I extend Dawkins' well-known *Battle of the Sexes*, by
introducing a population with mating structure. Previous models of
the battle of the sexes are mostly formulated in terms of replicator
dynamics. They specify dynamical systems on the basis of the payoff
matrices of the evolutionary population-game, by assuming that
strategies produce copies of themselves, proportional to their average
payoff values. In chapter 1, I model the sex war more
mechanistically, by expressing the costs of raising the offspring and
performing a prolonged courtship via a time delay for the
corresponding individuals, instead of via payoff matrices. I assume
that during such a time delay, an individual is not able to have new
matings. Only after sitting out the delay, can an individual (and its
offspring) reappear on the mating market. From these assumptions I
derive a pair-formation submodel, and a system of partial- or
delay-differential equations describing the dynamics of the game. The
dynamical behaviour of this model is shown to be qualitatively
different from that of the traditional models of the battle of the
sexes.

The pair-formation model can be reduced by a time-scale argument. This idea is developed in chapter 2, where I present an approximation of the delay-differential equations system in terms of ordinary differential equations. This simplified system can be analysed much more thoroughly than the original delay-differential equations system. A combination of analytical and numerical techniques shows that the model can give rise to two non-trivial asymptotically stable equilibrium points: an asymptotically stable interior equilibrium where both female strategies and both male strategies are present, and a stable boundary equilibrium where only one of the female strategies and both male strategies are present. The traditional approaches of modelling the battle of the sexes lead to a unique interior equilibrium, accompanied by cyclic dynamics. My analysis shows that they miss crucial properties of the pair formation-based dynamics. In other words, the addition of an elementary further assumption about individual life history fundamentally changes the model predictions.

In general, I conclude that in analysing evolutionary games one should pay careful attention to the specific mechanisms involved in the conflict, and to derive simple models for evolutionary games, starting from more complex, mechanistic building blocks. The wide-spread method of modelling games at a high phenomenological level, through payoff matrices, can be misleading. As far as further research is concerned, an interesting extension would be to study the co-evolution of female preference and male willingness. The quasi-steady state population-dynamical equilibria calculated by means of the time-scaled pair-formation submodel can serve as a basis for such an adaptive dynamics.

Tue Aug 10 12:25:42 MET DST 1999