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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