Evolutionary games on graphs

Summary We present a general framework for modelling adaptive trait dynamics in which we integrate various concepts and techniques from modern ESS-theory The concept of evolutionarily singular strategies is introduced as a generalization of the ESS-concept We give a full classification of the singular strategies in terms of ESSstability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategy’s neighbourhood Of particular interest is a type of singular strategy that is an evolutionary attractor from a great distance, but once in its neighbourhood a population becomes dimorphic and undergoes disruptive selection leading to evolutionary branching Modelling the adaptive growth and branching of the evolutionary tree can thus be considered as a major application of the framework A haploid version of Levene’s ‘soft selection’ model is developed as a specific example to demonstrate evolutionary dynamics and branching in monomorphic and polymorphic populations

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Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree

Understanding speciation is a fundamental biological problem. It is believed that many species originated through allopatric divergence, where new species arise from geographically isolated populations of the same ancestral species. In contrast, the possibility of sympatric speciation (in which new species arise without geographical isolation) has often been dismissed, partly because of theoretical difficulties. Most previous models analysing sympatric speciation concentrated on particular aspects of the problem while neglecting others. Here we present a model that integrates a novel combination of different features and show that sympatric speciation is a likely outcome of competition for resources. We use multilocus genetics to describe sexual reproduction in an individual-based model, and we consider the evolution of assortative mating (where individuals mate preferentially with like individuals) depending either on an ecological character affecting resource use or on a selectively neutral marker trait. In both cases, evolution of assortative mating often leads to reproductive isolation between ecologically diverging subpopulations. When assortative mating depends on a marker trait, and is therefore not directly linked to resource competition, speciation occurs when genetic drift breaks the linkage equilibrium between the marker and the ecological trait. Our theory conforms well with mounting empirical evidence for the sympatric origin of many species.

/pdf/on-the-origin-of-species-by-sympatric-speciation-98bziomw8u.pdf

On the origin of species by sympatric speciation

Evolutionary game dynamics is the application of population dynamical methods to game theory. It has been introduced by evolutionary biologists, anticipated in part by classical game theorists. In this survey, we present an overview of the many brands of deterministic dynamical systems motivated by evolutionary game theory, including ordinary differential equations (and, in particular, the replicator equation), differential inclusions (the best response dynamics), difference equations (as, for instance, fictitious play) and reaction-diffusion systems. A recurrent theme (the so-called `folk theorem of evolutionary game theory') is the close connection of the dynamical approach with the Nash equilibrium, but we show that a static, equilibrium-based viewpoint is, on principle, unable to always account for the long-term behaviour of players adjusting their behaviour to maximise their payoff.

https://www.ams.org/bull/2003-40-04/S0273-0979-03-00988-1/S0273-0979-03-00988-1.pdf

Evolutionary Game Dynamics

▪ Abstract Recent theories regarding the evolution of predator-prey interactions is reviewed. This includes theory about the dynamics and stability of both populations and traits, as well as theory predicting how predatory and anti-predator traits should respond to environmental changes. Evolution can stabilize or destabilize interactions; stability is most likely when only the predator evolves, or when traits in one or both species are under strong stabilizing selection. Stability seems least likely when there is coevolution and a bi-directional axis of prey vulnerability. When population cycles exist, adaptation may either increase or decrease the amplitude of those cycles. An increase in the defensive ability of prey is less likely to produce evolutionary counter-measures in its partner than is a comparable increase in attack ability of the predator. Increased productivity may increase or decrease offensive and defensive adaptations. The apparent predominance of evolutionary responses of prey to predat...

https://www.jstor.org/stable/pdfplus/221726.pdf

The Evolution of Predator-Prey Interactions: Theory and Evidence

We set out to explore a class of stochastic processes, called "adaptive dynamics", which supposedly capture some of the essentials of the long-term biological evolution. These processes have a strong deterministic component. This allows a classification of their qualitative features which in many aspects is similar to classifications from the theory of deterministic dynamical systems. But they also display a good number of clear-cut novel dynamical phenomena. 

The sample functions of an adaptive dynamics are piece-wise constant function from R_+ to the finite subsets of some "trait" space X in R^k. Those subsets we call "adaptive conditions". Both the range and the jumps of a sample function are governed by a function s, called "fitness", mapping the present adaptive condition and the trait value of a potential "mutant" to R. Sign(s) tell us which subsets of X qualify as adaptive conditions, which mutants can potentially "invade", leading to a jump in the sample function, and which adaptive condition(s) can result from such invasion. 

Fitness supposedly satisfy certain constraints derived from their population/community dynamical origin, such as the fact that all mutants which are equal to some "residents", i.e., element of the present adaptive condition, have zero fitness. Apart from that, we suppose that s is as smooth as can be possibly condoned by its community dynamical origin. Moreover, we assume that a mutant can differ but little from its resident "progenitor". 

In sections 1 and 2 we describe the biological background of our mathematical framework. In section 1 we deal with the position of our framework relative to present and past evolutionary research. In section 2 we discuss the community dynamical origin of s, and the reasons for making a number of specific simplifications relative to the full complexity seen in nature. 

In sections 3 and 4 we consider some general, mathematical as well as biological, conclusions that can be drawn from our framework in its simplest guise, that is, when we assume that X is 1-dimensional, and that the cardinality of the adaptive conditions stays low. The main result is a classification of the adaptively singular points. These points comprise both the adaptive point attractors, as well as the points where the adaptive trajectory can branch, thus attaining its characteristic tree-like shape. 

In section 5 we discuss how adaptive dynamics relate through a limiting argument to stochastic models in which individual organisms are represented as separate entities. It is only through such a limiting procedure that any class of population or evolutionary models can eventually be justified. Our basic assumptions are (i) clonal reproduction, i.e., the resident individuals reproduce faithfully without any of the complications of sex or Mendelian genetics, except for the occasional occurrence of a mutant, (ii) a large system size and an even rarer occurrence of mutations per birth event, (iii) uniqueness and global attractiveness of any interior attractor of the community dynamics in the limit of the infinite system size. 

In section 6 we try to delineate, by a tentative listing of "axioms", the largest possible class of processes that can result from the kind of limiting considerations spelled out in section 5. And in section 7 we heuristically derive some very general predictions about macro-evolutionary patterns, based on those weak assumptions only. 

In the final section 8 we discuss (i) how the results from the preceding sections may fit into a more encompassing view of biological evolution, and (ii) some directions for further research.

J.S. van Heerwaarden

Papers

Adaptive Dynamics: A Geometrical Study of the Consequences of Nearly Faithful Reproduction