# An Improved Lotka–Volterra Model Using Quantum Game Theory

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## Abstract

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## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Quantum Game Model

#### 2.1.1. Basics of the Quantum Game Model

#### 2.1.2. Hilbert Spaces and Space of States

#### 2.1.3. Entanglement and Disentanglement

#### 2.2. Calculation Steps for the Quantum Game Model

#### 2.2.1. Hypothesis

#### 2.2.2. Calculation Steps

## 3. An Improved L–V Model: Coexistence of Competition and Cooperation

#### 3.1. Equilibrium and Stability Conditions in Competitive Coexistence

#### 3.2. Equilibrium and Stability Conditions in Cooperative Coexistence

## 4. Results and Scenario Analysis

#### 4.1. Scenario Settings

#### 4.2. Scenario Simulation Results and Analysis

_{a}= 50, M

_{b}= 50). However, population sizes for players in competitive coexistence are always smaller than the initial environment capacity (M

_{a}= 50, M

_{b}= 50).

_{a}= 50, M

_{b}= 50); the entanglement intensity ($\gamma $) negatively affects the final population size for player A, but has no impact on the evolution of population size for player B; the final population size for player B is larger than for player A, even though the payoff for player A is higher than for player B, which reflects weaker players defeating stronger players in case 2. Second, in cooperative coexistence: the final population sizes for players surpasses the initial environment capacity; the entanglement intensity ($\gamma $) simultaneously and positively affects population sizes for the players; all final population sizes for player A are larger than for player B no matter the $\gamma $ value. Meanwhile, by contrast, final population sizes for players in cooperative coexistence are always larger than the initial environmental capacity, but always smaller than the initial environmental capacity in competitive coexistence. Furthermore, increasing entanglement intensity increases population sizes for players in cooperative coexistence, but constrains population sizes for players who initially take an independent strategy in competitive coexistence. In addition, in competitive coexistence, the final and stable population sizes for player B are larger than those for player A, even though player B has lower returns than player A, which reflects that, in some cases, weaker players can beat stronger players in competitive coexistence. However, in cooperative coexistence, the final population sizes for player A with higher payoffs are always larger than the final population sizes for player B, which illustrates that players with higher payoffs generally have more power in cooperative coexistence.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the quantum game calculation. (Note: this schematic diagram is mainly referred to Eisert et al. 1999).

**Figure 3.**Evolution trajectories for player population sizes in case 1. (Note: QQ is for case 1; 20, 45, and 70 are the different entanglement intensities; Coo and Com are the cooperation and competition coexistence, respectively; xA20 and xB20, xA45 and xB45, and xA70 and xB70 are the evolution trajectories of player A and B population sizes with entanglement intensity 20°, 45° and 70°, respectively).

**Figure 4.**Evolution trajectories for player population sizes in case 2. (Note: ca0 is case 2; 20, 45, and 70 equal different entanglement intensities; Coo and Com are cooperation and competition coexistence, respectively; xA20 and xB20, xA45 and xB45, and xA70 and xB70 are the evolution trajectories of player A and B population sizes with entanglement intensity 20°, 45° and 70°, respectively).

**Figure 5.**Evolution trajectories for player population sizes in case 3. (Note: ca180 is case 3; 20, 45, and 70 equal different entanglement intensities; Coo and Com are cooperation and competition coexistence, respectively; xA20 and xB20, xA45 and xB45, and xA70 and xB70 are the evolution trajectories of player A and B population sizes with entanglement intensity 20°, 45° and 70°, respectively).

Player B | |||

Cooperation ($\widehat{\mathit{C}}$) | Competition ($\widehat{\mathit{D}}$) | ||

Player A | Cooperation ($\widehat{\mathit{C}}$) | ${A}_{11}$, ${B}_{11}$ | ${A}_{12}$, ${B}_{12}$ |

Competition ($\widehat{\mathit{D}}$) | ${A}_{21}$, ${B}_{21}$ | ${A}_{22}$, ${B}_{22}$ |

Case | Strategy | Parameter | Quantum Payoff |
---|---|---|---|

1 | ${\widehat{U}}_{A}=\widehat{Q}$, ${\widehat{U}}_{B}=\widehat{Q}$ | ${\theta}_{A}=0$, ${\theta}_{B}=0$, ${\alpha}_{A}=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$,${\alpha}_{B}=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | $\overline{{R}_{A}}={A}_{11}$, $\overline{{R}_{B}}={B}_{11}$ |

2 | ${\widehat{U}}_{A}=\widehat{C}$, ${\widehat{U}}_{B}=\widehat{Q}$ | ${\theta}_{A}=0$, ${\theta}_{B}=0$, ${\alpha}_{A}=0$, ${\alpha}_{B}=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | $\overline{{R}_{A}}={A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)$, $\overline{{R}_{B}}={B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)$ |

