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Darwinian population genetics, or simply population genetics, is a central feature, if not the central feature, of the modern evolutionary synthesis, or neo-Darwinism. The modern evolutionary synthesis is a combination of Darwin's theory of the evolution of species by natural selection, Mendel's theory of genetics as the basis for biological inheritance, and mathematical population genetics. Put together by dozens of scientists throughout the 1930s and 1940s, Darwinian population genetics is our best model of the process that incrementally created all life on Earth, evolution and natural selection.
Population genetics is the study of the genetic distribution and change of allele frequency within a given species -- basically, which genes are more or less prominent within that species. This distribution and the way it changes can be understood comprehensively through five forces: natural selection, genetic drift, mutation, migration and nonrandom mating. Mathematical population genetics is a formalism that allows us to precisely predict future allele frequencies if complete information about these five forces is available. Of course, it never is, although approximations can be exceedingly useful. Population genetics is among the best-verified models in science.
Population genetics can explain the process by which new adaptations emerge and become fixed in a species, how speciation occurs, why some adaptive traits evolve more readily than others, the evolution of evolvability and many other matters of scientific interest. Because evolution mostly takes place over millions of years (i.e., evolutionary timescales), direct experimentation can be difficult. However, the principles of population genetics have been tested and verified with species whose replicative times are very short, such as bacteria, which can reproduce in half an hour given suitable space and nutrients.
The five evolutionary forces quantified and studied by population genetics are intuitively easy to understand. Natural selection occurs when an organism is killed off by its environment, rival species, or members of the same species, for whatever reason. If death occurs before reproduction, the organism is said to be evolutionarily unsuccessful. Whatever traits are associated with the premature death will eventually be selected out of the gene pool. Natural selection is perhaps the most powerful of the evolutionary forces, and the most widely understood.
Genetic drift occurs randomly in offspring. When a certain trait doesn't effect an organism's ability to survive or reproduce one way or the other, it may evolve and become fixed purely as a consequence of the genetic lottery. Mutations are also a side effect of the copying errors of DNA, and rarely become visible or significant, although occasionally they may result in an advantageous survival trait.
Migration occurs when members of a species move from one place to another, cutting off reproductive ties with other members of the species. When a part of a species becomes reproductively isolated from another part by a geographic barrier, the two groups eventually speciate into different variants. This was famously observed on the Galapagos Islands by Charles Darwin. Nonrandom mating is another very powerful force in population genetics. More attractive members of a species generally acquire more mates for longer periods, and experience greater reproductive success as a result.
Population genetics is a huge field which has been studied by many millions of scientists, not only biologists, and will continue to be studied by millions to come. Unfortunately, it is only taught at the most basic level in the public school systems.