Matrix games, introduced in Subsection 3.1.2, formed the core of the early work on evolutionary games. Most game theoretic models, notions of strategy dynamics, solution concepts and applications of ESS definitions occurred explicitly in the context of matrix games. For continuous strategies, modelers relied on either Nash solutions (Auslander et al., 1978), or model-specific interpretations of the ESS concept (Lawlor and Maynard Smith, 1976; Eshel, 1983). The bulk of developments in evolutionary game theory associated with matrix games pre-date the G-function, strategy dynamics, and the ESS maximum principle. For a review of these developments see Hines (1987), Hofbauer and Sigmund (1988), and Cressman (2003). In this chapter, we place matrix games within the context of G-functions and the more general theory of continuous evolutionary games. We reformulate the ESS frequency maximum principle developed in Section 7.5 for application to matrix games. This reformulation requires some additional terminology and new definitions.

Fitness for a matrix game is expressed in terms of strategy frequency and a matrix of payoffs. As with continuous games, the G-function in the matrix game setting must take on a maximum value at all of the strategies which make up the ESS coalition vector. The reformulated maximum principle is applicable to both the traditional bi-linear matrix game and a more general non-linear matrix game.

Because evolution occurs within an ecological setting, the concepts and models of population ecology are integral to evolutionary game theory. The organisms' environment and ecologies provide the “rules,” the context to which evolution responds. The transition from an ecological model to an evolutionary model can be made seamless. Examples include the Logistic growth model, Lotka–Volterra competition equations, models of predator–prey interactions, and consumer-resource models. In fact, any model or characterization of population dynamics can be reformulated as an evolutionary game. One need only identify evolutionary strategies that determine fitness and population growth rates. Conjoining an ecological model of population growth with heritable strategies puts the model in an evolutionary game setting. Not surprisingly, then, evolutionary game theory is well suited for addressing FF&F (fit of form and function) under all of nature's diverse ecological scenarios.

Games such as arms races, Prisoner's Dilemma, chicken, battle of the sexes, and wars of attrition have become standard bases for considering the evolution of many social behaviors (any issue of animal behavior offers examples of these or variants of these games). These games, however, are not unique to evolutionary ecology. They are products of and recurrent themes in economics, engineering, sociology, and political science. It is from these disciplines that game theory first emerged as the mathematical tools for understanding and solving conflicts of interest.

Natural selection produces strategies that are continually better than those discarded along the way to some evolutionary equilibrium. Intuitively this implies that eventually, natural selection should produce the “best” strategy for a given situation. The flowering time of a plant, the leg length of a coyote, or the filter feeding system of a clam should produce higher fitness than alternative strategies that are evolutionarily feasible (within the genetic, developmental, and physical constraints in the bauplan). In graphical form these products of natural selection should reside on peaks of the adaptive landscape. Yet we have seen in the previous chapter how, under Darwinian dynamics, natural selection may produce strategies that evolve to minimum points, maximum points, and saddlepoints on the adaptive landscape.

An evolutionary ecologist studying the traits of a species whose strategy has evolved to a convergent stable minimum on the adaptive landscape may, on reflection, be surprised. At this minimum, any individual with a strategy that deviates slightly from that produced by Darwinian dynamics has a higher, not lower, fitness than the resident strategy. While an evolutionarily stable minimum can result from Darwinian dynamics, this strategy is not the “correct” solution to the evolutionary game. In this chapter, we expand upon the original word definition of an evolutionarily stable strategy (ESS) as given by Maynard Smith: “An ESS is a strategy such that, if all members of a population adopt it, then no mutant strategy could invade the population under the influence of natural selection” (Maynard Smith, 1974).

The identity of plants and animals rendered by pre-historic artists in cave paintings, rock art, and sculpture is usually recognizable. With the development of language, plants and animals were given names, but a systematic categorization arose more recently when Carl Linnaeus introduced the idea of species as a binomial nomenclature of grouping organisms by genus and species. These early forms of pictorial, vernacular, and formal means of identifying groups of animals and plants were unaltered by later knowledge of evolution and phylogeny. Categorization simply recognizes the following three obvious properties of nature.

• First, individuals (like matter in the universe) tend to be clumped rather than randomly or uniformly spread across the space of all imaginable morphologies, physiologies, and behaviors. This clumping of individuals around discrete types can to us be conspicuous – it's hard to misidentify an elephant. However, for some species, it can be really tricky. For example, both humans and male hummingbirds find it nigh impossible to distinguish the species identity of certain female hummingbirds. But, whether tightly clumped (as a planet or asteroid) or only vaguely clumped (more nebula-like), individuals can and seem to be naturally ordered as discrete kinds.

• Second, long before Darwin, heritability was recognized by the fact that kinds tended to breed among themselves (assortative mating). Assortative mating can be socially or geographically imposed, or it can be due to physical constraints (elephants and hummingbirds cannot breed in any circumstance). Of course, hybrids occur; yet the very concept of a “hybrid” implies that organisms can be grouped by heritable characteristics and that crossings between groups produce novel, yet predictable, mixes of heritable traits.

