scholarly journals A hunter-gatherer–farmer population model: Lie symmetries, exact solutions and their interpretation

2018 ◽  
Vol 30 (2) ◽  
pp. 338-357 ◽  
Author(s):  
R. M. CHERNIHA ◽  
V. V. DAVYDOVYCH
Keyword(s):  
Exact Solutions ◽  
Population Model ◽  
Reaction Diffusion ◽  
Lie Symmetry ◽  
Lie Symmetries ◽  
System Modelling ◽  
Hunter Gatherers ◽  
Three Component Reaction ◽  
Biological Interpretation

The Lie symmetry classification of the known three-component reaction–diffusion system modelling the spread of an initially localized population of farmers into a region occupied by hunter-gatherers is derived. The Lie symmetries obtained for reducing the system in question to systems of ordinary differential equations (ODEs) and constructing exact solutions are applied. Several exact solutions of travelling front type are also found, their properties are identified and biological interpretation is discussed.

scholarly journals Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

2020 ◽  
pp. 1-21
Author(s):  
R. M. CHERNIHA ◽  
V. V. DAVYDOVYCH
Keyword(s):  
Asymptotic Behaviour ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Hunter Gatherers ◽  
Classical Symmetry ◽  
Classical Symmetries ◽  
Reaction Diffusion Model ◽  
Three Component Reaction ◽  
Biological Interpretation ◽  
First Time

Abstract Q-conditional (non-classical) symmetries of the known three-component reaction-diffusion (RD) system [K. Aoki et al. Theor. Popul. Biol. 50, 1–17 (1996)] modelling interaction between farmers and hunter-gatherers are constructed for the first time. A wide variety of Q-conditional symmetries are found, and it is shown that these symmetries are not equivalent to the Lie symmetries. Some operators of Q-conditional (non-classical) symmetry are applied for finding exact solutions of the RD system in question. Properties of the exact solutions (in particular, their asymptotic behaviour) are identified and possible biological interpretation is discussed.

scholarly journals Exact Solutions of a Mathematical Model Describing Competition and Co-Existence of Different Language Speakers

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 154 ◽  
Author(s):  
Roman Cherniha ◽  
Vasyl’ Davydovych
Keyword(s):  
Mathematical Model ◽  
Exact Solutions ◽  
System Modeling ◽  
Language Shift ◽  
Reaction Diffusion ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Traveling Fronts ◽  
Three Component Reaction ◽  
Symmetry Method

The known three-component reaction–diffusion system modeling competition and co-existence of different language speakers is under study. A modification of this system is proposed, which is examined by the Lie symmetry method; furthermore, exact solutions in the form of traveling fronts are constructed and their properties are identified. Plots of the traveling fronts are presented and the relevant interpretation describing the language shift that has occurred in Ukraine during the Soviet times is suggested.

scholarly journals Lie symmetries of generalized Kawahara equations

2020 ◽  
pp. 3-10
Author(s):  
O.O. Vaneeva ◽  
◽  
A.Yu. Zhalij ◽  
Keyword(s):  
Variable Coefficients ◽  
Lie Symmetry ◽  
Group Classification ◽  
Lie Symmetries

We carry out the group classification of a normalized class of generalized Kawahara equations with variable coefficients. Admissible transformations are studied, and the partition of the class into two normalized subclasses is performed. For each of these subclasses, the respective equivalence groupoids are found. As a result, all equations from the class admitting Lie symmetry extensions are revealed.

Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, II

2006 ◽  
Vol 17 (5) ◽  
pp. 597-605 ◽  
Author(s):  
ROMAN CHERNIHA ◽  
MYKOLA SEROV
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Applied Mathematics ◽  
Reaction Diffusion ◽  
Second Order ◽  
Lie Symmetries ◽  
Nonlinear Reaction ◽  
Diffusion Convection ◽  
Natural Way

New results concerning Lie symmetries of nonlinear reaction-diffusion-convection equations, which supplement in a natural way the results published in the European Journal of Applied Mathematics (9(1998) 527–542) are presented.

scholarly journals A reaction–diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modelling

2021 ◽  
pp. 1-18
Author(s):  
ROMAN M. CHERNIHA ◽  
VASYL V. DAVYDOVYCH
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Interesting Case ◽  
Lie Symmetry ◽  
Point Of View ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

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