## Research

The research theme at MaBS is the mathematical biology of evolution. Evolution is the unifying theory of the biological sciences, and our aim is to design advanced mathematical methods and models that account for the biological complexity involved in most evolutionary processes. Complexity arises on all levels of biological organization: molecular, organismal, and ecological. The key issues of evolutionary research, such as adaptation and speciation, are usually addressed in special sub-disciplines for each of these levels, i.e. molecular population genetics, quantitative genetics, and evolutionary ecology. We work on all three fields with the special goal to create an integrative approach, with a combination of models, concepts, and methods.

### Topics

#### Population Genetics of Adaptation

#### Population Genetics of Speciation

#### Evolutionary consequences of gene interactions (epistasis)

#### Frequency-dependent selection and interspecific interactions

### Methods

Our mathematical methods are quite diverse and follow the needs of the biological problem that is addressed. Often techniques from various mathematical fields are combined.

#### Stochastics

In molecular population genetics, evolution is modelled as a stochastic process. We use time-forward approaches based on branching processes (e.g. Hermisson et al. 2002) and diffusions (e.g. Hermisson and Pennings 2005) and time-backward approaches, which use the coalescent (e.g. Hermisson and Pennings 2006a ,b).

#### Differential equations

The deterministic models in quantitative genetics and evolutionary ecology are formalized as systems of differential and difference equations. We use various techniques from these fields to analyse the equilibrium structure (existence, stability, domains of attraction) of biological models (e.g. Hermisson et al. 2003).

#### Adaptive dynamics

Adaptive dynamics is an increasingly popular toolbox to model phenotypic evolution under realistic ecololgical scenarios (Metz et al. 1996; Dieckmann and Law 1996: J. Math. Biol. 34: 579-612; Geritz et al. 1998: Evol. Ecol. 12: 35-57). It combines elements from evolutionary game theory, the theories of dynamical systems and stochastic processes. For an exhaustive literature survey on adaptive dynamics see here.

#### Statistical methods

Statistical tests and methods link empirical data to patterns that are predicted from theoretical models. In the context of molecular data, we have used statistical analysis (Stoletzki et al. 2005). and have developed tests (Pennings and Hermisson 2006b).

#### Computer simulations

In complex biological situations, computer simulations are always needed to complement and validate the analytical analysis. Our simulation tools use various different techniques, including time forward ('Wright-Fisher') simulations using multinomial sampling (e.g. Kopp and Hermisson 2007), individual based simulations (e.g. Carter et al. 2005), and time-backward simulations using the coalescent (Pennings and Hermisson 2006b).