# Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type

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## Abstract

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## 1. Introduction

## 2. Model Equations and Perturbation Theory

## 3. Semiclassical Approximation for the Cauchy Problem

#### 3.1. Semiclassical Solution of the Non-Local Fisher–KPP Equation

#### 3.2. The First-Order Correction to the Perturbation Solution of the Non-Local Fisher–KPP Equation

## 4. Symmetry Operators

## 5. Example

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Murray, J.D. Mathematical Biology. I. An Introduction, 3rd ed.; Springer: New York, NY, USA, 2001; p. XXIII, 553. [Google Scholar]
- Fisher, R.A. The wave of advance of advantageous genes. Annu. Eugen.
**1937**, 7, 255–369. [Google Scholar] [CrossRef] - Kolmogorov, A.N.; Petrovskii, I.G.; Piskounov, N.S. A study of the diffusion equation with increase in the amount of substance and its application to a biology problem. Bull. Univ. Moscow Ser. Int. A
**1937**, 1, 1–16. [Google Scholar] - Fuentes, M.A.; Kuperman, M.N.; Kenkre, V.M. Nonlocal interaction effects on pattern formation in population dynamics. Phys. Rev. Lett.
**2003**, 91, 158104. [Google Scholar] [CrossRef] [PubMed] - d’Onofrio, A.; Gandolfi, A. Mathematical Oncology 2013. Modeling and Simulation in Science, Engineering and Technology; Springer: New York, NY, USA, 2014; p. 334. [Google Scholar]
- Levchenko, E.A.; Shapovalov, A.V.; Trifonov, A.Y. Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation. J. Phys. A Math. Theor.
**2014**, 47. [Google Scholar] [CrossRef] - Levchenko, E.A.; Shapovalov, A.V.; Trifonov, A.Y. Asymptotics semiclassically concentrated on curves for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation. J. Phys. A Math. Theor.
**2016**, 49. [Google Scholar] [CrossRef] - Shapovalov, A.V.; Obukhov, V.V. Some aspects of nonlinearity and self-organization in biosystems on examples of localized excitations in the DNA molecule and generalized Fisher–KPP model. Symmetry
**2018**, 10, 53. [Google Scholar] [CrossRef] - Bluman, G.W.; Cole, J.D. Similarity Methods for Differential Equations; Series Title: Applied Mathematical Sciences, V. 13; Springer: New York, NY, USA, 1974; p. IX, 333. [Google Scholar]
- Ovsyannikov, L.V. Group Analysis of Differential Equations; Academic Press: New York, USA, 1982; p. 432. [Google Scholar]
- Ibragimov, N.H. Transformation Groups Applied to Mathematical Physics; Mathematics and Its Applications; Soviet Series; D. Reidel Publishing: Dordrecht, The Netherlands, 1985; p. XV, 394. [Google Scholar]
- Olver, P.J. Applications of Lie Groups to Differential Equations; Series Title: Graduate Texts in Mathematics, V. 107; Springer: New York, NY, USA, 1993; p. XXVIII, 513. [Google Scholar]
- Akhatov, I.S.; Gazizov, R.K.; Ibragimov, N.H. Nonlocal symmetries: a heuristic approach. J. Sov. Math.
**1991**, 55, 1401–1450. [Google Scholar] [CrossRef] - Chetverikov, V.N.; Kudryavtsev, A.G. Modeling integro-differential equations and a method for computing their symmetries and conservation laws. Am. Math. Soc. Transl.
**1995**, 167, 1–22. [Google Scholar] - Zawistowski, Z.J. Symmetries of integro-differential equations. Proc. Inst. Mat. NAS Ukraine
**2002**, 43, 263–270. [Google Scholar] [CrossRef] - Meleshko, S.V. Methods for Constructing Exact Solutions of Partial Differential Equations: Mathematical and Analytical Techniques with Applications to Engineering; Springer: New York, NY, USA, 2005. [Google Scholar]
- Fushchych, W.; Shtelen, W. On nonlocal transformations. Lett. Nuovo Cim.
**1985**, 44, 40–42. [Google Scholar] [CrossRef] - Levchenko, E.A.; Shapovalov, A.V.; Trifonov, A.Y. Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation. J. Math. Anal. Appl.
**2012**, 395, 716–726. [Google Scholar] [CrossRef] [Green Version] - Maslov, V.P. Operational Methods; MIR Pub.: Moscow, Russia, 1976; p. 559. [Google Scholar]
- Maslov, V.P. The Complex WKB Method for Nonlinear Equations. I. Linear Theory; Series: Progress in Mathematical Physics (Book 16); Birkhäuser: Basel, Switzerland, 1994; p. 304. [Google Scholar]
