TY  - THES
AB  - The Moran model is a widespread  model for a finite population in the field of population genetics.
In the first part of the thesis, we consider a Moran model with recombination. 
Models of recombination take a special role between
linear and nonlinear models.
Although there is abundant interaction and
hence nonlinearity, the deterministic system that describes the
frequencies of all possible types may be exactly transformed
into a linear one.  The
underlying linear structure even allows an explicit
solution. In certain important special
cases (notably, in so-called single-crossover dynamics in continuous time), 
this solution is surprisingly simple and immediately plausible.

Elucidating underlying linear structures in the corresponding stochastic
system has only
started very recently. In the single-crossover case,  
Baake and Herms (2006) observed that the expected type frequencies in the 
finite system follow those in the deterministic model. This and other results now lead to
the question whether in the general recombination scheme 
the dynamics of the expectations may be embedded into a higher but finite dimensional 
space, such that
they are given by a finite system of differential equations. We show that the system of moments closes here after a finite number of steps, without any need for approximations. Surprisingly, this property is lost when resampling  is included.  

In the second part of the thesis, we consider a Moran model with selection and mutation. Starting from this, we trace back the ancestral lines of single
individuals. We are interested in the stationary distribution of the corresponding
ancestral types. Fearnhead and Taylor already showed two approaches to this problem. Fearnhead's (2002) is based on the ancestral selection graph (Krone/Neuhauser
1997),  Taylor's (2007) relies on a description of the full population backward in time by means of a diffusion equation.
Kluth (2010) derived a recursion that describes this stationary distribution directly from the model. For the diffusion limit,
this leads to the same equation as Taylor's approach.

We can solve this recursion explicitly and may solve Taylor's equation directly from this recursion and thus link his solution 
to the Moran model and its graphical representation. This leads to a better understanding of the model and allows us to propose 
a new particle model.
DA  - 2012
LA  - eng
PY  - 2012
TI  - The Moran model of population genetics : case studies with recombination and selection
UR  - https://nbn-resolving.org/urn:nbn:de:hbz:361-25138439
Y2  - 2024-11-22T05:16:04
ER  -