  We propose a change in focus from the prevalent paradigm based on the
branching property as a tool to analyze the structure of population models, to
one based on the self-similarity property, which we also introduce for the
first time in the setting of measure-valued processes. By extending the
well-known Lamperti transformation for self-similar Markov processes to the
Banach-valued case we are able to generalize celebrated results in population
genetics that describe the frequency-process of measure-valued stable branching
processes in terms of the subfamily of Beta-Fleming-Viot processes. In our work
we describe the frequency process of populations whose total size evolves as
any positive self-similar Markov process in terms of general
$\Lambda$-Fleming-Viot processes. Our results demonstrate the potential power
of the self-similar perspective for the study of population models in which the
reproduction dynamics of the individuals depend on the total population size,
allowing for more complex and realistic models.
