A central problem in evolutionary biology is the occurrence in the fossil record of new species of organisms. Darwin's view, in The Origin of Species, was that speciation is the result of gradual accumulations of changes in body-plan and behaviour. Mayr asked why gene-flow failed to prevent speciation, and his answer was the classical allopatric theory in which a small founder population becomes geographically isolated and evolves independently of the main group. An alternative class of mechanisms, sympatric speciation, assumes that no such isolation occurs. These mechanisms overcome the stabilising effect of gene-flow by invoking selection effects, for example sexual selection and assortative mating. We interpret sympatric speciation as a form of symmetry-breaking bifurcation, and model it by a system of nonlinear ODEs that is 'all-to-all coupled', that is, equivariant under the action of the symmetric group SN. We show that such bifurcations can be interpreted as speciation events in which the dominant long-term behaviour is divergence into two species. Generically this divergence occurs by jump bifurcation - 'punctuated equilibrium' in the terminology of evolutionary biology. Despite the discontinuity of such a bifurcation, mean phenotypes change smoothly during such a speciation event. So, arguably, do mean-field genotypes related to continous characters. Our viewpoint is that speciation is driven by natural selection acting on organisms, with the role of the genes being secondary: to ensure plasticity of phenotypes. This view is supported, for example, by the evolutionary history of African lake cichlids, where over 400 species (with less genetic diversity than humans) have arisen over a period of perhaps 200,000 years. Sympatric speciation of the kind we discuss is invisible to classical mean-field genetics, because mean-field genotypes vary smoothly. Our methods include numerical simulations and analytic techniques from equivariant bifurcation theory. We focus on two main models: the generic cubic-order truncation of a symmetry-breaking bifurcation in an SN-equivariant system of ODEs, and the BirdSym system introduced by Elmhirst in which the biological interpretation of variables is more explicit. We relate our conclusions to field observations of various organisms, including Darwin's finches. We also offer a biological interpretation of our models, in which speciation is represented as an emergent property of a complex system of entities at the organism level. We briefly review questions about selection at the level of groups or species in the light of this interpretation.
