# 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

- Volterra, V. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. Acad. Lincei
**1926**, 2, 31–113. [Google Scholar] - Britton, N.F. Essential Mathematical Biology; Springer: Berlin, Germany, 2003. [Google Scholar]
- Lotka, A.J. Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc.
**1920**, 42, 1595–1599. [Google Scholar] [CrossRef] - Murray, J.D. Mathematical Biology II; Springer: Berlin, Germany, 2003. [Google Scholar]
- Cherniha, R.; Davydovych, V. Nonlinear Reaction-Diffusion Systems—Conditional Symmetry, Exact Solutions and Their Applications in Biology; Lecture Notes in Mathematics; Springer: Berlin, Germany, 2017; ISBN 978-3-319-65465-2. [Google Scholar]
- Hou, X.; Leung, A.W. Traveling wave solutions for a competitive reaction diffusion system and their asymptotics. Nonlinear Anal. Real World Appl.
**2008**, 9, 2196–2213. [Google Scholar] [CrossRef] - Leung, A.W.; Hou, X.; Feng, W. Traveling wave solutions for Lotka–Volterra system re-visited. Discret. Contin. Dyn. Syst.
**2011**, 15, 171–196. [Google Scholar] [CrossRef] - Cherniha, R.; Dutka, V. A diffusive Lotka–Volterra system: Lie symmetries, exact and numerical solutions. Ukr. Math. J.
**2004**, 56, 1665–1675. [Google Scholar] [CrossRef] - Rodrigo, M.; Mimura, M. Exact solutions of a competition-diffusion system. Hiroshima Math. J.
**2000**, 30, 257–270. [Google Scholar] - Hung, L.-C. Exact traveling wave solutions for diffusive Lotka Volterra systems of two competing species. Jpn. J. Ind. Appl. Math.
**2012**, 29, 237–251. [Google Scholar] [CrossRef] - Cherniha, R.; Davydovych, V. Conditional symmetries and exact solutions of the diffusive Lotka Volterra system. Math. Comput. Model.
**2011**, 54, 1238–1251. [Google Scholar] [CrossRef] - Cherniha, R. Lie symmetries of nonlinear two-dimensional reaction-diffusion Systems. Rep. Math. Phys.
**2000**, 46, 63–76. [Google Scholar] [CrossRef] - Hou, Z.; Lisena, B.; Pireddu, M.; Zanolin, F. Lotka–Volterra and Related Systems: Recent Developments in Population Dynamics; Walter de Gruyter: Berlin, Germany, 2013; ISBN 978-3-11-026951-2. [Google Scholar]
- Bao, X.X.; Li, W.T.; Wang, Z.C. Time periodic traveling curved fronts in the periodic Lotka–Volterra competition-diffusion system. J. Dyn. Differ. Equ.
**2017**, 29, 981–1016. [Google Scholar] [CrossRef] - Hetzer, G.; Shen, W. Convergence in almost periodic competition diffusion systems. J. Math. Anal. Appl.
**2001**, 262, 307–338. [Google Scholar] [CrossRef] - Hutson, V.; Mischaikow, K.; Poláčik, P. The evolution of dispersal rates in a heterogeneous time-periodic environment. J. Math. Biol.
**2001**, 43, 501–533. [Google Scholar] [CrossRef] [PubMed] - Struk, O.O.; Tkachenko, V.I. On impulsive Lotka–Volterra systems with diffusion. Ukr. Math. J.
**2002**, 54, 629–646. [Google Scholar] [CrossRef] - Ovsiannikov, L.V. The Group Analysis of Differential Equations; Academic: New York, NY, USA, 1982. [Google Scholar]
- Ovsiannikov, L.V. Group relations of the equation of non-linear heat conductivity. Dokl. Akad. Nauk SSSR
**1959**, 125, 125492–125495. [Google Scholar] - Zhdanov, R.Z.; Lahno, V.I. Group classification of heat conductivity equations with a nonlinear source. J. Phys. A Math. Gen.
**1999**, 32, 7405. [Google Scholar] [CrossRef] - Kingston, J.G. On point transformations of evolution equations. J. Phys. A Math. Gen.
**1991**, 24, L769–L774. [Google Scholar] [CrossRef] - Kingston, J.G.; Sophocleous, C. On form-preserving point transformations of partial differential equations. J. Phys. A Math. Gen.
**1998**, 31, 1597–1619. [Google Scholar] [CrossRef] - Gazeau, J.P.; Winternitz, P. Symmetries of variable coefficient Korteweg-de Vries equations. J. Math. Phys.
**1992**, 33, 4087–4102. [Google Scholar] [CrossRef] - Cherniha, R.; Davydovych, V.; King, J.R. Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model. ArXiv, 2017; arXiv:1704.07696. [Google Scholar]
- Cherniha, R.; King, J.R. Non-linear reaction diffusion systems with variable diffusivities: Lie symmetries, ansatze and exact solutions. J. Math. Anal. Appl.
**2005**, 308, 11–35. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M.; Rassokha, I. Lie symmetries and form preserving transformations of reaction-diffusion-convection equations. J. Math. Anal. Appl.
**2008**, 342, 1363–1379. [Google Scholar] [CrossRef] - Vaneeva, O.O.; Popovych, R.O.; Sophocleous, C. Enhanced group analysis and exact solutions of variable coefficient semilinear diffusion equations with a power source. Acta Appl. Math.
**2009**, 106, 1–46. [Google Scholar] [CrossRef] - Cherniha, R.; Serov, M.; Pliukhin, O. Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications; CRC Press: Boca Raton, FL, USA, 2018; ISBN 9781498776172. [Google Scholar]
- Arrigo, D.J. Symmetry Analysis of Differential Equations: An Introduction; John Wiley and Sons: Hoboken, NJ, USA, 2015; ISBN 9781118721445. [Google Scholar]
- Bluman, G.W.; Cheviakov, A.F.; Anco, S.C. Applications of Symmetry Methods to Partial Differential Equations; Springer: New York, NY, USA, 2010; ISBN 978-0-387-68028-6. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations; Springer: Berlin, Germany, 1986; ISBN 978-1-4684-0274-2. [Google Scholar]
- Nie, L.; Teng, Z.; Hu, L.; Peng, J. Permanence and stability in non-autonomous predator-prey Lotka–Volterra systems with feedback controls. Comput. Math. Appl.
**2009**, 58, 436–448. [Google Scholar] [CrossRef]

