# Ecological Dynamics and Evolution of Cooperation in Vehicular Ad Hoc Networks

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. The Proposed Model Based on Evolutionary Public Goods Game

#### 3.1. Formulation of the Model

#### 3.2. Using the Game to Inform the Model

## 4. Dynamical Analysis

**Scenario**(

**i**) Consider a system without any cooperators. In such situation ($x=0$), according to Equation (6), the average payoff of the defector will be ${\mathsf{\Pi}}_{D}=0$, hence the rate of change of defector frequencies is $\dot{y}\left(t\right)<0$. This results in decreasing defector frequency in the population and, consequently, it goes extinct.

**Scenario**(

**ii**) In the absence of defectors in the population ($y=0$), according to Equation (6), the average payoff of cooperators is given by:

**Scenario**(

**iii**) The system can enable cooperators to survive even when defectors free-ride on contributions of the cooperators. In order to investigate this scenario, a new variable $\omega =\frac{x}{x+y}$ is introduced. By using Equation (8), the changing rate of $\omega $ and z are given by:

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Bifurcation diagram for population dynamics of the vehicles in the absence of the defectors. Darker (brighter) region indicates points that it can be moves into the basin of attraction of the unstable (stable) point $x=0$ ($x=1$). The dynamics has been illustrated for $N=8$, $r=3$, and $c=1$.

**Figure 2.**Types of vehicles dynamics in the absence of the defectors with regards to the parameters c and r. Darker (brighter) region indicates points that it can be moves into the basin of attraction of the unstable (stable) point $x=0$ ($x=1$). In panel (

**a**) the dynamics have been illustrated for $N=8$, $r=3$, and $d=0.5$, while in panel (

**b**) the dynamics has been shown for $N=8$, $c=1$, and $d=0.5$.

**Figure 3.**An example of the population dynamics in the presence of cooperators and defectors. This game has one polymorphic stable point between $\omega $ and z$(\omega =1,z=0.2)$. The red rectangle shows valid region of the dynamics. Red triangular and green solid circular represent saddle and stable points in the system, respectively.

**Figure 4.**Roots of function $\left(\frac{rc(1-{z}^{N})}{N(1-z)}-(rc-1){z}^{N-1}-c\right)$ for $r<2$ (panel (

**a**)) and $r\ge 2$ and $c>1$ (panel (

**b**)) for different values of N.

**Figure 5.**Types of evolutionary dynamics with regards to the model’s parameters. The game has three types of polymorphic equilibria. 1-Panel (

**a**): The game has no any polymorphic stable equilibrium point 2-Panel (

**b**): The game has one polymorphic equilibrium point 3-Panel (

**c**): The game has two polymorphic equilibria including stable and saddle points. Red triangular and green solid circular represent saddle and stable points in the system, respectively.

**Figure 6.**Types of evolutionary dynamics with regards to d. Each graph is plotted for different values of d. 1-Panel (

**a**): d = 1.6, 2-Panel (

**b**): d = 0.8, 3-Panel (

**c**): d = 0.4, 4-Panel (

**d**): d = 0.2, 5-Panel (

**e**): d = 0.01. Decreasing the value of d reduce the chance of cooperation between vehicles in the small population densities.

**Figure 7.**Types of evolutionary dynamics with regards to n. 1-Panel (

**a**): n = 3, 2-Panel (

**b**): n = 5. High value of n results in decreasing the chance of cooperation between vehicles in the small population densities.

**Figure 8.**Types of evolutionary dynamics with regards to c. 1-Panel (

**a**): c = 2.25, 2-Panel (

**b**): c = 5, 3-Panel (

**c**): c = 50. Low value of c increases the chance of cooperation between vehicles.

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**MDPI and ACS Style**

Salimi Sartakhti, J.; Stodt, F.
Ecological Dynamics and Evolution of Cooperation in Vehicular Ad Hoc Networks. *Telecom* **2023**, *4*, 236-248.
https://doi.org/10.3390/telecom4020014

**AMA Style**

Salimi Sartakhti J, Stodt F.
Ecological Dynamics and Evolution of Cooperation in Vehicular Ad Hoc Networks. *Telecom*. 2023; 4(2):236-248.
https://doi.org/10.3390/telecom4020014

**Chicago/Turabian Style**

Salimi Sartakhti, Javad, and Fatemeh Stodt.
2023. "Ecological Dynamics and Evolution of Cooperation in Vehicular Ad Hoc Networks" *Telecom* 4, no. 2: 236-248.
https://doi.org/10.3390/telecom4020014