Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces

Strategy Selection in Structured Populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix View the MathML source. We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.MathematicsOrganismic and Evolutionary Biolog

Nowak et al. reply

Our paper challenges the dominant role of inclusive fitness theory in the study of social evolution1. We show that inclusive fitness theory is not a constructive theory that allows a useful mathematical analysis of evolutionary processes. For studying the evolution of cooperation or eusociality we must instead rely on evolutionary game theory or population genetics. The authors of the five comments offer the usual defense of inclusive fitness theory, but do not take into account our new results.MathematicsOrganismic and Evolutionary Biolog

Demographic noise can reverse the direction of deterministic selection

Deterministic evolutionary theory robustly predicts that populations displaying altruistic behaviours will be driven to extinction by mutant cheats that absorb common benefits but do not themselves contribute. Here we show that when demographic stochasticity is accounted for, selection can in fact act in the reverse direction to that predicted deterministically, instead favouring cooperative behaviors that appreciably increase the carrying capacity of the population. Populations that exist in larger numbers experience a selective advantage by being more stochastically robust to invasions than smaller populations, and this advantage can persist even in the presence of reproductive costs. We investigate this general effect in the specific context of public goods production and find conditions for stochastic selection reversal leading to the success of public good producers. This insight, developed here analytically, is missed by both the deterministic analysis as well as standard game theoretic models that enforce a fixed population size. The effect is found to be amplified by space; in this scenario we find that selection reversal occurs within biologically reasonable parameter regimes for microbial populations. Beyond the public good problem, we formulate a general mathematical framework for models that may exhibit stochastic selection reversal. In this context, we describe a stochastic analogue to r-K theory, by which small populations can evolve to higher densities in the absence of disturbance.Comment: 25 pages, 12 figure

On the origin of biological construction, with a focus on multicellularity

Biology is marked by a hierarchical organization: all life consists of cells; in some cases, these cells assemble into groups, such as endosymbionts or multicellular organisms; in turn, multicellular organisms sometimes assemble into yet other groups, such as primate societies or ant colonies. The construction of new organizational layers results from hierarchical evolutionary transitions, in which biological units (e.g., cells) form groups that evolve into new units of biological organization (e.g., multicellular organisms). Despite considerable advances, there is no bottom-up, dynamical account of how, starting from the solitary ancestor, the first groups originate and subsequently evolve the organizing principles that qualify them as new units. Guided by six central questions, we propose an integrative bottom-up approach for studying the dynamics underlying hierarchical evolutionary transitions, which builds on and synthesizes existing knowledge. This approach highlights the crucial role of the ecology and development of the solitary ancestor in the emergence and subsequent evolution of groups, and it stresses the paramount importance of the life cycle: only by evaluating groups in the context of their life cycle can we unravel the evolutionary trajectory of hierarchical transitions. These insights also provide a starting point for understanding the types of subsequent organizational complexity. The central research questions outlined here naturally link existing research programs on biological construction (e.g., on cooperation, multilevel selection, self-organization, and development) and thereby help integrate knowledge stemming from diverse fields of biology

Strategy abundance in evolutionary many-player games with multiple strategies

Evolutionary game theory is an abstract and simple, but very powerful way to model evolutionary dynamics. Even complex biological phenomena can sometimes be abstracted to simple two-player games. But often, the interaction between several parties determines evolutionary success. Rather than pair-wise interactions, in this case we must take into account the interactions between many players, which are inherently more complicated than the usual two-player games, but can still yield simple results. In this manuscript we derive the composition of a many-player multiple strategy system in the mutation-selection equilibrium. This results in a simple expression which can be obtained by recursions using coalescence theory. This approach can be modified to suit a variety of contexts, e.g. to find the equilibrium frequencies of a finite number of alleles in a polymorphism or that of different strategies in a social dilemma in a cultural context.Comment: 15 pages, 6 figures, Journal of Theoretical Biology (2011

Calculating Evolutionary Dynamics in Structured Populations

Evolution is shaping the world around us. At the core of every evolutionary process is a population of reproducing individuals. The outcome of an evolutionary process depends on population structure. Here we provide a general formula for calculating evolutionary dynamics in a wide class of structured populations. This class includes the recently introduced “games in phenotype space” and “evolutionary set theory.” There can be local interactions for determining the relative fitness of individuals, but we require global updating, which means all individuals compete uniformly for reproduction. We study the competition of two strategies in the context of an evolutionary game and determine which strategy is favored in the limit of weak selection. We derive an intuitive formula for the structure coefficient, σ, and provide a method for efficient numerical calculation

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net