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Exact nonclassical symmetry solutions of Lotka–Volterra-type population systems

Part of: Parabolic equations and systems Partial differential equations Representations of solutions

Published online by Cambridge University Press:  25 November 2022

P. Broadbridge Open the ORCID record for P. Broadbridge [Opens in a new window] ,
R. M. Cherniha  and
J. M. Goard
P. Broadbridge*
Affiliation:
School of Engineering and Mathematical Sciences and Institute of Mathematics for Industry-Kyushu University, La Trobe University, Bundoora, VIC 3086, Australia
R. M. Cherniha
Affiliation:
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 01004 Kyiv, Ukraine
J. M. Goard
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
*
*Correspondence author. Email: P.Broadbridge@latrobe.edu.au
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Abstract

New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

Keywords

Nonlinear reaction–diffusion systemLotka–Volterrapredator–preynonclassical symmetryintegrabilityexact solution

MSC classification

Primary: 35K40: Second-order parabolic systems
Secondary: 92D25: Population dynamics (general) 35K57: Reaction-diffusion equations 35C05: Solutions in closed form 35B06: Symmetries, invariants, etc.
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Special Issue 5: Symmetries and Differential Equations , October 2023 , pp. 998 - 1016
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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