is present and viable hybrids can be formed. The production of inviable hybrids is a form of reproductive character displacement, under which most definitions is the completion of a speciation event. Research has well established the fact that interspecific mate discrimination occurs to a greater extent between sympatric populations than it does in purely allopatric populations; however, other factors have been proposed to account for the observed patterns. Reinforcement in allopatry has been shown to occur in nature (evidence for speciation by reinforcement), albeit with less frequency than a classic allopatric speciation event. A major difficulty arises when interpreting reinforcement's role in allopatric speciation, as current phylogenetic patterns may suggest past gene flow. This masks possible initial divergence in allopatry and can indicate a "mixed-mode" speciation event—exhibiting both allopatric and sympatric speciation processes. Mathematical models Developed in the context of the genetic basis of reproductive isolation, mathematical scenarios model both prezygotic and postzygotic isolation with respect to the effects of genetic drift, selection, sexual selection, or various combinations of the three. Masatoshi Nei and colleagues were the first to develop a neutral, stochastic model of speciation by genetic drift alone. Both selection and drift can lead to postzygotic isolation, supporting the fact that two geographically separated populations can evolve reproductive isolation—sometimes occurring rapidly. Fisherian sexual selection can also lead to reproductive isolation if there are minor variations in selective pressures (such as predation risks or habitat differences) among each population. (See the Further reading section below). Mathematical models concerning reproductive isolation-by distance have shown that populations can experience increasing reproductive isolation that correlates directly with physical, geographical distance. This has been exemplified in models of ring species; however, it has been argued that ring species are a special case, representing reproductive isolation-by distance, and demonstrate parapatric speciation instead—as parapatric speciation represents speciation occurring along a cline. Other models Various alternative models have been developed concerning allopatric speciation. Special cases of vicariant speciation have been studied in great detail, one of which is peripatric speciation, whereby a small subset of a species population becomes isolated geographically; and centrifugal speciation, an alternative model of peripatric speciation concerning expansion and contraction of a species' range. Other minor allopatric models have also been developed are discussed below. Peripatric Peripatric speciation is a mode of speciation in which a new species is formed from an isolated peripheral population. If a small population of a species