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

Published online by Cambridge University Press:  25 November 2022

P. Broadbridge*
School of Engineering and Mathematical Sciences and Institute of Mathematics for Industry-Kyushu University, La Trobe University, Bundoora, VIC 3086, Australia
R. M. Cherniha
Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 01004 Kyiv, Ukraine
J. M. Goard
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
*Correspondence author. Email:


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.

© The Author(s), 2022. Published by Cambridge University Press

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