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

TL;DR

This paper constructs and analyzes Q-conditional symmetries of a three-component reaction-diffusion model, leading to new exact solutions with biological interpretations, distinct from classical Lie symmetries.

Contribution

It introduces the first explicit construction of Q-conditional symmetries for a known reaction-diffusion system modeling ecological interactions.

Findings

Found a wide variety of Q-conditional symmetries not equivalent to Lie symmetries.

Derived exact solutions using these symmetries.

Analyzed the asymptotic behavior and biological implications of solutions.

Abstract

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