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 and for testing purposes, species density data are used in conjunction with data derived from liquid-chromatography mass spectrometry of proteins. 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 in physics, a theoretical development is proposed based on a newly identified fact, namely that scaling parameters for magnetization exactly correspond to imaginary parts of the Riemann zeta function\'s nontrivial zeros. A master torus and a Lagrangian/Hamiltonian are derived expressing fractal structures as a solution for diminishing divergent terms in renormalization. We will also focus on an application of our developed model. We extend current PzDom model to the so-called exPzDom model to qualify population dynamics as a topological matter as a whole, not focusing on hierarchy. The indicators in the exPzDom model adhere well to the empirical dynamics of SARS-CoV-2 infected people and align appropriately with actual policies instituted by the Japanese government. In our patch with zeta dominance (PzDom) model or its extended version (exPzDom), calculations only require knowledge of the density of individuals over time.
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