A functional framework for the implementation of genetic algorithms: Comparing Haskell and Standard ML
Metastability is a common phenomenon. Many evolutionary processes, both natural and arti cial, alternate between periods of stasis and brief periods of rapid change in their behavior. In this paper an analytical model for the dynamics of a mutation-only genetic algorithm (GA) is introduced that identi es a new and general mechanism causing metastability in evolutionary dynamics. The GA’s population dynamics is described in terms of ows in the space of tness distributions. The trajectories through tness distribution space are derived in closed form in the limit of in nite populations. We then show how nite populations induce metastability, even in regions where tness does not exhibit a local optimum. In particular, the model predicts the occurrence of “ tness epochs” — periods of stasis in population tness distributions — at nite population size and identi es the locations of these tness epochs with the ow’s hyperbolic xed points. This enables exact predictions of the metastable tness distributions during the tness epochs, as well as giving insight into the nature of the periods of stasis and the innovations between them. All these results are obtained as closed-form expressions in terms of the GA’s parameters. An analysis of the Jacobian matrices in the neighborhood of an epoch’s metastable tness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space’s topological structure. More general quantitative features of the dynamics — tness uctuation amplitudes, epoch stability, and speed of the innovations — are also determined from the Jacobian eigenvalues. The analysis shows how quantitative predictions for a range of dynamical behaviors, that are speci c to the nite-population dynamics, can be derived from the solution of the in nite-population dynamics. The theoretical predictions are shown to agree very well with statistics from GA simulations. We also discuss the connections of our results with those from population genetics and molecular evolution theory. c © 1999 Elsevier Science B.V. All rights reserved.