Abstract
The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of exact solutions is examined in detail in order to provide an application for describing the competition of two species in population dynamics. The results obtained are compared with those published earlier as well.
Highlights
Exact Solutions of the Diffusive Two-About 100 years ago, A
Volterra [2] independently developed a mathematical model, which nowadays serves as the mathematical background for population dynamics, ecology, chemical reactions, etc
Their model is based on a system of ordinary differential equations (ODEs) with quadratic nonlinearities
Summary
The most powerful methods for the construction of exact solutions for non-integrable nonlinear partial differential equations (PDEs) are symmetry-based methods. Following Fushchych’s proposal dating back to the 1980s [24,25], we use the terminology ‘Qconditional symmetry’ instead of ‘nonclassical symmetry’ (see a discussion concerning terminology in Chapter 3 of [22]) This method was suggested 50 years ago, its successful applications for solving nonlinear systems of PDEs were accomplished only in the 2000s, and the majority of such papers were published during the last 10 years (see [17,18,26,27,28,29,30]).
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