Lie symmetry classification and exact solutions of a diffusive Lotka-Volterra system with convection

TL;DR

This paper applies Lie symmetry analysis to a reduced diffusive Lotka-Volterra system with convection, classifying symmetries and deriving exact solutions, including radially symmetric and Weierstrass function solutions, relevant for modeling viscous fingering in chemical reactions.

Contribution

It provides a complete Lie symmetry classification of the system and constructs new exact solutions, enhancing understanding of the model's symmetry properties and solution space.

Findings

Widest Lie algebras occur with linear stream functions

Exact solutions include radially symmetric and Weierstrass function solutions

Some solutions illustrate spatiotemporal concentration evolution

Abstract

A mathematical model for description of the viscous fingering induced by a chemical reaction is under study. This complicated five-component model is reduced to a three-component diffusive Lotka-Volterra system with convection by introducing a stream function. The system obtained is examined by the classical Lie method. A complete Lie symmetry classification is derived via a rigorous algorithm. In particular, it is proved that the widest Lie algebras of invariance occur when the stream function generate a linear velocity field. The most interesting cases (from the symmetry and applicability point of view) are further studied in order to derive exact solutions. A wide range of exact solutions are constructed for radially-symmetric stream functions. These solutions include time-dependent and radially symmetric solutions as well as more complicated solutions expressed in terms of the Weierstrass function. It was shown that some of exact solutions can be used for demonstration of spatiotemporal evolution of concentrations corresponding to two reactants and their product.