Evolutionary game theory and the adaptive dynamics approach: adaptation where individuals interact

Abstract

Evolutionary game theory and the adaptive dynamics approach have made invaluable contributions to understanding how gradual evolution leads to adaptation when individuals interact. Here, we review some of the basic tools that have come out of these contributions to model the evolution of quantitative traits in complex populations. We collect together mathematical expressions that describe directional and disruptive selection in class- and group-structured populations in terms of individual fitness, with the aims of bridging different models and interpreting selection. In particular, our review of disruptive selection suggests there are two main paths that can lead to diversity: (i) when individual fitness increases more than linearly with trait expression; (ii) when trait expression simultaneously increases the probability that an individual is in a certain context (e.g. a given age, sex, habitat, size or social environment) and fitness in that context. We provide various examples of these and more broadly argue that population structure lays the ground for the emergence of polymorphism with unique characteristics. Beyond this, we hope that the descriptions of selection we present here help see the tight links among fundamental branches of evolutionary biology, from life history to social evolution through evolutionary ecology, and thus favour further their integration. This article is part of the theme issue ‘Half a century of evolutionary games: a synthesis of theory, application and future directions’.

Evolutionary game theory of continuous traits from a causal perspective

“…In evolutionary models we are commonly interested in the evolutionary response (change in the population mean phenotype x*) resulting from such a fitness function, given certain simplifying assumptions. There are alternative ways of transitioning from the fitness function to an expression for evolutionary change, for example, focusing on population genetics or quantitative genetics (see [17][18][19][20][21]). We will use a quantitative genetic approach here because it can be presented concisely while illuminating some key simplifying assumptions made in evolutionary game theory.…”

“…It has since been merged with several other aspects of evolutionary theory, such as models where one explicitly considers the evolutionary dynamic as a sequence of successive allele replacements (going by a variety of names e.g. adaptive dynamics, invasion analysis, or trait substitution sequence models: [14][15][16][17][18]), quantitative genetics [19][20][21] and kin selection [17,[22][23][24]. While EGT was initially focused on finding stable endpoints [1,3] for trait evolution, these subsequent extensions bring a dynamic aspect to evolutionary game theory, characterize equilibria (as well as limitations of equilibrium concepts: [25,26]) in richer detail, and connect game theory to social evolution theory.…”

Half a century of evolutionary games: a synthesis of theory, application and future directions

“…To a modern audience, the 1973 article of Maynard Smith and Price may not even be immediately recognizable as evolutionary game theory, with its computer simulations of games with a small set of discrete strategies. Leafing through the articles in this theme issue makes it apparent that the variety of methodological approaches in evolutionary game theory today is very broad, ranging from calculus-based study of continuous traits [ 11 , 19 , 20 ] to computer simulations [ 21 , 22 ]. Evolutionary game theory of continuous, quantitative traits also connects seamlessly with kin selection and modelling of evolution in class- and group-structured populations [ 11 ], and when we consider long-term evolution, evolutionary game theory is also relatively robust to complexities and constraints of the genetic system [ 8 ].…”

“…Leafing through the articles in this theme issue makes it apparent that the variety of methodological approaches in evolutionary game theory today is very broad, ranging from calculus-based study of continuous traits [ 11 , 19 , 20 ] to computer simulations [ 21 , 22 ]. Evolutionary game theory of continuous, quantitative traits also connects seamlessly with kin selection and modelling of evolution in class- and group-structured populations [ 11 ], and when we consider long-term evolution, evolutionary game theory is also relatively robust to complexities and constraints of the genetic system [ 8 ]. Furthermore, Lehtonen & Otsuka [ 19 ] argue that while the aim of evolutionary game theory models is typically to identify evolutionary endpoints, the mathematical components (partial derivatives and their combinations) arising in continuous, quantitative evolutionary games can themselves be useful in the causal interpretation of the evolutionary model and of fitness and selection, and that they can be interpreted in the ‘path coefficient’ formalism of Sewell Wright [ 23 ] and in the modern framework of causal modelling [ 24 ].…”

“…In models of continuous, quantitative traits the ESS criterion is typically formulated in terms of a second derivative test which checks for local evolutionary stability (e.g. [ 11 , 12 ]). Phrased in terms of alleles, once an allele coding for an ESS value has reached fixation, no other mutant allele can increase in frequency (e.g.…”