• […]

A bauplan for a group of evolutionarily identical individuals together with their environment represents the essential elements needed to construct a G-function. The bauplan has two aspects. First, it describes a set of evolutionarily feasible strategies, and, second, it specifies the intrinsic ecological properties, aptitudes, trade-offs, and limitations of this group. The environment provides a setting within which the bauplan produces species that evolve, diversify, and persist. For instance, hornbills, a frugivorous bird of African forests, differ in size, wing morphology, and bill characteristics. However, all members of the group are easily identifiable as hornbills quite distinct from other birds. It is reasonable to assume that all species of hornbills share the same bauplan and, hence, within the same environment their fitness is determined from a single G-function. Toucans of Central and South America occupy similar ecological niches to hornbills. These birds have radiated along similar morphological lines to hornbills. Yet they have a distinct bauplan from hornbills and from other bird groups. In an evolutionary game involving both hornbills and toucans, two G-functions would be required.

When modeling evolution, one usually has some taxa (such as hornbills or toucans) along with an environmental setting in mind. In this chapter, we undertake the practical task of bringing together the mathematical notation of Chapter 2 and the G-function, Definition 3.2.1, to formulate the required fitness generating functions. We will consider a number of different evolutionary games.

The ecological theater and evolutionary play were Hutchinson's words for evoking the interplay between ecological and evolutionary dynamics. Yet evolutionary ecology, that harmonious blend between what is evolutionarily feasible and ecologically acceptable, has often been difficult to achieve conceptually. The genes of traditional genetic models of evolution are not easily integrated with the individuals of population dynamic models. The G-function provides a conceptual bridge between population dynamics and evolutionary changes in strategy frequency. The structure of the G-function and the underlying population dynamics provide the ecological theater while the strategy dynamics, species archetypes, and macroevolutionary production of novel G-functions provide the evolutionary theater. Evolutionary game theory is broadly applicable to modeling questions in evolutionary ecology. Here we explore a subset of topics. The topics are chosen because of familiarity, broad interest in ecology and evolution, and for illustrating the formulation and application of the G-function approach.

Habitat selection

Behaviors allow animals to adapt to temporal and spatial variabilities in hazards and opportunities. Feeding behaviors are grouped roughly into decisions of what to eat (diet choice), how thoroughly to exploit a depleting feeding opportunity (patch use), and where to seek resources (habitat selection). There is a hierarchy of temporal and spatial scales in going from choosing a food item in the food patch up to habitat selection where an organism seeks a place to live and forage. Habitats may vary in the quality of climate, physical structure, predation risk, and resource productivity.

Bernstein et al. (1983) coined the term “the Darwinian dynamic” to describe the dynamical process underlying natural selection. Michod (1999) adds “Darwinian dynamics are systems of equations that satisfy Darwin's conditions of variability, heritability, and the struggle to survive and reproduce.” We take this same view. In fact, for several years, the authors have been collaborating on a particular unifying approach to Darwinian dynamics that puts the study of evolution in a sound mathematical framework by recognizing that natural selection is an evolutionary game. The objective of this book is to explain how the evolutionary game approach along with the concept of a fitness generating function (called a G-function) is used to formulate the equations for Darwinian dynamics. We then show how to use these equations to predict and/or simulate the outcome of evolution. The G-function also produces an adaptive landscape that is useful in analyzing results and drawing conclusions.

After 20 years of development, with our work spread over numerous publications, it was difficult, even for us, to see the whole picture. This book allowed us to draw together and unify our work within one cover. It should be a good reference for anyone interested in the mathematics of evolution. It can also function as a textbook. Working out the details of the examples provides ample homework problems.

This is a book quite unlike any other publication intended for the study of evolution. It might be thought of as mathematical Darwinism. Darwin used logical verbal arguments to understand evolution.

Darwin used lengthy, sometimes discursive, yet convincing, verbal arguments in his Origin of Species. Darwin's postulates, as discussed in Chapter 1 and upon which his theory is built, apply broadly to the explanation or understanding of evolution. However, as verbal concepts they are limited to persuasion with few formal predictive capabilities. For example, one can understand why Darwin's finches have particular beak characteristics (Weiner, 1994). One can even predict that natural selection will tend to increase beak size during periods of drought. However, one cannot use verbal Darwinian arguments to predict the exact beak size appropriate to a particular species of finch. In fact, the ability to make such a prediction based on pure Darwinian principles is impossible unless these principles can be translated into a mathematical language. Only then can his theory be used not only to explain, but also to make predictions. Furthermore, without a mathematical framework, it is difficult or impossible to understand how a trait such as cooperation evolves.

Making Darwin's theory rigorous and predictive has been an achievement of population genetics and quantitative genetics approaches to evolution. These approaches often get bogged down in the genetic details and, consequently, lose a sense of the ecological interactions that take place to determine evolution by natural selection. Furthermore, while the genetic approach may determine those “selfish” genes that are propagated through time, it is the trait that the genes code for that actually is selected.

