97-03-08

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Abstract:

Complex Adaptations and the Structure of Recombination Spaces

## Guenter P. Wagner and Peter F. Stadler

According to the Darwinian theory of evolution,
adaptation results from spontaneously generated genetic variation and
natural selection.
Mathematical models of this process can be seen as describing a dynamics
on an algebraic structure which in turn is defined by the processes which
generate genetic variation (mutation and/or recombination). The theory of
complex adaptive system has shown that the properties of the algebraic
structure induced by mutation and recombination is more important for
understanding the dynamics than the differential equations themselves.
This has motivated new directions in the mathematical analysis of
evolutionary models in which the algebraic properties induced by mutation
and recombination are at the center of interest. In this paper we
summarize some new results on the algebraic properties of recombination
spaces. It is shown that the algebraic structure induced by recombination
can be represented by a map from the pairs of types to the power set of
the types. This construct is called P-structure. Utilizing this approach
deep commonalties between the recombination spaces defined by string
recombination models and the corresponding point mutation spaces can be
shown. Each fitness landscape which is elementary for point mutation is
also elementary for string recombination. This is an unexpected result
because of the fundamentally different nature of mutation and
recombination processes

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