# Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients

## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Theorem**

**1.**

**Sketch**

**of**

**the**

**proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Sketch**

**of**

**the**

**proof**

**of**

**Theorem**

**3.**

**Remark**

**1.**

## 3. Exact Solutions of the DLV System

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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d | Reaction Terms | Basic Operators of Maximal Algebra of Invariance | |
---|---|---|---|

1. | $d\left(t\right)$ | ${b}_{1}\left(t\right){u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}v{\partial}_{v}$ |

${b}_{2}\left(t\right)uv$ | |||

2. | $d\left(t\right)$ | 0 | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}{\partial}_{u}+\int b\left(t\right)dt\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}$ |

$b\left(t\right)uv$ | |||

3. | ${d}_{0}$ | $u\left({\beta}_{1}{t}^{k}u+{\gamma}_{1}{t}^{l}v\right)$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2(k+1)u{\partial}_{u}-2(l+1)v{\partial}_{v}$ |

$v\left({\beta}_{2}{t}^{k}u+{\gamma}_{2}{t}^{l}v\right)$ | |||

4. | ${d}_{0}$ | $\beta {t}^{k}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2(k+1)u{\partial}_{u},\phantom{\rule{4pt}{0ex}}v{\partial}_{v}$ |

${t}^{k}uv$ | |||

5. | ${d}_{0}$ | 0 | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2(k+1)u{\partial}_{u},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}{\partial}_{u}+\int {t}^{k}dt\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}$ |

${t}^{k}uv$ | |||

6. | ${d}_{0}$ | 0 | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2\left((k+1)u+\beta {(k+1)}^{2}\right){\partial}_{u}-$ |

$v\left({t}^{k}u+{t}^{l}\mathrm{exp}\left(\beta {t}^{k+1}\right)v\right)$ | $2\left(l+1+\beta (k+1){t}^{k+1}\right)v{\partial}_{v}$ | ||

7. | ${d}_{0}$ | 0 | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-4\beta {\partial}_{u}-2\left(l+1+2\beta \mathrm{ln}t\right)v{\partial}_{v}$ |

$v\left({t}^{-1}u+{t}^{l}\mathrm{exp}\left(\beta {\mathrm{ln}}^{2}t\right)v\right)$ | |||

8. | ${d}_{0}$ | ${t}^{-1}{cos}^{-2}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2k(k+2\mathrm{tan}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}-$ |

$v\left({\displaystyle \frac{\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}}{{cos}^{2}\left(k\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}u+{\displaystyle \frac{{t}^{l}}{{cos}^{\beta}\left(k\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}v\right)$ | $2\left(l+1+k\beta \mathrm{tan}\left(k\mathrm{ln}t\right)\right)v{\partial}_{v}$ | ||

9. | ${d}_{0}$ | ${t}^{-1}{cos}^{-2}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2k(k+2\mathrm{tan}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}-$ |

$\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}{cos}^{-2}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}uv$ | $2k\beta \mathrm{tan}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}$ | ||

10. | ${d}_{0}$ | ${t}^{-1}{\mathrm{cosh}}^{-2}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2k(k+2\mathrm{tanh}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}-$ |

$v\left({\displaystyle \frac{\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}}{{\mathrm{cosh}}^{2}\left(k\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}u+\frac{{t}^{l}}{{\mathrm{cosh}}^{\beta}\left(k\mathrm{ln}t\right)}\phantom{\rule{0.166667em}{0ex}}v\right)$ | $2\left(l+1-k\beta \mathrm{tanh}\left(k\mathrm{ln}t\right)\right)v{\partial}_{v}$ | ||

11. | ${d}_{0}$ | ${t}^{-1}{\mathrm{cosh}}^{-2}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2k(k+2\mathrm{tanh}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}+$ |

$\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}{\mathrm{cosh}}^{-2}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}uv$ | $2k\beta \mathrm{tanh}\left(k\mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}$ | ||

12. | ${d}_{0}$ | ${t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2(-1+2{\mathrm{ln}}^{-1}t\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}+$ |

$v\left(\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}u+{t}^{l}{\mathrm{ln}}^{-\beta}t\phantom{\rule{0.166667em}{0ex}}v\right)$ | $2\left(\beta {\mathrm{ln}}^{-1}t-l-1\right)v{\partial}_{v}$ | ||

13. | ${d}_{0}$ | ${t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+$ |

$\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}uv$ | $2(-1+2{\mathrm{ln}}^{-1}t\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}+2\beta {\mathrm{ln}}^{-1}t\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}$ |

Reaction Terms | Basic Operators of Maximal Algebra of Invariance | |
---|---|---|

1. | $u\left({t}^{k}\mathrm{ln}t\phantom{\rule{0.166667em}{0ex}}u+{t}^{k}\phantom{\rule{0.166667em}{0ex}}v\right)$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2(k+1)u{\partial}_{u}-2\left(u+(k+1)v\right){\partial}_{v}$ |

$v\left({t}^{k}\mathrm{ln}t\phantom{\rule{0.166667em}{0ex}}u+{t}^{k}\phantom{\rule{0.166667em}{0ex}}v\right)$ | ||