3 | ${\widehat{U}}_{A}=\widehat{D}$, ${\widehat{U}}_{B}=\widehat{Q}$ | ${\theta}_{A}=\pi $, ${\theta}_{B}=0$, ${\alpha}_{A}=0$, ${\alpha}_{B}=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ | $\overline{{R}_{A}}={A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)$, $\overline{{R}_{B}}={B}_{21}+\left({B}_{12}-{B}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)$ |

4 | ${\widehat{U}}_{A}=\widehat{D}$, ${\widehat{U}}_{B}=\widehat{D}$ | ${\theta}_{A}=\pi $, ${\theta}_{B}=\pi $, ${\alpha}_{A}=0$, ${\alpha}_{B}=0$ | $\overline{{R}_{A}}={A}_{22}$, $\overline{{R}_{B}}={B}_{22}$ |

5 | ${\widehat{U}}_{A}=\widehat{C}$, ${\widehat{U}}_{B}=\widehat{C}$ | ${\theta}_{A}=0$, ${\theta}_{B}=0$, ${\alpha}_{A}=0$, ${\alpha}_{B}=0$ | $\overline{{R}_{A}}={A}_{11}$, $\overline{{R}_{B}}={B}_{11}$ |

6 | ${\widehat{U}}_{A}=\widehat{D}$, ${\widehat{U}}_{B}=\widehat{C}$ | ${\theta}_{A}=\pi $, ${\theta}_{B}=0$, ${\alpha}_{A}=0$, ${\alpha}_{B}=0$ | $\overline{{R}_{A}}={A}_{21}$, $\overline{{R}_{B}}={B}_{21}$ |

Equilibrium | $\mathbf{det}\mathit{J}$ | $\mathit{t}\mathit{r}\mathit{J}$ | Stability Condition |
---|---|---|---|

${E}_{4}\left(\frac{{M}_{A}\overline{{R}_{A}}\left(\overline{{R}_{B}}-1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1},\frac{{M}_{B}\overline{{R}_{B}}\left(\overline{{R}_{A}}-1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1}\right)$ | $\frac{{r}_{A}{r}_{A}\left(\overline{{R}_{A}}-1\right)\left(\overline{{R}_{B}}-1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1}$ | $\frac{\overline{{R}_{A}}{r}_{A}\left(\overline{{R}_{B}}-1\right)+\overline{{R}_{B}}{r}_{B}\left(\overline{{R}_{A}}-1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1}$ | $\mathrm{sgn}\left(\overline{{R}_{A}}\overline{{R}_{B}}-1\right)$ = $\mathrm{sgn}\left(\overline{{R}_{B}}-1\right)$ = $\mathrm{sgn}\left(\overline{{R}_{A}}-1\right)$$=+1$ |

Case | Equilibrium | Stability Condition |
---|---|---|

1 | ${\mathit{E}}_{4}\left(\frac{{\mathit{M}}_{\mathit{A}}{\mathit{A}}_{11}\left({\mathit{B}}_{11}-1\right)}{{\mathit{A}}_{11}{\mathit{B}}_{11}-1},\frac{{\mathit{M}}_{\mathit{B}}{\mathit{B}}_{11}\left({\mathit{A}}_{11}-1\right)}{{\mathit{A}}_{11}{\mathit{B}}_{11}-1}\right)$ | $\mathbf{sgn}\left({\mathit{A}}_{11}{\mathit{B}}_{11}-1\right)=\mathbf{sgn}\left({\mathit{A}}_{11}-1\right)=\mathbf{sgn}\left({\mathit{B}}_{11}-1\right)=+1$ |

2 | ${E}_{4}\left(\begin{array}{l}\frac{{M}_{A}\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)}{\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1},\\ \frac{{M}_{B}\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)}{\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1}\end{array}\right)$ | $\begin{array}{l}\mathrm{sgn}\left(\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)\\ =\mathrm{sgn}\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)-1\right)\\ =\mathrm{sgn}\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)-1\right)=+1\end{array}$ |

3 | ${E}_{4}\left(\begin{array}{l}\frac{{M}_{A}\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)}{\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1},\\ \frac{{M}_{B}\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)}{\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1}\end{array}\right)$ | $\begin{array}{l}\mathrm{sgn}\left(\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)\\ =\mathrm{sgn}\left(\left(\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)\right)\\ =\mathrm{sgn}\left(\left(\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)\right)=+1\end{array}$ |

Equilibrium | $\mathbf{det}\mathit{J}$ | $\mathit{t}\mathit{r}\mathit{J}$ | Stability Condition |
---|---|---|---|

