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Adaptive dynamics is a mathematical framework for studying evolution. It extends evolutionary game theory to account for more realistic ecological dynamics and it can incorporate both frequency- and density-dependent selection. This is a practical guide to adaptive dynamics that aims to illustrate how the methodology can be applied to the study of specific systems. The theory is presented in detail for a single, monomorphic, asexually reproducing population. We explain the necessary terminology to understand the basic arguments in models based on adaptive dynamics, including invasion fitness, the selection gradient, pairwise invasibility plots (PIP), evolutionarily singular strategies, and the canonical equation. The presentation is supported with a worked-out example of evolution of arrival times in migratory birds. We show how the adaptive dynamics methodology can be extended to study evolution in polymorphic populations using trait evolution plots (TEPs). We give an overview of literature that generalises adaptive dynamics techniques to other scenarios, such as sexual, diploid populations, and spatially-structured populations. We conclude by discussing how adaptive dynamics relates to evolutionary game theory and how adaptive-dynamics techniques can be used in speciation research.

The basic principle of evolution, survival of the fittest

Adaptive dynamics is a set of techniques developed more recently, largely during the 1990s, for understanding the long-term consequences of small mutations in the traits expressing the phenotype. The number of papers using adaptive dynamics techniques is increasing steadily as adaptive dynamics is gaining ground as a versatile tool for evolutionary modeling, with applications in a range of diverse areas including speciation and diversification ([

To facilitate the analysis of systems with eco-evolutionary feedbacks, adaptive dynamics makes use of terminology and concepts that are not found in traditional ecological and evolutionary textbooks. Using adaptive dynamics to analyze even rather simple models can therefore prove daunting for the first time. Previous literature describes and discusses the tools for adaptive dynamics analysis from different perspectives. The foundational papers by Metz

Responding to the need for a broadly accessible introduction to adaptive dynamics, this paper provides a step-by-step guide on how adaptive dynamics can be used to analyze eco-evolutionary models. The paper is aimed at students and researchers wanting to learn adaptive dynamics to the level necessary to follow the arguments made in adaptive-dynamics studies. In the next section we introduce the fundamental concepts behind adaptive dynamics. Then, in

The two fundamental ideas of adaptive dynamics are (i) that the resident population can be assumed to be in a dynamical equilibrium when a new mutant type appears and (ii) that the eventual fate of such mutant invasions can be inferred from the initial growth rate of the mutant population while it is still rare compared with the resident type. When used in concert, these two assumptions amount to a separation of the slower evolutionary time scale from the faster ecological time scale. The initial growth rate of the mutant is generally known as its

The first step in an adaptive-dynamics analysis is to identify the traits that are undergoing evolutionary change. These trait should be specified quantitatively, at the individual level ([

With the demographic model in place, it is possible to determine the invasion fitness, the initial growth rate of a rare mutant invading a resident population. Depending on model complexity, this may be straightforward or very challenging, but once determined all techniques of adaptive dynamics can be applied independent of the underlying model. In the next section we will introduce these techniques for monomorphic populations.

Brief explanation of the principal terms used in adaptive dynamics.

Term | Description |
---|---|

Differential equation describing a deterministic approximation of evolutionary dynamics with small mutational steps. | |

Singular strategy that, within a neighborhoods, is approached gradually. | |

Singular strategy that is both convergence stable and evolutionarily stable. | |

Population with individuals having either of two distinct trait values. | |

Trait value at which the selection gradient vanishes. | |

Trait value that cannot be invaded by any nearby mutant. | |

The expected growth rate of a rare mutant. | |

Population consisting of individuals with only one distinct trait value. | |

Graphical illustration of invasion success of potential mutants when the population is monomorphic. | |

The expected rate at which an individual produces offspring. Can be determined by dividing the population growth rate by the number of individuals. | |

Population with individuals having either of several distinct trait values. | |

Slope of the invasion fitness at the resident trait value. Gives information on the direction and speed of evolutionary change. | |

Graphical illustration of invasion success of potential nearby mutants when the population is dimorphic. |

To see how the fundamental concepts introduced above are used, it makes sense to start with the simplest case of a monomorphic population,

The invasion fitness

Plot of the invasion fitness

We will always assume that the resident is at its demographic attractor, and as a consequence

The selection gradient is defined as the slope of the invasion fitness at

To illustrate the concepts introduced above, consider a population of

Next, assume that the birth-rate is subject to evolutionary change without constraints. For this end, we extend the model to include two populations, a resident population

