Adaptive dynamics: the continuity argument

Population regulation and adaptive dynamics of cross-feeding

The particular importance of evolutionary studies in microbial experimental systems is that starting from the level of the metabolism of individual cells, the adaptive dynamics can be followed step by step by biochemical, genetic, and population dynamical tools. Moreover, the coincidence of evolutionary and ecological time scales helps to clarify the mutual role of ecological and evolutionary principles in predicting adaptive dynamics in general. Ecological principles define the ecological conditions under which adaptive branching can occur. This paper overviews and interprets the results of empirical and modeling studies of the evolution of metabolic cross-feeding in glucose-limited E.coli chemostats and batch cultures in the context of theories of robust coexistence and adaptive dynamics. Empirical results consistently demonstrate that the interactions between cells are mediated by the changing metabolite concentrations in the cultures and modeling confirms that these changes may control the adaptive dynamics of the clones. In consequence, the potential results of evolution can be predicted at the functional level by evolutionary flux balance analysis (evoFBA), while the genetic changes are more contingent. evoFBA follows the scheme of adaptive dynamics theory by calculating the feedback environment that changes during the evolutionary process and provides a promising tool to further investigate adaptive divergence in small microbial communities. Three general conclusions close the paper.

Evolution of Cooperation in Social Dilemmas with Assortative Interactions

Cooperation in social dilemmas plays a pivotal role in the formation of systems at all levels of complexity, from replicating molecules to multi-cellular organisms to human and animal societies. In spite of its ubiquity, the origin and stability of cooperation pose an evolutionary conundrum, since cooperation, though beneficial to others, is costly to the individual cooperator. Thus natural selection would be expected to favor selfish behavior in which individuals reap the benefits of cooperation without bearing the costs of cooperating themselves. Many proximate mechanisms have been proposed to account for the origin and maintenance of cooperation, including kin selection, direct reciprocity, indirect reciprocity, and evolution in structured populations. Despite the apparent diversity of these approaches they all share a unified underlying logic: namely, each mechanism results in assortative interactions in which individuals using the same strategy interact with a higher probability than they would at random. Here we study the evolution of cooperation in both discrete strategy and continuous strategy social dilemmas with assortative interactions. For the sake of tractability, assortativity is modeled by an individual interacting with another of the same type with probability r and interacting with a random individual in the population with probability 1−r, where r is a parameter that characterizes the degree of assortativity in the system. For discrete strategy social dilemmas we use both a generalization of replicator dynamics and individual-based simulations to elucidate the donation, snowdrift, and sculling games with assortative interactions, and determine the analogs of Hamilton’s rule, which govern the evolution of cooperation in these games. For continuous strategy social dilemmas we employ both a generalization of deterministic adaptive dynamics and individual-based simulations to study the donation, snowdrift, and tragedy of the commons games, and determine the effect of assortativity on the emergence and stability of cooperation.

The Price Equation, Gradient Dynamics, and Continuous Trait Game Theory

A recent article convincingly nominated the Price equation as the fundamental theorem of evolution and used it as a foundation to derive several other theorems. A major section of evolutionary theory that was not addressed is that of game theory and gradient dynamics of continuous traits with frequency-dependent fitness. Deriving fundamental results in these fields under the unifying framework of the Price equation illuminates similarities and differences between approaches and allows a simple, unified view of game-theoretical and dynamic concepts. Using Taylor polynomials and the Price equation, I derive a dynamic measure of evolutionary change, a condition for singular points, the convergence stability criterion, and an alternative interpretation of evolutionary stability. Furthermore, by applying the Price equation to a multivariable Taylor polynomial, the direct fitness approach to kin selection emerges. Finally, I compare these results to the mean gradient equation of quantitative genetics and the canonical equation of adaptive dynamics.

Evolutionary dynamics of a smoothed war of attrition game

In evolutionary game theory the War of Attrition game is intended to model animal contests which are decided by non-aggressive behavior, such as the length of time that a participant will persist in the contest. The classical War of Attrition game assumes that no errors are made in the implementation of an animal׳s strategy. However, it is inevitable in reality that such errors must sometimes occur. Here we introduce an extension of the classical War of Attrition game which includes the effect of errors in the implementation of an individual׳s strategy. This extension of the classical game has the important feature that the payoff is continuous, and as a consequence admits evolutionary behavior that is fundamentally different from that possible in the original game. We study the evolutionary dynamics of this new game in well-mixed populations both analytically using adaptive dynamics and through individual-based simulations, and show that there are a variety of possible outcomes, including simple monomorphic or dimorphic configurations which are evolutionarily stable and cannot occur in the classical War of Attrition game. In addition, we study the evolutionary dynamics of this extended game in a variety of spatially and socially structured populations, as represented by different complex network topologies, and show that similar outcomes can also occur in these situations.