# Masaryk University

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# Dynamic selection system and replicator equation

Authors 2013 Conference abstract Some connections between dynamic equation systems $$n_j^\Delta =n_jg_j({\boldsymbol{n}}),\qquad x_i^\Delta=x_i\big(f_i({\boldsymbol{x}})\ominus\bar{f}({\boldsymbol{x}})\big)$$ are presented. In particular, the second system can be inferred from the first one with equal dimension and it is equivalent with the system of the first form with a lower dimension. These results generalizes and unifies the known ones. The first system models a dynamic of interacting populations, i.e. a kind of natural selection; the second system describes an evolution of gene (replicator) frequencies. Hence, the analysis reveals a link between ecology and evolution - at least on the level of mathematical models. Some properties of the dynamic replicator equation solution are also presented and interpreted.