Bifurcation Structure in Diversity Dynamics

We propose a measure of total population diversity D of an evolving population of genetically specied individuals. Total diversity D is the sum of two components, within-gene diversityWg and betweengene diversity Bg . We observe the dynamics of diversity in the context of a particular model, a twodimensional 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 e ciency 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 di erent 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 Wg and Bg re ects the typical features of these di erent classes of diversity dynamics as well as corresponding di erences 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 adaptive 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 arti cial life [?]. One reason for the impressive e ects in many arti cial life models is their \emergent" architecture: The system'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 computation agents (basically, individual computer programs) that interact with each other and with their environment in a way that allows natural selection implicitly to shape their strategies for achieving various global computational goals [?, ?, ?, ?, ?]. We de ne statistical \macrovariables"| loosely akin to thermodynamic macrovariables like pressure or temperature|that re ect fundamental aspects of a system's adaptive behavior. Then we try to identify simple laws relating these macrovariables to other fundamental system parameters and we try to use these macrovariables to identify and explain basic classi cations of systems. An obvious but striking feature of complex adaptive systems is the evolutionary dynamics of population diversity. How can population diversity be de ned and measured? How does diversity change as a population evolves? How do diversity dynamics vary as a function of other fundamental system parameters, such as mutation rate and selection pressure? Does population diversity de ne qualitatively di erent kinds of evolving systems? The present study addresses these questions (see also [?, ?]). 2 A Simple Model of Evolution The model used here is designed to be simple yet able to capture the essential features of an evolutionary process [?, ?, ?, ?, ?]. The model consists of organisms (sometimes called \bugs") moving about in a two dimensional world. The only thing that exists in the world besides the organisms is food. Food is put into the world in heaps that are concentrated at particular locations, with levels decreasing with distance from a central location. Food is refreshed periodically in time and randomly in space. The frequency and size of the heaps are variable parameters in the simulation. The food represents energy for the organisms. Organisms interact with the food eld by eating it at their current site at each time step, decrementing the food value in the environment and incrementing their internal food supply. Organisms must continually replenish their internal food supply to survive. Surviving and moving expend energy. Organisms pay a tax just for living and a movement tax proportional to the distance traveled. If a organism's internal food supply drops to zero, it dies and disappears from the world. On the other hand, an organism can remain alive inde nitely if it can continue to nd enough food. Any evolutionary learning that occurs in the model is the e ect of the one stress of continually nding enough food to remain alive. A good strategy for ourishing in this model would be to e ciently acquire and manage vital energetic resources. It is important to note that selection and adaptation in the model are \intrinsic" or \indirect" in the sense that survival and reproduction is determined solely by the contingencies involved in each organisms nding and expending food. No externally-speci ed tness function governs the evolutionary dynamics [?, ?]. The organisms in this model follow individually di erent strategies for nding food (and hence are sometimes called \strategic bugs" [?]). The behavioral disposition of bugs is genetically hardwired. A behavioral strategy is simply a map taking an organism's current sensory state|information about its present local environment (the ve site von Neumann neighborhood)|to a vector indicating the magnitude and direction of its subsequent movement: S : (s1; :::; s5)! ~v = (r; ): (1) A bug's sensory state has two bits of resolution for each site in its its local environment, allowing the bug to recognize four food levels at each site (least food, somewhat more food, much more food, most food). Its behavioral repertoire is also nite, with four bits of resolution for magnitude r (zero, one, ..., fteen steps) and three bits for direction (north, northeast, east, ...). A unit step in the NE, SE, SW, or NW direction is de ned as movement to the next diagonal site, so its magnitude is p 2 times greater than a unit step in the N, E, S, or W direction. Each movement vector ~v thus produces a displacement (x; y) in a square space of possible spatial destinations from a bug's current location. The graph of the strategy map S may be thought of as a look-up table with 2 entries, each entry taking one of 2 possible values. This look-up table represents an organism's overall behavioral strategy. The entries are input-output pairs that link a speci c behavior (output) with each sensory state (input) that an organism could possibly encounter. The input entries in the look-up table represent genes or loci, and the movement vectors assigned to them represent alleles. Since bugs have 1024 genes or loci, each containing one out of a possible 128 alleles or behaviors, the total number of di erent genomes is 128. Although nite, this space of genomes allows for evolution in a huge space of genetic possibilities, which simulates the much larger number of possibilities in the biological world. When a bug's internal food supply crosses a threshold, it produces some number of o spring by asexual budding. After reproduction, the parental food supply is divided equally among the new children and the parent(s). Parental genes are inherited with some probability of mutation. Point mutations of the genes change the output values of entries in a child's look-up table. The mutation rate determines the probability with which individual loci mutate during reproduction. At the limit of = 1, every allele has probability one of mutating and thus each child's alleles are chosen completely randomly. While mutation rate is an explicit parameter of the model, selection pressure is controlled indirectly by adjusting other explicit parameters. The parameter output noise, P0, is de ned as the probability that the behavior actually performed by a bug on a given occasion in a given local environment will be chosen randomly from the 2 possible behaviors, rather than determined by the bug's genes. If P0 = 1, then natural selection has no opportunity to \test" the usefulness of the behavioral traits encoded in a bug's genome. The bugs are still subject to di erential survival and reproduction, and so there is a sort of \selection," but the alleles or traits transmitted in reproduction re ect only random genetic drift. There is heritable genetic variation but no heritable phenotypic variation, so natural selection plays no role in shaping the evolution of either genotypes or phenotypes. In simulations reported in this paper, output noise P0 was set to either zero or one, thus creating pairs of simulations in which all model parameters were identical except for the presence or absence of selection's e ects on the course of evolution. This model is a very abstract and idealized representation of a population of evolving organisms, and has many biologically unrealistic respects. Nevertheless, our working hypothesis is that this model captures many fundamental aspects of evolving systems, and is thus a useful way to investigate the essential aspects of more complex evolving systems. 3 Measures of Diversity How might population diversity be measured? (To simplify terminology, in what follows \diversity" always means population diversity.) Our proposal, very roughly, is to represent the population as a cloud of points in an abstract genetic space, and then de ne its diversity as the spread of that cloud. In the present model, an allele is a movement vector, a spatial displacement triggered by the sensory state corresponding to a given local environment. An individual's genotype is a complete set of spatial displacements, one for each possible sensory state. To capture the total population diversity, D, then, collect all the displacements of all bugs in all environments into a cloud, and measure the spread or variance of that cloud. (One can de ne related measures of diversity based on information-theoretic uncertainty rather than variance [?].) More explicitly, we de ne total diversity as the mean squared deviation between the average movement of the whole population, averaged over all individuals and over all sensory states, and the individual movements of particular individuals subject to particular conditions, i.e.,