# Evolutionary Game Theory: Darwinian Dynamics and the G Function Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Evolutionary Game Theory: Darwinian Dynamics and G Functions

#### 2.1. Introduction to G Functions

#### 2.2. Population and Strategy Dynamics

**u**) and population densities (

**x**), we can plot the per capita growth rate as a function of the focal individual’s strategy (v). Unlike Wright’s fitness landscape, the adaptive landscape changes over time, with changes in densities and strategies of organisms in the population continually molding its shape.

#### 2.3. G Function Modeling Recipe

**u**, affect the values of key parameters in the model. Once we are able to represent these parameters as functions of v,

**u**, and

**x**, we have our G function. This model can then be simulated using the Darwinian dynamics developed in the previous section to observe the ecological and evolutionary dynamics of the species.

## 3. Example: Predator–Prey (Multiple G Functions)

**u**, $\mathit{\mu}$,

**x**, and

**y**. Here, we choose to allow the carrying capacity (K), the competition coefficient (a), and the predation term (b) to vary based on the strategies of the predator and prey. Namely, we assume that a strategy of $\mathit{v}=0$ maximizes a prey’s carrying capacity. Deviations from this, for example as a result of evolving to predatory pressures, decrease the carrying capacity in a Gaussian fashion:

## 4. Example: Combination Therapy in Cancer (Vector-Valued Strategies)

**v**,

**u**, and

**x**. Naturally, since our strategies capture drug resistance, we allow ${\mathit{s}}_{1}$ and ${\mathit{s}}_{2}$ to vary as a function of

**v**. Namely, assuming a Michaelis-Menten form, we let:

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Example snapshot of adaptive landscape. This figure depicts a temporal snapshot of a possible adaptive landscape. The x-axis represents the trait value under consideration and the y-axis represents the fitness of the individual. The adaptive landscape then depicts the fitness conferred to an individual adopting a given trait value at a given time. In reality, adaptive landscapes are highly dynamic, changing in response to species’ population sizes and strategy frequencies.

**Figure 3.**Predator–prey control simulations. The first panel depicts the population and strategy dynamics of predator (orange/gold) and prey (blue/cyan). The second panel is a snapshot of the adaptive landscape at the eco-evolutionary equilibrium. The third panel is a 3D adaptive landscape, showing the changes in the adaptive landscape over time. Note that the adaptive landscape stays relatively constant over time and the ESS for both prey and predator occur when their focal strategies are 0.

**Figure 4.**Predator–prey high maximal capture probability simulations. The first panel shows the population and strategy dynamics of predator (orange/gold) and prey (blue/purple). The second panel shows the adaptive landscape at equilibrium and the third panel is the 3D adaptive landscape. This time, we see a highly dynamic adaptive landscape, paralleling the oscillatory behavior noticed in the strategy dynamics. Though the strategy equilibria for predator and prey still coincide, the equilibrium is an ESS for the predator but is a convergent stable minimum for the prey, indicating the possibility for invasibility and speciation of the prey.

**Figure 5.**Predator–prey speciation event simulations. The first panel shows the population and strategy dynamics of predator (green), prey 1 (blue/purple), and prey 2 (orange/gold). The second panel shows the adaptive landscape at equilibrium and the third panel is the 3D adaptive landscape (for visualization purposes, fitness values above 0.05 were truncated). Again, the adaptive landscape greatly changes over time, particularly in the first half of the simulation. Speciation clearly occurs, with the prey occupying strategy peaks on either side of the predator’s (generalist) strategy.

Variable | Interpretation |
---|---|

G | Per Capita Growth Rate of a Member |

v | Strategy of Focal Member |

u | Vector of Strategies of all Members |

x | Vector of Population Sizes of all Members |

U | Evolutionary Strategy Set |

Parameter | Interpretation | Value |
---|---|---|

${\mathit{K}}_{\mathit{m}}$ | Carrying Capacity | 100 |

${\mathit{b}}_{\mathit{m}}$ | Maximal Capture Probability | 0.15 |

c | Conversion Efficiency: Prey to Predator | 0.25 |

${\mathit{r}}_{1,2}$ | Intrinsic Proliferation Rate | 0.25 |

${\mathit{k}}_{1,2}$ | Evolvability | 0.5 |

${\varphi}_{\mathit{k}}^{2}$ | Range of Resources | 2 |

${\varphi}_{\mathit{a}}^{2}$ | Species Niche Width | 4 |

${\varphi}_{\mathit{b}}^{2}$ | Breadth of Predation | 10 |

${\sigma}_{\mathit{i}}^{2}$ | Evolvability | 0.5 |

Parameter | Interpretation | Value |
---|---|---|

${\mathit{K}}_{\mathit{m}}$ | Carrying Capacity | 100 |

r | Intrinsic Proliferation Rate | 0.25 |

${\sigma}_{1,2}$ | Evolvability Variance | 0.5 |

${\delta}_{1,2}$ | Evolvability Covariance | [−0.25, 0, 0.25] |

${\lambda}_{1,2}$ | Initial Drug Resistance | 1 |

${\mathit{b}}_{1,2}$ | Impact of Drug Resistance on Death | 0.3 |

${\mathit{m}}_{1,2}$ | Drug Dosage/Efficacy | 0.3 |

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Bukkuri, A.; Brown, J.S.
Evolutionary Game Theory: Darwinian Dynamics and the *G* Function Approach. *Games* **2021**, *12*, 72.
https://doi.org/10.3390/g12040072

**AMA Style**

Bukkuri A, Brown JS.
Evolutionary Game Theory: Darwinian Dynamics and the *G* Function Approach. *Games*. 2021; 12(4):72.
https://doi.org/10.3390/g12040072

**Chicago/Turabian Style**

Bukkuri, Anuraag, and Joel S. Brown.
2021. "Evolutionary Game Theory: Darwinian Dynamics and the *G* Function Approach" *Games* 12, no. 4: 72.
https://doi.org/10.3390/g12040072