What will happen if

Three qualitatively different singular strategies. (a) A local fitness maximum representing a possible endpoint of evolutionary change. (b) A local fitness minimum at which evolutionary branching can occur. (c) A degenerate case where the criteria from

The evolutionary dynamics can be studied graphically. Recall that the invasion fitness represents the fitness landscape as experienced by a rare mutant. In a large (effectively infinite) population, only mutants with trait values

Pairwise invasibility plots (PIPs) of the example birth-death model in

Four logical combinations of evolutionary stability and convergence stability for a singular strategy. (a) Evolutionarily stable and convergence stable. A possible endpoint of evolution: the strategy can be attained gradually and then it will resist any invaders successfully. (b) Evolutionarily stable but not convergence stable. Such singular strategies should rarely be realized in nature: although the strategy cannot be invaded once it is realized, evolution starting from any nearby strategy will gradually lead away from the singular strategy. (c) Convergence stable but not evolutionarily stable. A scenario where a population can become dimorphic: the singular strategy can be established gradually, but then it can be invaded by mutants both above and below the resident strategy at the same time. (d) Neither evolutionarily stable nor convergence stable. As in (b), a monomorphic population will evolve away from the singular strategy. It is still possible that a dimorphic population will arise, if coexistence is supported (see

The evolutionarily singular strategies in a PIP are found where the boundary of the region of positive invasion fitness intersects the diagonal. The singular strategy in

Singular strategies can be located and classified once the invasion fitness is known. To locate singular strategies, it is sufficient to find the points for which the selection gradient vanishes,

The evolutionary process can be envisaged as a sequence of successfully established mutations. This process is strictly directional in large populations as only mutants with positive invasion fitness can invade. The most common way to model directional evolutionary change is a differential equation on the form,

The most common assumption in adaptive dynamics is mutation-limited evolution with small mutational steps. Under these assumptions, Dieckmann and Law [

While the canonical equation is often used to study evolutionary change, this is by no means necessary. In many cases, the eventual evolutionary outcome is independent of mutational step size and can for monomorphic evolutionary dynamics be determined directly from the pairwise invasibility plot. For polymorphic populations, Ito and Dieckmann [

To illustrate how the introduced techniques for analyzing evolutionary change can be used in practice, we here analyze a simplified variant of a model from Johansson and Jonzén [

We write

To answer this question, we first assume that there are more individuals than available territories and that individuals survive from one year to the next with probability

We now consider an invasion by a mutant bird with population size

Assuming that the resident is at its demographic attractor and that the mutant is so rare that it has a negligible influence on per capita growth rates, we have

Next, we determine the selection gradient. Differentiating the invasion fitness with respect to the mutant trait value

To move further, we make the specific assumption that the competitive ability decreases exponentially with arrival time and that the reproductive success of territory owners is given by a Gaussian function (see

In many territorial animals, there is a prior residence effect by which the first individual to occupy a territory often gets the upper hand in the competition. (a) In the model in

We conclude that, due to the individual advantage of arriving earlier relative to the rest of the population, the evolutionarily stable strategy occurs some time before the arrival date

The normal outcome of a successful invasion is that the mutant replaces the resident. However, other outcomes are also possible ([

Illustration of the graphical method for obtaining the region of coexistence. (a) A pairwise invasibility plot from the Snowdrift game [

The invasion fitness is generalized to

The emergence of protected dimorphism near singular strategies during the course of evolution is not unusual, but its significance depends on whether selection is stabilizing (invasion-fitness maximum) or disruptive (invasion-fitness minimum). In the latter case, the traits of the two morphs will diverge in a process often referred to as evolutionary branching. Metz

Evolution after evolutionary branching is illustrated using trait evolution plots. These show the region of coexistence, the direction of evolutionary change, and whether strategies at which the selection gradient vanishes are fitness maxima or minima. Evolution may well drive the dimorphic population outside the region of coexistence, in which case one morph goes extinct and the population once again becomes monomorphic.

Levene’s soft selection model studied by Geritz

An intersections of two isoclines is known as a singular coalition ([

The boundaries of the region of coexistence are extinction threshold for morphs, and hence for a dimorphic population

The isoclines defined by

Evolutionary branching in a morph

It is late afternoon and only hours remain until we have to cease writing and submit this manuscript for publication. Thankfully, most of what we have wanted to say has already been presented in the previous sections and all that remains is now to give our personal views on two selected topics and to offer recommendations for further reading on the rapidly developing field of adaptive dynamics.

