Lotka-Volterra equation and replicator dynamics: new issues in classification

Abstract

Replicator dynamics serves for modelling many biological processes, e.g. evolution of animal behaviour, but also selection in population genetics, and even prebiotic evolution. The Lotka-Volterra system is used in mathematical ecology to describe the interaction of two populations over time. Here, predator/prey situations can be modelled as well as competition for a resource. After a short account on applications and ramifications of planar classification results, a lacuna is closed which appeared in an earlier publication on classification (Biol Cybern 48:201---211, 1983). The now complete list of possible phase portraits under the replicator dynamics as well as under the Lotka-Volterra system is specified and contains, up to flow reversal, 49 qualitatively different cases for the former, and 110 or 67 for the latter dynamics, depending on whether or not one discriminates between different asymptotic slope behaviour. Furthermore, a systematic investigation of the flow under the replicator dynamics exhibits a variety of non-robust models which illustrate dynamic aspects of some solution concepts in evolutionary game theory, a field that is receiving widespread interest in the recent literature.