# The Hitchhiker’s Guide to Adaptive Dynamics

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## Abstract

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## 1. Introduction

## 2. Fundamental Concepts

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

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

Continuously stable strategy (CSS) | Singular strategy that is both convergence stable and evolutionarily stable. |

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

Evolutionarily singular strategy | Trait value at which the selection gradient vanishes. |

Evolutionarily stable strategy (ESS) | Trait value that cannot be invaded by any nearby mutant. |

Invasion fitness | The expected growth rate of a rare mutant. |

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

Pairwise invasibility plot (PIP) | Graphical illustration of invasion success of potential mutants when the population is monomorphic. |

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

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

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

Trait evolution plot (TEP) | Graphical illustration of invasion success of potential nearby mutants when the population is dimorphic. |

## 3. Monomorphic Evolution

#### 3.1. Invasion Fitness and the Selection Gradient

**Figure 1.**Plot of the invasion fitness ${s}_{r}\left(m\right)$, the expected growth rate of a rare mutant in the environment set by the resident (solid lines), as a function of the mutant trait value m, for two illustrative cases. The dashed lines denote the local tangent of ${s}_{r}\left(m\right)$ at $m=r$ where its slope corresponds to the selection gradient ${s}_{r}^{\prime}\left(r\right)$. (a) The population is monomorphic and consists only of individuals with trait value ${r}_{1}$. Mutants with higher trait values have positive expected growth rate and can hence invade. (b) A mutant with trait value ${r}_{2}$ has invaded and successfully replaced the resident. Since the population now consists of individuals with a new trait value, ${r}_{2}$, the fitness landscape itself has changed. Note that the invasion fitness vanishes exactly when the mutant trait value equals that of the resident, i.e., $m={r}_{1}$ and $m={r}_{2}$ for panel (a) and (b) respectively.

#### 3.2. Deriving the Invasion Fitness and the Selection Gradient from a Demographic Model

#### 3.3. Evolutionarily Singular Strategies and the Fitness Landscape

**Figure 2.**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 Section 3.5 fail because the second order derivative of ${s}_{r}\left(m\right)$ vanishes. These cases are without real-world significance, since finite evolutionary steps will lead evolution past these points.

#### 3.4. Pairwise Invasibility Plots

**Figure 3.**Pairwise invasibility plots (PIPs) of the example birth-death model in Section 3.2 with birth rate as the evolving trait value. Green regions represent combinations $(r,m)$ for which a mutant with trait value m can invade a resident population consisting of individuals with trait value r, i.e., for which ${s}_{r}\left(m\right)>0$. (a) PIP without constraints in birth rate, as given by (1). The birth rate can evolve to ever higher values. (b) PIP with costs of higher birth rate, as given by (2) with $c\left(r\right)={10}^{-1}exp\left(r\right)$. The singular point is both evolutionarily and convergence stable and is located at $r\approx 2.3$. Note that diagonal corresponds to ${s}_{r}\left(r\right)=0$.

**Figure 4.**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 Section 5), but it is likely to happen through a large mutational step rather than the gradual process of evolutionary branching.

#### 3.5. Evolutionary Stability Analysis

#### 3.6. Modeling Gradual Evolution

## 4. Example: Evolution of Arrival Time of Migratory Birds

**Figure 5.**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 Section 4, it is assumed that reproductive output E (orange line) is maximized for arrival at day ${x}_{\mathrm{opt}}$. Early-arriving individuals have a higher competitive ability C (blue line), which makes them more able to obtain territories. (b) Two individuals of blue tit (Cyanistes caeruleus), a partial migratory bird, competing over a nest box in the beginning of the breeding season (photo: Niclas Jonzén).

## 5. Polymorphic Evolution

**Figure 6.**Illustration of the graphical method for obtaining the region of coexistence. (a) A pairwise invasibility plot from the Snowdrift game [10]. (b) The same pairwise invasibility plot mirrored over the main diagonal. (c) The first two panels overlaid in which the region of coexistence is visible as the green (dark gray) area. Note that protected dimorphisms are possible even though the singular strategy is evolutionarily stable and selection thus stabilizing.

#### 5.1. Invasion Fitness and Selection Gradients in Polymorphic Populations

#### 5.2. Evolutionary Branching

#### 5.3. Trait Evolution Plots

**Figure 7.**Levene’s soft selection model studied by Geritz et al. [27]. (a) Pairwise invasibility plot showing the evolutionary dynamics for a monomorphic population. Since selection at the convergence stable singular strategy is disruptive, the population eventually becomes dimorphic with evolutionary dynamics given by the trait evolution plot. (b) Trait evolution plot showing the direction of evolutionary change. Thick lines are evolutionarily stable isoclines, where directional selection in one of the two morphs ceases. In this case, the trait evolution plot shows the final evolutionary outcome to be a stable protected dimorphism located at the intersection of the two isoclines. The green area is the region of coexistence, as described in Figure 6.

#### 5.4. Evolutionarily Singular Coalitions

#### 5.5. Connection of the Isoclines to the Boundary

#### 5.6. Further Evolutionary Branching

## 6. Discussion

#### 6.1. Relation to Evolutionary Game Theory

#### 6.2. Role in Speciation Research

#### 6.3. Recommendations for Further Reading

`mathstat.helsinki.fi/~kisdi/addyn.htm`. Adaptive dynamics is an active area of research, so be sure to check the forward citations for the latest developments.

Type of generalization | References |
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Explicit genetics and standing genetic variation | [39,58,59,60,61] |

Mathematical underpinnings | [42,62,63,64,65] |

Multiple species | [9,66,67,68] |

Multiple traits and function-valued traits | [43,69,70,71] |

Physiologically structured populations | [55,56] |

Sexually-reproducing populations | [72] |

Spatially-structured populations | [73,74,75] |

Stochastic environments | [76,77,78] |

Trade-off analysis | [79,80,81,82,83] |

## Acknowledgments

## A. Appendix: Local Classification of Singular Strategies

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^{1.}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 On the origin of species.^{2.}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.^{3.}Metz [34] argues that the name evolutionarily stable strategy (ESS) is a partial misnomer as the strategy does not need to be evolutionarily attracting. Since the ESS concept is deeply ingrained, it has been proposed that the meaning of the acronym should be altered to evolutionarily steady strategy. An ESS that is also evolutionarily attracting is called a continuously stable strategy (CSS).^{4.}The fixed point is stable since the slope of ${n}_{t+1}$ seen as a function of ${n}_{t}$ at ${n}_{t}={n}^{*}$ is exactly equal to p, which is positive and less than 1 in magnitude.

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Brännström, Å.; Johansson, J.; Von Festenberg, N. The Hitchhiker’s Guide to Adaptive Dynamics. *Games* **2013**, *4*, 304-328.
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**AMA Style**

Brännström Å, Johansson J, Von Festenberg N. The Hitchhiker’s Guide to Adaptive Dynamics. *Games*. 2013; 4(3):304-328.
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**Chicago/Turabian Style**

Brännström, Åke, Jacob Johansson, and Niels Von Festenberg. 2013. "The Hitchhiker’s Guide to Adaptive Dynamics" *Games* 4, no. 3: 304-328.
https://doi.org/10.3390/g4030304