Punctuated equilibrium as the default mode of evolution of large populations on fitness landscapes dominated by saddle points in the weak-mutation limit

Abstract Punctuated equilibrium is a mode of evolution in which phenetic change occurs in rapid bursts that are separated by much longer intervals of stasis during which mutations accumulate but no major phenotypic change occurs. Punctuated equilibrium has been originally proposed within the framework of paleobiology, to explain the lack of transitional forms that is typical of the fossil record. Theoretically, punctuated equilibrium has been linked to self-organized criticality (SOC), a model in which the size of ‘avalanches’ in an evolving system is power-law distributed, resulting in increasing rarity of major events. We show here that, under the weak-mutation limit, a large population would spend most of the time in stasis in the vicinity of saddle points in the fitness landscape. The periods of stasis are punctuated by fast transitions, in ln N e time ( N e , effective population size), when a new beneficial mutation is fixed in the evolving population, which moves to a different saddle, or on much rarer occasions, from a saddle to a local peak. Thus, punctuated equilibrium is the default mode of evolution under a simple model that does not involve SOC or other special conditions. Significance The gradual character of evolution is a key feature of the Darwinian worldview. However, macroevolutionary events are often thought to occur in a non-gradualist manner, in a regime known as punctuated equilibrium, whereby extended periods of evolutionary stasis are punctuated by rapid transitions between states. Here we analyze a mathematical model of population evolution on fitness landscapes and show that, for a large population in the weak-mutation limit, the process of adaptive evolution consists of extended periods of stasis, which the population spends around saddle points on the landscape, interrupted by rapid transitions to new saddle points when a beneficial mutation is fixed. Thus, punctuated equilibrium appears to be the default regime of biological evolution.