Early in the 20th century, the mathematician G.H. Hardy and the physician Wilhelm Weinberg, independently developed probabilistic models of genetic variation at the population level. The Hardy-Weinberg principle states that the gene pool of a population will remain constant indefinitely (i.e., it will be in equilibrium) unless a "disturbing" influence is introduced. In other words, the genetic variation introduced through the processes of meiosis and random fertilization does not affect a population's overall gene pool.

A theoretical non-evolving population is said to be in Hardy-Weinberg equilibrium. For a population to reach and maintain this equilibrium, it must meet specific criteria: it must be infinitely large, exhibit random mating patterns, have a constant, unchanging gene pool (no net mutation), have no migration into or out of the population, and have no natural or sexual selection occurring within the population.

Hardy-Weinberg calculations identify the allelic and genotypic frequencies expected from generation to generation, when a population is in Hardy-Weinberg equilibrium. For a population to reach and maintain this equilibrium, it must meet criteria (e.g. random mating patterns, no mutation, no selection) that are essentially unattainable outside of a laboratory setting. Thus, evolution occurs when natural populations deviate from Hardy-Weinberg equilibrium, causing shifts from the expected allelic and genotypic distributions. The major mechanisms that drive these shifts are selection, mutation, migration, genetic drift, and non-random mating.

Most populations are limited in size, and many can be very small. In small populations, dramatic changes in allele frequency can occur simply by chance. This is an evolutionary mechanism called genetic drift. The smaller the population, the greater the effect of genetic drift on the population's gene pool. When only a random subset of individuals contribute to the next generation, the statistical properties of sampling cause random deviations in allele frequency and consequent genotypic frequency.

Imagine a small population with only two alleles, "a" and "b." Then imagine that, by chance, a tree fell on a random set of individuals and that the survivors happened to be homozygous for "a."  Obviously, only the survivors will reproduce, so the "b" allele will no longer be present in future generations. In this case, the "a" allele is said to be fixed in the population. That is, every individual is homozygous for the "a" allele. This dramatic case illustrates fixation in a single generation. In large populations, genetic drift is more likely to alter the allelic (and genotypic) frequencies from one generation to the next. Through time, however, genetic drift can cause fixation even in large populations.

Notice that in this example, the "a" allele became overrepresented entirely by chance: a tree fell and eliminated all members of the population carrying the "b" allele. As we have seen, evolutionary processes are not always adaptive (resulting from natural selection). Genetic drift can fix any allele, regardless of whether it is harmful, neutral, or advantageous. Suppose the homozygous "aa" individuals in our example had some disease that reduced their fecundity (number of offspring per mating), relative to individuals with at least one "b" allele. Natural selection would have favored an increase in the "b" allele, but the strong effects of genetic drift (from the effects of the falling tree) took over and fixed the disadvantageous allele within the population.

Genetic distinctiveness arises through processes that introduce new alleles and genes into a population (mutation and migration) and processes that alter the frequencies of genes that already exist within a population (natural selection, sexual selection, and genetic drift).

A mutation is a random change in an organism's genetic material. Heritable mutations (those that are carried by the gametes) can introduce new alleles and genes into a population and, therefore, provide raw material for the evolutionary process.

Migration, or the movement of organisms from one place to another, can also cause changes in the gene pools of different populations. Through migration, new alleles can be introduced or taken away from a population, or the frequencies of alleles and genotypes in a population can be altered.

Through natural selection, heritable traits (and the alleles that confer those traits) that are beneficial to reproductive success become more common in a population while those that are disadvantageous become increasingly rare.

Sexual selection is a special form of selection that leads to the development of sexual dimorphic traits (traits that differ between the sexes). Through sexual selection, alleles that confer an advantage in the ability to obtain mates become more common in a population.

Genetic drift refers to changes in a population's allele frequencies that occur due to chance. In general, the smaller the population, the greater the impact of genetic drift. 
These processes, which drive evolution and speciation, can be reviewed in the BioEd Online presentation entitled "Biological Evolution." In addition, information about different species concepts, as well as mechanisms of reproductive isolation, can be found in the presentation "Species Concepts."

Allopatric speciation is the evolution of reproductive isolation in populations that are separated by geographic barriers. What constitutes a geographic barrier is not strictly defined, rather, it can be understood as any environmental factor that prevents or dramatically reduces gene flow between two populations. Thus, while geographic barriers most commonly result from large-scale climatic and geological events (mountain formation, glaciation, continental drift), they can also result from strict habitat preferences that "microgeographically" isolate populations.

Geographic isolation prevents gene flow among previously interbreeding populations and allows them to evolve independently. This almost inevitably leads to divergence between the two populations over time as distinct evolutionary changes accumulate: different mutations arise in the different populations, genetic drift fixes different genes in the populations, and the populations undergo different adaptive changes in response to natural selection. Over time, the two populations may become reproductively incompatible or isolated, essentially as a by-product of the genetic divergence of other traits. Eventually, the two (initially identical) species will no longer interbreed, even if they are brought back into contact with one another under natural conditions. Allopatric speciation is considered to be the most common of the known modes of speciation.

Peripatric speciation is a type of allopatric speciation. In both of these geographic modes of speciation, a single population becomes divided into two independent populations that can no longer interbreed because they are separated by some physical barrier. Because of the separation, the two populations evolve independently and eventually become reproductively incompatible. The distinction between allopatric and peripatric speciation is the relative sizes of the populations involved. In allopatric speciation, a population is separated into two relatively large independent populations. In contrast, in peripatric speciation, only a small fraction of the original population becomes geographically isolated. Peripatric speciation originally was known as founder effect speciation because it can occur when a few individuals (the founders) colonize a new habitat, such as an island, thereby establishing a new population away from the parent population.

The same genetic forces drive the evolution of reproductive isolation in both allopatric and peripatric speciation: mutation, natural selection, and genetic drift. However, in peripatric speciation, evolution may occur on a faster time scale because small populations are more susceptible to the random effects of genetic drift. In addition, it is likely that there will be a period of rapid selection and adaptation if the colonists' new habitat is substantially different from their original environment. For example, the founder effect has been used to explain the rapid speciation of Hawaiian fruit flies (Drosophila). Approximately 500 species of Drosophila currently inhabit the Hawaiian archipelago, and all are believed to have descended from a common ancestor that reached the island of Kauai (the oldest of the volcanic islands) over five million years ago. Some believe that as new Hawaiian islands formed as a result of volcanic activity, they were colonized by small groups of Drosophila from the older islands. Because these founders represented only a fraction of the genetic variability present in the parent species, it is likely that genetic drift played a large role in the evolution of genetic distinctiveness between species of Hawaiian Drosophila. In addition, the founders evolved through the process of natural selection to adapt to aspects of the new environment, such as the endemic plant hosts.

The founder effect has been invoked in the speciation events of the Hawaiian Drosophila because so many species were formed in such a relatively short time; the standard allopatric model does not predict that several hundred speciation events should occur in only a few million years. However, the importance of the founder-induced speciation is debated. Some have suggested that it rarely occurs, especially compared to the more standard allopatric model of speciation.