Evolution dynamics in terraced NK landscapes

We consider populations of agents evolving in the fitness landscape of an extended NK model with a tunable amount of neutrality. We study the statistics of the jumps in mean population fitness which occur in the ‘punctuated equilibrium’ regime and show that, for a wide range of landscapes parameters, the number of events in time t is Poisson distributed, with the time parameter replaced by the logarithm of time. This simple log-Poisson statistics likewise describes the number of records in any sequence of t independently generated random numbers. The implications of such behavior for evolution dynamics are discussed. PACS numbers: 87.10.+e, 87.15.Aa, 05.40.-a Introduction. Evolutionary dynamics can be described as a stochastic process unfolding in a ‘fitness landscape’[1], a process which can be simulated by means of genetic algorithms [2]. The dynamical behavior of such algorithms is controlled by a number of parameters[3] as e.g. the population size, the rate of mutation and the strenght of selection. A pervasive dynamical regime is the so-called ‘punctuated equilibrium’[4] or ‘epochal behavior’, where relevant measures of evolution, as e.g. the mean fitness, remain constants for long periods during which the fitness distributions of the individuals is strongly peaked. Occasionally, a fitter mutant appears and quickly ‘takes over’ the population (see e.g. Fig. 1). Real experiments performed on bacterial colonies evolving in a controlled environment have shown that the fitness and cell size increase at a decelerating rate [5]. A similar slowing down is discussed by Kauffman[6] in the ‘long jump’ dynamics of the NK model, while Aranson et al.[7] find a logarithmic growth of the average fitness for quasi-species evolving in a rugged fitness landscape. In a macroevolutionary context, Raup and Sepkoski[8] suggested that the noticeable decay of the extinction rate[9, 10] might stem from the properties of an underlying optimization process. This idea was taken up in the ‘reset’ model[10, 11], where jumps in the average fitness of populations are linked to fitness record achieved during evolution[12]. Such a link between small and grand scale evolution is rather controversial: If macroevolutionary events are mainly driven extrinsically, e.g. by meteorite impacts[13], any patterns in the fossil record must ultimately stem from the mechanics of celestial bodies in the solar system[14]. Conversely, if, as recently proposed by several authors[15], the fossil record is mainly shaped by complex interactions within the biotic system, macro-evolutionary patterns must emerge from population dynamics. Our main interest lies in the statistical properties of the evolutionary jumps underlying fitness changes. Since the role of neutral mutations[16] for evolution is well established, and since punctuated equilibria exist even in the absence of local fitness