The effects of different factors (such as mutation, random genetic drift, and selection) on biological populations within one or a few generations are typically very small, and the time-scale for most evolutionary change is rather long. This makes it very difficult to observe or measure these changes directly. Moreover, even the simplest biological organisms, such as bacteria, have thousands of genes that can be in many different states (alleles). This results in an enormous potential for genetic diversity. For example, the number of different genetic combinations that are possible with L genes, each with just two alleles, is 2L. If L = 1,000 (which is a reasonable estimate for simple bacteria), there are 21,000 ~ 10301 genetic combinations. This number is much larger than estimates of the number of elementary particles in our universe. In many cases biological intuition is not helpful in evaluating the quantitative or qualitative effects of different biological factors on genetic systems that are that complex. Difficulties in direct observations and enormous genetic

complexity make it necessary to use mathematical models and methods to analyze evolutionary change at the genetic level.

Theoretical population genetics provides a mathematical foundation for the study of evolutionary genetics. The common procedure of theoretical population genetics is to start with some simple mathematical models that, although not fully realistic, can be completely analyzed and then refined into more realistic models that can be used to answer specific evolutionary questions. There are two general areas of theoretical population genetics: a prospective theory and a retrospective theory. The prospective theory takes the current state of a population (species, ecosystem) as given and tries to predict the relevant biological properties in the future. The prospective theory mostly uses methods developed within the mathematical theory of the dynamical systems (deterministic and stochastic). In the retrospective theory, one observes the current state of a population (species, ecosystem) and asks how it got here. The retrospective theory relies heavily upon statistical methods. Theoretical population genetics is essential for interpreting genetic variation, for predicting evolutionary change, and for reconstructing evolutionary history. It also provides a foundation for understanding the evolution of different characteristics, such as life histories and genome structure.

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