A Dynamical Model for Selection

Reviewing the formalism of [44] for modeling selection, the dynamics of the model is described in terms of the fitness distribution p f) of the population which is expressed as an expansion in cumulants. The cumulants of a distribution p f) are defined through 

The first four cumulants of a fitness distribution p f) are then 

They represent the mean, variance, skew, and curtosis of the fitness distribution. To give an intuitive picture, the first two cumulants roughly capture the infinite population size limit of the model. The higher cumulants, skew and curtosis, are important to describe the dynamics of a finite population where, e.g., selection causes the fitness distribution to quickly become skewed and thus deviate from a Gaussian. An evolving population can, at each time step, be approximated by a set of these variables. Its dynamics can then be viewed in terms of the evolution of the cumulants. In the following, the dynamics of an evolving population will 

Consider first the - from this viewpoint - simplest operator selection. Its dynamics is solely determined by the fitness distribution of the population. Here, Boltzmann selection is considered in a population of P strings with fitnesses / and a = 1. P. A member with fitness / is chosen from the population with the probability 

When truncating the expansion after the fourth cumulant, a closed expression is obtained which can be iterated to give the evolution of the population under selection. Furthermore, using the rescaled selection parameter 

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