Bifurcation Structure in Diversity Dynamics 2 a Simple Model of Evolution

We propose a measure of total population diversity D of an evolving population of genetically speci-ed individuals. Total diversity D is the sum of two components, within-gene diversity W g and between-gene diversity B g. We observe the dynamics of diversity in the context of a particular model, a two-dimensional world with organisms competing for resources and evolving by natural selection acting implicitly on genetic changes in their movement strategies. We examine how diversity dynamics and population performance|measured as the eeciency with which the population extracts energetic resources from its environment|depend on mutation rate and the presence or absence of selection. Systematic exploration of mutation rates reveals a bifurcation into qualitatively diierent classes of diversity dynamics, whether or not selection is present. Class I: At low mutation rates, diversity dynamics exhibit \punctuated equilibria"|periods of static diversity values broken by rapid changes. Class II: At intermediate mutation rates, diversity undergoes large random uctuations without always approaching any evident equilibrium value. Class III: At high mutation rates, diversity is stable, with small uctuations around an equilibrium value. Optimal population performance occurs within a range of mutation rates that straddles the border between class I and class II. The relationships among diversity D and its components W g and B g reeects the typical features of these diierent classes of diversity dynamics as well as corresponding diierences in the gene pool, which ranges from genetically similar individuals in class I to genetically dissimilar individuals in class III. The fact that class I dynamics occur whether or not selection is present suggests that stochastically branching trait transmission processes have an intrinsic tendency to exhibit punctuated equilibria in population diversity over a critical range of branching (mutation) rates. 1 The Evolution of Diversity Complex adaptive systems are embodied in many settings , ranging from ecological populations of organisms, through immune systems of antigens and antibodies, even to networks of neurons in the brain. By abstracting away the diverse details, one can model complex adap-tive systems at a level of generality that might reveal fundamental principles governing broad classes of such systems|this we take to be the working hypothesis of artiicial life ?]. One reason for the impressive eeects in many artiicial life models is their \emergent" architecture: The sys-tem's global adaptive behavior emerges unpredictably from an explicitly modeled population of low-level individuals. We have been studying a class of models consisting of a population of …