- Belov, V.V.; Dobrokhotov, S.Y. Semiclassical Maslov asymptotics with complex phases. I. General appoach. Teor. Mat. Fiz.
**1992**, 130, 215–254. [Google Scholar] - Trifonov, A.Y.; Shapovalov, A.V. The one-dimensional Fisher–Kolmogorov equation with a nonlocal nonlinearity in a semiclassical approximetion. Russ. Phys. J.
**2009**, 52, 899–911. [Google Scholar] [CrossRef] - Shapovalov, A.V.; Trifonov, A.Y. An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation. Int. J. Geom. Methods Mod. Phys.
**2018**, 15, 1850102. [Google Scholar] [CrossRef] [Green Version] - Shapovalov, A.V.; Obukhov, V.V. Influence of the environment on pattern formation in the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii-Piskunov model. Russ. Phys. J.
**2018**, 6, 1093–1099. [Google Scholar] [CrossRef] - Tarasov, S.A.; Zarubaev, V.V.; Gorbunov, E.A.; Sergeeva, S.A.; Epstein, O.I. Activity of ultra-low doses of antibodies to gamma-interferon against lethal influenza A (H1N1) 2009 virus infection in mice. Antivir. Res.
**2012**, 93, 219–224. [Google Scholar] [CrossRef] [PubMed] - Nicoll, J.; Gorbunov, E.A.; Tarasov, S.A.; Epstein, O.I. Subetta treatment increases adiponectin secretion by mature human adipocytes in vitro. Int. J. Endocrinol.
**2013**, 2013, 925874. [Google Scholar] [CrossRef] - Epstein, O. The spatial homeostasis hypothesis. Symmetry
**2018**, 10, 103. [Google Scholar] [CrossRef] - Lisok, A.L.; Shapovalov, A.V.; Trifonov, A.Y. Symmetry and Intertwining Operators for the Nonlocal Gross–Pitaevskii Equation. Symmetry Integr. Geom. Methods Appl.
**2013**, 9, 1–21. [Google Scholar] [CrossRef] - Levchenko, E.A.; Trifonov, A.Y.; Shapovalov, A.V. Symmetry operators of the nonlocal Fisher–Kolmogorov– Petrovskii–Piskunov equation with a quadratic operator. Russ. Phys. J.
**2014**, 56, 1415–1426. [Google Scholar] [CrossRef] - Haas, F.; Eliasson, B. Time-dependent variational approach for Bose–Einstein condensates with nonlocal interaction. J. Phys. B At. Mol. Opt. Phys.
**2018**, 51, 175302. [Google Scholar] [CrossRef] - Bamba, K.; Capozziello, S.; Nojiri, S.; Odintsov, S.D. Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests. Astrophys. Space Sci.
**2012**, 342, 155–228. [Google Scholar] [CrossRef] - Simbawa, E. Mechanistic model for cancer growth and response to chemotherapy. Comput. Math. Methods Med.
**2017**, 2017. [Google Scholar] [CrossRef] - Bagrov, V.G.; Belov, V.V.; Trifonov, A.Y. Semiclassical trajectory-coherent approximation in quantum mechanics: I. High order corrections to multidimensional time-dependent equations of Schrödinger type. Ann. Phys.
**1996**, 246, 231–280. [Google Scholar] [CrossRef] - Belov, V.V.; Trifonov, A.Y.; Shapovalov, A.V. The trajectory-coherent approximation and the system of moments for the Hartree type equation. Int. J. Math. Math. Sci.
**2002**, 32, 325–370. [Google Scholar] [CrossRef] [Green Version] - Tikhonov, A.N.; Samarskii, A.A. Equations of Mathematical Physics; Pergamon Press: Oxford, UK, 1963; p. 765. [Google Scholar]
- Miller, W. Symmetry and Separation of Variables; Cambridge University Press: Cambridge, UK, 1984; p. 318. [Google Scholar]
- Patera, J.; Sharp, R.T.; Winternitz, P.; Zassenhaus, H. Invariants of real low dimension Lie algebras. J. Math. Phys.
**1976**, 6, 986–994. [Google Scholar] [CrossRef] - Konyaev, A.Y. Classification of Lie algebras with generic orbits of dimension 2 in the coadjoint representation. Sb. Math.
**2014**, 205, 45–62. [Google Scholar] [CrossRef] - Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach Science Publishers: Langhorne, PA, USA, 1993; p. xxxvi, 976. [Google Scholar]

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**MDPI and ACS Style**

Shapovalov, A.V.; Trifonov, A.Y.
Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type. *Symmetry* **2019**, *11*, 366.
https://doi.org/10.3390/sym11030366

**AMA Style**

Shapovalov AV, Trifonov AY.
Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type. *Symmetry*. 2019; 11(3):366.
https://doi.org/10.3390/sym11030366

**Chicago/Turabian Style**

Shapovalov, Alexander V., and Andrey Yu. Trifonov.
2019. "Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type" *Symmetry* 11, no. 3: 366.
https://doi.org/10.3390/sym11030366