### Abstract

We propose an approximate analytical approach to a (1 + 1) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher-Kolmogorov-Petrovskii- Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel-Kramers-Brillouin (WKB)-Maslov semiclassical approximation is applied to the generalized nonlocal Fisher-KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

Original language | English |
---|---|

Article number | 366 |

Journal | Symmetry |

Volume | 11 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Mar 2019 |

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### Keywords

- Nonlocal Fisher-KPP model
- Perturbation method
- Reaction-diffusion
- Released activity
- Semiclassical approximation
- Symmetries

### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- Mathematics(all)
- Physics and Astronomy (miscellaneous)

### Cite this

**Approximate solutions and symmetry of a two-component nonlocal reaction-diffusion population model of the Fisher-KPP type.** / Shapovalov, Alexander V.; Trifonov, Andrey Yu.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Approximate solutions and symmetry of a two-component nonlocal reaction-diffusion population model of the Fisher-KPP type

AU - Shapovalov, Alexander V.

AU - Trifonov, Andrey Yu

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We propose an approximate analytical approach to a (1 + 1) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher-Kolmogorov-Petrovskii- Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel-Kramers-Brillouin (WKB)-Maslov semiclassical approximation is applied to the generalized nonlocal Fisher-KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

AB - We propose an approximate analytical approach to a (1 + 1) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher-Kolmogorov-Petrovskii- Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel-Kramers-Brillouin (WKB)-Maslov semiclassical approximation is applied to the generalized nonlocal Fisher-KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature.

KW - Nonlocal Fisher-KPP model

KW - Perturbation method

KW - Reaction-diffusion

KW - Released activity

KW - Semiclassical approximation

KW - Symmetries

UR - http://www.scopus.com/inward/record.url?scp=85067278873&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067278873&partnerID=8YFLogxK

U2 - 10.3390/sym11030366

DO - 10.3390/sym11030366

M3 - Article

VL - 11

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 3

M1 - 366

ER -