Speciation: a Case Study in Symmetric Bifurcation Theory

Symmetric bifurcation theory is the study of how the trajectories of symmetric vector fields behave as parameters are varied. We introduce some of the basic ideas of this theory in the context of dynamical system models of speciation in evolution. Abstractly, these models are dynamical systems that are equivariant under the natural permutation action of the symmetric group SN on R kN for some integers N, k ≥ 1. The general theory, which is group-theoretic in nature, makes it possible to analyse such systems in a systematic manner. The results explain several phenomena that can be observed in simulations of specific equations. In particular, in steady-state bifurcation, primary branches involve bifurcation to two-species states; such bifurcations are generically jumps; and the weighted mean phenotype of the organisms changes smoothly, whereas the standard deviation jumps. In particular, classical mean-field genetics, which focusses on allele proportions in the population, cannot detect this kind of speciation event.