By Julian Hofrichter, Jürgen Jost, Tat Dat Tran
The current monograph develops a flexible and profound mathematical standpoint of the Wright--Fisher version of inhabitants genetics. This famous and intensively studied version incorporates a wealthy and lovely mathematical constitution, that is exposed right here in a scientific demeanour. as well as techniques through research, combinatorics and PDE, a geometrical viewpoint is introduced in via Amari's and Chentsov's details geometry. this idea permits us to calculate many amounts of curiosity systematically; likewise, the hired worldwide standpoint elucidates the stratification of the version in an unheard of demeanour. in addition, the hyperlinks to statistical mechanics and massive deviation thought are explored and built into strong instruments. Altogether, the manuscript presents a superior and extensive operating foundation for graduate scholars and researchers attracted to this field.
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Extra resources for Information Geometry and Population Genetics: The Mathematical Structure of the Wright-Fisher Model
Example text
Instead of letting members of the population produce offspring, we simply replace them by other individuals from the population. Thus, at every generation, for each individual in the population, randomly some individual is chosen, possibly the original individual itself, that replaces it. If we do that for all individuals simultaneously, we obtain a process that is equivalent to the Wright– Fisher process. But then, instead of updating all individuals simultaneously, we can also do that sequentially.
X/ C o. x/ will later on determine the drift terms in the Kolmogorov equations. x/ C o. /: xj / C o. x/ will become the coefficients of the diffusion term in the Kolmogorov equations. ıXt /˛ D o. 11), the second and higher 1 moments are the same, up to terms of order o. 13). Besides selection and mutation, there is another important ingredient in models of population genetics, recombination. That will be treated in Chap. 5. 6 The Case of Two Alleles Before embarking upon the mathematical treatment of the general Wright–Fisher model in subsequent chapters, it might be useful to briefly discuss the case where we only have two alleles, A0 and A1 .
TM/ ! V; Z/ 7! 5) and it satisfies the following product rule: DV . f Z/ D V. 6) Thus, when we multiply V by a smooth function f , we can simply pull that function out, but when we multiply Z, we also need to differentiate that function f . Again, tensor calculus expresses things in local coordinates. 7) The Christoffel symbols, however, do not transform as tensors, in contrast to the curvature tensor to be defined below. 8) 52 3 Geometric Structures and Information Geometry for vector fields X; Y; Z on M.