The mathematical description of biological evolution by means of natural selection is a multi-faceted subject. Disciplines as widely divergent as population genetics, quantitative genetics, evolutionary ecology, game theory and life-history theory provide various methods to tackle evolutionary questions. In recent decades, evolutionary game theory and the analysis of invasibility, which predict the fate of rare mutant types that are numerically dominated by resident types, have gained solid ground for analysing frequency- and density-dependent selection.

Evolution can be seen as the ongoing ousting of resident types by successful novel mutant strategies. Mutants can invade if their fitness is positive, that is, if they can increase in numbers when introduced in an environment that is set by the resident type. Hence, fitness by definition depends on the prevailing environment, which in its turn is determined by the phenotypic composition of the resident population. Recent developments such as the emerging theory of adaptive dynamics integrate classification schemes of evolutionarily stable strategies in dynamical systems theory. By restricting the attention to clonal genetics, and separating the time-scales of convergence to a population-dynamical attractor and of the generation of new successful mutations, the environmental condition can be expressed in terms of the resident genotype. In this way, invasion functions can be derived that depend on the type of the mutant and on the composition of the resident population. These invasion functions serve, together with mutation, as the generators of the dynamical systems describing the evolutionary process.

**This thesis** deals with three problems in the framework of
evolutionary game dynamics and complex population dynamics, with
invasibility analysis as a unifying principle. It consists of three
parts, and each part in turn consists of two chapters. The first part
deals with evolutionary game dynamics, where the model
*Battle of the Sexes* is taken as an example. It shows
that the evolutionary predictions of traditional game-dynamic models,
in which the costs and benefits of a strategy are summarized by simple
entries from a payoff matrix, may be quite different from models that
explicitly consider the mechanisms controlling mating in a structured
population. This part stresses the importance of incorporating
life-history information in game-dynamic model building. In part two,
the semelparous life-history of Pacific salmon is used to illustrate
an analysis of invasion in periodically fluctuating populations.
Contrary to the steady-state situation, successful invasion can
inevitably lead to an attractor switch, causing the expulsion of the
former invader. This intriguing example may prove to be the exception
to the rule, but in any case underlines the need for precision when
defining invasibility of strategies in the presence of multiple
attractors. The third part argues that the strategies brought forth
by evolution should be defined as strategies that are both
non-invasible and attainable. It raises and answers the question
whether (and, if so, how) they can also be found by maximizing simple
optimization criteria. This part makes yet another plea for more
mechanistic model building, emphasizing the significance of the
precise way in which density dependence limits population growth.

Tue Aug 10 12:25:42 MET DST 1999