${E}_{4}\left(\frac{{M}_{A}\overline{{R}_{A}}\left(\overline{{R}_{B}}+1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1},\frac{{M}_{B}\overline{{R}_{B}}\left(\overline{{R}_{A}}+1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1}\right)$ | $\frac{{r}_{A}{r}_{A}\left(\overline{{R}_{A}}+1\right)\left(\overline{{R}_{B}}+1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1}$ | $\frac{\overline{{R}_{A}}{r}_{A}\left(\overline{{R}_{B}}+1\right)+\overline{{R}_{B}}{r}_{B}\left(\overline{{R}_{A}}+1\right)}{\overline{{R}_{A}}\overline{{R}_{B}}-1}$ | $\mathrm{sgn}\left(\overline{{R}_{A}}\overline{{R}_{B}}-1\right)$$=+1$ |

Case | Equilibrium | Stability Condition |
---|---|---|

1 | ${\mathit{E}}_{4}\left(\frac{{\mathit{M}}_{\mathit{A}}{\mathit{A}}_{11}\left({\mathit{B}}_{11}+1\right)}{{\mathit{A}}_{11}{\mathit{B}}_{11}-1},\frac{{\mathit{M}}_{\mathit{B}}{\mathit{B}}_{11}\left({\mathit{A}}_{11}+1\right)}{{\mathit{A}}_{11}{\mathit{B}}_{11}-1}\right)$ | $\mathbf{sgn}\left({\mathit{A}}_{11}{\mathit{B}}_{11}-1\right)$$=+1$ |

2 | ${E}_{4}\left(\begin{array}{l}\frac{{M}_{A}\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)+1\right)}{\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1},\\ \frac{{M}_{B}\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)+1\right)}{\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1}\end{array}\right)$ | $\begin{array}{l}\mathrm{sgn}\left({A}_{11}+\left({A}_{22}-{A}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{11}+\left({B}_{22}-{B}_{11}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\\ =+1\end{array}$ |

3 | ${E}_{4}\left(\begin{array}{l}\frac{{M}_{A}\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)+1\right)}{\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1},\\ \frac{{M}_{B}\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left(\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)+1\right)}{\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1}\end{array}\right)$ | $\begin{array}{l}\mathrm{sgn}\left(\left({A}_{21}+\left({A}_{12}-{A}_{21}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)\left({B}_{12}+\left({B}_{21}-{B}_{12}\right){\mathrm{sin}}^{2}\left(\gamma \right)\right)-1\right)\\ =+1\end{array}$ |

Variable in Theoretical Model | Corresponding Variable in Simulation Model | Initial Value | Variable in Theoretical Model | Corresponding Variable in Simulation Model | Initial Value |
---|---|---|---|---|---|

${\mathit{A}}_{11}$ | A11 | 1.5 | ${\mathit{M}}_{\mathit{A}}$ | M_{a} | 50 |

${\mathit{A}}_{12}$ | A12 | 0.8 | ${\mathit{x}}_{\mathit{B}}$ | x_{B} | 10 |

${\mathit{A}}_{21}$ | A21 | 1.8 | ${\mathit{r}}_{\mathit{B}}$ | r_{B} | 0.8 |

${\mathit{A}}_{22}$ | A22 | 1.3 | ${\mathit{M}}_{\mathit{B}}$ | M_{b} | 50 |

${\mathit{B}}_{11}$ | B11 | 1.4 | ${\mathit{\theta}}_{\mathit{A}}$ | c_{a} | 180 |

${\mathit{B}}_{12}$ | B12 | 1.7 | ${\mathit{\theta}}_{\mathit{B}}$ | c_{b} | 0 |

${\mathit{B}}_{21}$ | B21 | 0.7 | ${\mathit{\alpha}}_{\mathit{A}}$ | alf_{a} | 0 |

${\mathit{B}}_{22}$ | B22 | 1.2 | ${\mathit{\alpha}}_{\mathit{B}}$ | alf_{b} | 90 |

${\mathit{x}}_{\mathit{A}}$ | xA | 10 | $\mathit{\gamma}$ | GA | 40° |

${\mathit{r}}_{\mathit{A}}$ | rA | 0.5 |

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**MDPI and ACS Style**

Huang, D.; Delang, C.O.; Wu, Y.; Li, S. An Improved Lotka–Volterra Model Using Quantum Game Theory. *Mathematics* **2021**, *9*, 2217.
https://doi.org/10.3390/math9182217

**AMA Style**

Huang D, Delang CO, Wu Y, Li S. An Improved Lotka–Volterra Model Using Quantum Game Theory. *Mathematics*. 2021; 9(18):2217.
https://doi.org/10.3390/math9182217

**Chicago/Turabian Style**

Huang, Dingxuan, Claudio O. Delang, Yongjiao Wu, and Shuliang Li. 2021. "An Improved Lotka–Volterra Model Using Quantum Game Theory" *Mathematics* 9, no. 18: 2217.
https://doi.org/10.3390/math9182217