Gradients versus Cycling in Genetic Selection Models

We review the hierarchy of (continuous time) selection models starting with the classical Fisher's viability selection model, and its generalizations when allowing mutations, recombination, sex-dependent viabilities, fertility selection and different mortality rates. We analyse the question in which way Fisher's "Fundamental Theorem of Natural Selection" and Kimura's Maximum Principle can be extended to these more general situations. It turns out that in many cases this is principally impossible since the dynamics becomes cycling or even chaotic