Structured Populations and Games over Generations
The actions and state of an individual in one generation can affect the state of offspring in the next generation, and hence the ability of these offspring to leave offspring themselves. This chapter deals with games in this multigenerational setting. Projection matrices are used to keep track of the state and number of descendants in successive years and generations. Invasion fitness is then defined in terms of the leading eigenvalue of the projection matrix. Simple examples illustrate these concepts and show how to apply them. Reproductive value is a function that measures how the ability to leave descendants in future generations depends on the current state. The Nash equilibrium condition is reformulated in terms of reproductive value maximization. This new criterion is used to justify the fitness proxy used in the analysis of sex allocation earlier in the book. The analysis is extended to the case where offspring may inherit aspects of their mother’s quality, with a focus on the question of whether high-quality mothers should produce sons or daughters. As a second application of reproductive value maximization, the co-evolution of female preference for a particular male trait and the trait itself is analysed, with the evolution of tail length in the widowbird as an illustrative application. Mean lifetime reproductive success is used as a fitness proxy in much of the book. Its use is finally justified in this chapter, where the fitness proxy is used to analyse the evolutionarily stable age of first reproduction in a population.