2. | $u\left({\displaystyle \frac{{t}^{k-1}(1-k\mathrm{ln}t)}{{\mathrm{ln}}^{2}t}}\phantom{\rule{0.166667em}{0ex}}u+{t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}v\right)$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2\left({\mathrm{ln}}^{-1}t-k\right)u{\partial}_{u}+2\frac{2v+{t}^{k}u-\mathrm{ln}t}{\mathrm{ln}t}{\partial}_{v}$ |

$v\left({\displaystyle \frac{{t}^{k-1}(1-k\mathrm{ln}t)}{{\mathrm{ln}}^{2}t}}\phantom{\rule{0.166667em}{0ex}}u+{t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}v\right)$ | ||

3. | $u\left({\displaystyle \frac{{t}^{k-1}\left(k+l\mathrm{tan}\left(l\mathrm{ln}t\right)\right)}{cos\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}u+{\displaystyle \frac{{t}^{-1}}{{cos}^{2}\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}v\right)$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2\left(l\mathrm{tan}\left(l\mathrm{ln}t\right)+k\right)u{\partial}_{u}-$ |

$v\left({\displaystyle \frac{{t}^{k-1}\left(k+l\mathrm{tan}\left(l\mathrm{ln}t\right)\right)}{cos\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}u+{\displaystyle \frac{{t}^{-1}}{{cos}^{2}\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}v\right)$ | $2\left(2l\mathrm{tan}\left(l\mathrm{ln}t\right)v+{\displaystyle \frac{{l}^{2}{t}^{k}}{cos\left(l\mathrm{ln}t\right)}}u+{l}^{2}\right){\partial}_{v}$ | |

4. | $u\left({\displaystyle \frac{{t}^{k-1}\left(k-l\mathrm{tanh}\left(l\mathrm{ln}t\right)\right)}{\mathrm{cosh}\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}u+{\displaystyle \frac{{t}^{-1}}{{\mathrm{cosh}}^{2}\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}v\right)$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2\left(l\mathrm{tanh}\left(l\mathrm{ln}t\right)-k\right)u{\partial}_{u}+$ |

$v\left({\displaystyle \frac{{t}^{k-1}\left(k-l\mathrm{tanh}\left(l\mathrm{ln}t\right)\right)}{\mathrm{cosh}\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}u+{\displaystyle \frac{{t}^{-1}}{{\mathrm{cosh}}^{2}\left(l\mathrm{ln}t\right)}}\phantom{\rule{0.166667em}{0ex}}v\right)$ | $2\left(2l\mathrm{tanh}\left(l\mathrm{ln}t\right)v+{\displaystyle \frac{{l}^{2}{t}^{k}}{\mathrm{cosh}\left(l\mathrm{ln}t\right)}}u+{l}^{2}\right){\partial}_{v}$ | |

5. | $b\left(t\right){u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}u{\partial}_{v},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}(1+\int b\left(t\right)dt\phantom{\rule{0.166667em}{0ex}}u){\partial}_{v}$ |

$b\left(t\right)uv$ | ||

6. | ${t}^{k}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}u{\partial}_{v},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}(1+\int {t}^{k}dt\phantom{\rule{0.166667em}{0ex}}u){\partial}_{v},$ |

${t}^{k}uv$ | $2t{\partial}_{t}+x{\partial}_{x}-2(k+1)u{\partial}_{u}$ | |

7. | ${t}^{-1}{cos}^{-2}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}u{\partial}_{v},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}(\beta +\mathrm{tan}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{v},$ |

${t}^{-1}{cos}^{-2}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}uv$ | $2t{\partial}_{t}+x{\partial}_{x}-2\beta \mathrm{tan}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}-$ | |

$2({\beta}^{2}+2\beta \mathrm{tan}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}$ | ||

8. | ${t}^{-1}{\mathrm{cosh}}^{-2}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}u{\partial}_{v},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}(\beta +\mathrm{tanh}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{v},$ |

${t}^{-1}{\mathrm{cosh}}^{-2}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}uv$ | $2t{\partial}_{t}+x{\partial}_{x}+2\beta \mathrm{tanh}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}+$ | |

$2({\beta}^{2}+2\beta \mathrm{tanh}\left(\beta \mathrm{ln}t\right)\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}$ | ||

9. | ${t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}{u}^{2}$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}u{\partial}_{v},\phantom{\rule{4pt}{0ex}}v{\partial}_{v},\phantom{\rule{4pt}{0ex}}(1-{\mathrm{ln}}^{-1}t\phantom{\rule{0.166667em}{0ex}}u){\partial}_{v},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+$ |

${t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}uv$ | $2(-1+2{\mathrm{ln}}^{-1}t\phantom{\rule{0.166667em}{0ex}}u){\partial}_{u}+2{\mathrm{ln}}^{-1}t\phantom{\rule{0.166667em}{0ex}}v{\partial}_{v}$ |

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Davydovych, V.
Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients. *Symmetry* **2018**, *10*, 41.
https://doi.org/10.3390/sym10020041

**AMA Style**

Davydovych V.
Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients. *Symmetry*. 2018; 10(2):41.
https://doi.org/10.3390/sym10020041

**Chicago/Turabian Style**

Davydovych, Vasyl’.
2018. "Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients" *Symmetry* 10, no. 2: 41.
https://doi.org/10.3390/sym10020041