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

/pdf/evolutionarily-singular-strategies-and-the-adaptive-growth-9tna03ynki.pdf

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.

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

We study the dynamics of a population of residents that is being invaded by an initially rare mutant. We show that under relatively mild conditions the sum of the mutant and resident population sizes stays arbitrarily close to the initial attractor of the monomorphic resident population whenever the mutant has a strategy sufficiently similar to that of the resident. For stochastic systems we show that the probability density of the sum of the mutant and resident population sizes stays arbitrarily close to the stationary probability density of the monomorphic resident population. Attractor switching, evolutionary suicide as well as most cases of "the resident strikes back" in systems with multiple attractors are possible only near a bifurcation point in the strategy space where the resident attractor undergoes a discontinuous change. Away from such points, when the mutant takes over the population from the resident and hence becomes the new resident itself, the population stays on the same attractor. In other words, the new resident "inherits" the attractor from its predecessor, the former resident.

Invasion dynamics and attractor inheritance.

We review mechanisms that lead to cyclic evolution with alternating levels of diversity. Such cycles involve directional evolution towards a so-called evolutionary branching point, where selection becomes disruptive and splits the population into two strategies. Coevolution of these strategies eventually leads to the extinction of one of them. The remaining strategy evolves back to the evolutionary branching point, and a new cycle begins. There are a number of different evolutionary mechanisms that can produce this kind of cycles including chance extinction, switching between population dynamical attractors, and coevolution with an ecologically distinct species. We also present an example for branching-extinction cycles where the direction of evolution changes between monomorphic and dimorphic populations solely due to the different levels of diversity. The latter cycles exhibit a novel feature: Even though extinction is deterministic in the sense that it is unavoidable and always occurs at the same trait values, it is random which of the two coexisting strategies goes extinct. As a result, long and short cycles alternate in a random sequence.

https://www.mv.helsinki.fi/home/kisdi/pdf/FEJ-11.PDF

Red Queen Evolution by Cycles of Evolutionary Branching and Extinction

We use the theory of adaptive dynamics to construct and analyse a generic example of cycling evolution with alternating levels of polymorphism. A monomorphic population evolves towards larger trait values until it reaches a so-called evolutionary branching point. Disruptive selection at the branching point splits the population into two strategies. In the dimorphic population the strategies undergo parallel coevolution towards smaller trait values. Finally one of the two strategies goes extinct, and the remaining single strategy evolves upwards again to the branching point. The reversal of the direction of evolution is brought about by the changing level of polymorphism. Extinction is deterministic, i.e., it occurs inevitably and always at the same trait values; which of the two strategies goes extinct is, however, random. The present model is discussed in relation to other mechanisms for evolutionary cycles involving branching and extinction.

We introduce a notion of attractor to community-dynamical processes as they are studied in biological models and their computer simulations. This attractor concept is modeled after the Conley-Ruelle attractor. It incorporates the fact that in an immigration-free community-dynamical process populations can go extinct at low values of their densities.

F.J.A. Jacobs

Papers

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

Invasion dynamics and attractor inheritance.

Red Queen Evolution by Cycles of Evolutionary Branching and Extinction

Red Queen Evolution by Cycles of Evolutionary Branching and Extinction

On the Concept of Attractor in Community-Dynamical Processes