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

$v\left(\right)open="("\; close=")">{\beta}_{2}{t}^{k}u+{\gamma}_{2}{t}^{l}v$ | |||

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 | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2\left(\right)open="("\; close=")">(k+1)u+\beta {(k+1)}^{2}$ |

$v\left(\right)open="("\; close=")">{t}^{k}u+{t}^{l}\mathrm{exp}\left(\right)open="("\; close=")">\beta {t}^{k+1}$ | $2\left(\right)open="("\; close=")">l+1+\beta (k+1){t}^{k+1}$ | ||

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

$v\left(\right)open="("\; close=")">{t}^{-1}u+{t}^{l}\mathrm{exp}\left(\right)open="("\; close=")">\beta {\mathrm{ln}}^{2}t$ | |||

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

$v\left(\right)open="("\; close=")">{\displaystyle \frac{\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}}{{cos}^{2}\left(\right)open="("\; close=")">k\mathrm{ln}t}}\phantom{\rule{0.166667em}{0ex}}v$ | $2\left(\right)open="("\; close=")">l+1+k\beta \mathrm{tan}\left(k\mathrm{ln}t\right)$ | ||

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

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

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

$v\left(\right)open="("\; close=")">{\displaystyle \frac{\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}}{{\mathrm{cosh}}^{2}\left(\right)open="("\; close=")">k\mathrm{ln}t}}\phantom{\rule{0.166667em}{0ex}}v$ | $2\left(\right)open="("\; close=")">l+1-k\beta \mathrm{tanh}\left(k\mathrm{ln}t\right)$ | ||

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

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

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

$v\left(\right)open="("\; close=")">\beta \phantom{\rule{0.166667em}{0ex}}{t}^{-1}{\mathrm{ln}}^{-2}t\phantom{\rule{0.166667em}{0ex}}u+{t}^{l}{\mathrm{ln}}^{-\beta}t\phantom{\rule{0.166667em}{0ex}}v$ | $2\left(\right)open="("\; close=")">\beta {\mathrm{ln}}^{-1}t-l-1$ | ||

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

$v\left(\right)open="("\; close=")">{t}^{k}\mathrm{ln}t\phantom{\rule{0.166667em}{0ex}}u+{t}^{k}\phantom{\rule{0.166667em}{0ex}}v$ | ||

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

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

3. | $u\left(\right)open="("\; close=")">{\displaystyle \frac{{t}^{k-1}\left(\right)open="("\; close=")">k+l\mathrm{tan}\left(\right)open="("\; close=")">l\mathrm{ln}t}{}}cos\left(\right)open="("\; close=")">l\mathrm{ln}t$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2\left(\right)open="("\; close=")">l\mathrm{tan}\left(\right)open="("\; close=")">l\mathrm{ln}tu{\partial}_{u}-$ |

$v\left(\right)open="("\; close=")">{\displaystyle \frac{{t}^{k-1}\left(\right)open="("\; close=")">k+l\mathrm{tan}\left(\right)open="("\; close=")">l\mathrm{ln}t}{}}cos\left(\right)open="("\; close=")">l\mathrm{ln}t$ | $2\left(\right)open="("\; close=")">2l\mathrm{tan}\left(\right)open="("\; close=")">l\mathrm{ln}tu+{l}^{2}$ | |

4. | $u\left(\right)open="("\; close=")">{\displaystyle \frac{{t}^{k-1}\left(\right)open="("\; close=")">k-l\mathrm{tanh}\left(\right)open="("\; close=")">l\mathrm{ln}t}{}}\mathrm{cosh}\left(\right)open="("\; close=")">l\mathrm{ln}t$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2\left(\right)open="("\; close=")">l\mathrm{tanh}\left(\right)open="("\; close=")">l\mathrm{ln}tu{\partial}_{u}+$ |

$v\left(\right)open="("\; close=")">{\displaystyle \frac{{t}^{k-1}\left(\right)open="("\; close=")">k-l\mathrm{tanh}\left(\right)open="("\; close=")">l\mathrm{ln}t}{}}\mathrm{cosh}\left(\right)open="("\; close=")">l\mathrm{ln}t$ | $2\left(\right)open="("\; close=")">2l\mathrm{tanh}\left(\right)open="("\; close=")">l\mathrm{ln}tu+{l}^{2}$ | |

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

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

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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