Exact solutions for evolutionary strategies on harmonic landscapes

In this paper two different evolutionary strategies are tested by means of harmonic landscapes. Both strategies are based on ensembles of searchers, spreading over the search space according to laws inspired by nature. The main difference between the two prototypes is given by the underlying selection mechanism, governing the increase or decrease of the local population of searchers in certain regions of the search space. More precisely, we compare a thermodynamic strategy, which is based on a physically motivated local selection criterion, with a biologically motivated strategy, which features a global selection scheme (i.e., global coupling of the searchers). Confining ourselves to a special class of initial conditions, we show that, in the simple case of harmonic test potentials, both strategies possess particular analytical solutions of the same type. By means of these special solutions, the velocities of the two strategies can be compared exactly. In the last part of the paper, we extend the scope of our discussion to a mixed strategy, combining local and global selection.