The following observations about patterns in nature have captured the imagination of humans for millennia.

1. Fit of form and function (FF&F): different organisms appear remarkably well suited and engineered for their particular environments. The high-crowned molars of zebras and white rhinoceros act as mulching mowers for grinding grass, and protect against the inevitable wear imposed by the silica content of grass. Black rhinos, on the other hand, have lower crowned molars favoring efficient mastication of leaves and foliage. None of these animals has the sharp and stabbing canines like those of lions. Distinct species of organisms apply themselves to different ecological tasks using their appropriate sets of tools. For example, zebras and white rhinoceros feed on grass, black rhinos browse leaves from shrubs, and lions kill and eat zebras.

2. Diversity of life: we share this planet with a phenomenal array of different life forms. These forms range from delicate mosses and annual flowering plants to awesome whales and fearsome sharks. While many of these forms differ in subtle ways, most can be readily recognized and categorized as types or species quite distinct from others. This is possible because the extant denizens of our planet do not exhibit a continuum of morphological variation from bacteria to redwood tree. Rather, the morphologies and characteristics of living organisms cluster like conspicuous and discrete galaxies in morpho-space.

3. Procession of life: despite the variety and discreteness of life, organisms seem connected by design rules of increasing levels of complexity.

Nature is not an art gallery with a static display of species. Rather, Nature is a rich dynamical system involving both ecological changes in population sizes and evolutionary changes in species' number and strategies. This is never so apparent as when humans modify the environment or attempt the daunting task of managing species. Habitat destruction, competing land use objectives, harvesting, and even relatively benign changes to environments can threaten the viability of a species' population. At the other end of the management spectrum, many exotic invasive species, diseases, and agricultural pest species defy control efforts. Understandably, most management and conservation of species focus on ecological dimensions and principles. Yet, all species have evolutionary dynamics as well. Rapid evolutionary responses are commonplace (Ashley et al., 2003). Antibiotic resistance in bacteria can evolve within months or years. Pesticide resistance and herbicide resistance in insects and weedy plants can occur within years or decades. Within the span of a few decades (and less), the hand weeding of plants from Asian rice paddies has selected for weed seedlings that mimic the look of rice plants. The mechanical seed-sorting technologies of mechanized agriculture have produced weed ecotypes that mimic the flowering phenology and seed characteristics of grain crops. Newly established island populations of rodents show rapid evolutionary changes (usually increases) in body size and concomitant changes in skull morphology. Red squirrels have expanded their range south a few hundred miles into central Indiana, USA.

Darwinian dynamics couples population dynamics with strategy dynamics to model the evolutionary process. So far, we have focused on ecological models by using the G-function to express the population dynamics. In this chapter we obtain strategy dynamics using the same G-function by assuming heritable variation as a distribution of strategies around the mean strategy used by each species in the population.

Any theory of evolution that includes natural selection is incomplete, unless it includes both population dynamics and strategy dynamics. The resulting Darwinian dynamics captures the full rich behavior of the evolutionary processes. Its use clarifies two important features of evolutionary stability: resistance to invasion and dynamic attainability (Eshel and Motro, 1981; Eshel, 1983; Taylor, 1989; Christiansen, 1991; Takada and Kigami, 1991; Taylor, 1997).

The concept of a strategy dynamic requires us to deal with issues of mutation, heritable variation, and whether species represent asexual lineages or populations of sexually interbreeding individuals. The strategy ui of a species is no longer considered fixed. Rather, it describes a population's mean strategy value that contains some variability in value among the individuals of the population. The introduction of strategy dynamics with the population dynamics leads to a new time scale. In addition to the ecological time scale there is now an evolutionary time scale. Because population dynamics and strategy dynamics may occur on different time scales, it is useful to make this distinction. We will show that population dynamics generally, but not always, occur on a faster time scale than strategy dynamics.

The definition of an ESS in Chapter 6 requires that an ESS be convergent stable and resistant to invasion by alternative strategies. The Darwinian dynamics discussed in Chapter 5 provides convergent stability. In this chapter, an ESS maximum principle is obtained that characterizes the property of resistance to invasion. We show that, when u and x* have values corresponding to an ESS, the G-function must take on a maximum with respect to the focal strategy, v, when v is set equal to one of the strategies of the ESS. Like the Nash equilibrium, an ESS is a no-regret strategy for the G-function in the sense that at an ESS no individual can gain a fitness advantage by unilaterally changing strategy. This property of the ESS, as expressed in terms of the G-function, can be used as a necessary condition for solving for candidate ESS solutions. This necessary condition is formalized in this chapter as the ESS maximum principle. This principle also describes the property of adaptation, the true sense of FF&F. An adaptation is a strategy that maximizes individual fitness as determined from the G-function given the circumstances, and these circumstances include the strategies and population sizes of others.

We use the term ESS candidate to refer to any solutions obtained using the ESS maximum principle. While an ESS candidate will be resistant to invasion, it need not satisfy the convergent stability property required by the ESS definition. Thus, convergence stability must be checked by some other method.