The archetypal situation in evolutionary game theory is a population of individuals choosing between two or more pure strategies. The classical hawk-dove game, introduced by Maynard Smith and Price [

One of the most exciting findings of adaptive dynamics is evolutionary branching, the process by which an initially monomorphic population can become dimorphic through small mutational steps (see

We do not wish to place ourselves in the line of fire. Rather, we want to highlight the usefulness of adaptive dynamics in elucidating the ecological conditions that support species coexistence over evolutionary timescales. In the classical picture of speciation, an ancestral population becomes spatially separated. The two populations evolve in different directions and, over time, they might accumulate Dobzhansky–Muller incompatibilities. If, at some later time, these populations come into secondary contact, the potentially accumulated incompatibilities would severely reduce the fitness of any hybrid offspring, so-called post-zygotic isolation. Selection might then promote mechanisms that prevent the formation of hybrid offspring, so-called pre-zygotic isolation. In this scenario, two new species have emerged from one.

The above process, known as allopatric speciation, could conceivably work if the population becomes spatially segregated and the environments that the two subpopulations encounter are sufficiently different, for example if they contain different resources, henceforth apples and pears. Should the two environments contain both apples and pears, which we find more likely, the story would be more complicated. It would take a stroke of luck to ensure that one subpopulation specializes on apples with the other specializing on pears. An outcome that we find more plausible is that two subpopulations remain generalists, consuming both apples and pears. They might still acquire Dobzhansky–Muller incompatibilities and pre-zygotic isolation could potentially evolve upon secondary contact. What comes next, however, depends on the ecology. Under disruptive selection, we might see the emergence of two ecologically differentiated species. The period in allopatry has then become the tool that facilitates evolutionary branching, but the ecological side of the story remains unchanged.

Complete spatial segregation is probably unlikely and several mechanisms exist that can enable speciation in the presence of some degree of gene flow ([

A good introductory text to adaptive dynamics is Diekmann [

The canonical equation is introduced by Dieckmann and Law [

Several extensions of adaptive dynamics have been considered over the last two decades.

Salient extensions of adaptive dynamics theory ordered according to topic.

Type of generalization | References |
---|---|

Explicit genetics and standing genetic variation | [ |

Mathematical underpinnings | [ |

Multiple species | [ |

Multiple traits and function-valued traits | [ |

Physiologically structured populations | [ |

Sexually-reproducing populations | [ |

Spatially-structured populations | [ |

Stochastic environments | [ |

Trade-off analysis | [ |

Acknowledgments We thank Bernd Blasius and Thilo Gross for inviting us to the Stanislaw Lem workshop on evolution at Lviv, Ukraine in 2005. Following this workshop, we started writing the Hitchhiker’s Guide to Adaptive Dynamics. At this time, Å.B. was a postdoctoral fellow in Nara, Japan, supported by a generous grant from the Japan Society for the Promotion of Science. This support is gratefully acknowledged. We also want to thank Ulf Dieckmann, Hans Metz, Karl Sigmund, and Fugo Takasu for countless discussions about adaptive dynamics and its applications, as well as Niclas Jonzén for kindly allowing us to reproduce his photo of two blue tits competing for a nest box. Finally, we very much appreciate the kind and supportive statements we have received from readers of early versions of this manuscript.

We here give a brief mathematical derivation of criteria that can be used to check whether a singular strategy is evolutionarily stable, convergence stable, and whether protected dimorphisms (two coexisting populations with different trait values) are possibility near the singular strategy. The arguments build on Metz

As stated in

If a singular strategy is convergence stable but not evolutionarily stable, selection is disruptive near the singular strategy and evolutionary branching will eventually occur. However, even with stabilizing selection, protected dimorphism may occur near a singular strategy provided there are points near the singular strategy where both

To be precise, the phrase “survival of the fittest” was coined by the philosopher Herbert Spencer and adopted by Darwin from the fifth edition of

In structured population models, there will be an initial transient phase during which the per capita growth rate depends on the population structure, whether the population is structured in space, size, stage, or according to another characteristic. The invasion fitness then has to be defined as the long-term per capita growth rate of the mutant population.

Metz [

The fixed point is stable since the slope of