Lie and conditional symmetries of the three-component diffusive Lotka - Volterra system

Lie and Q-conditional symmetries of the classical three-component diffusive Lotka - Volterra system in the case of one space variable are studied. The group-classification problems for finding Lie symmetries and Q-conditional symmetries of the first type are completely solved. Notably, non-Lie symmetries (Q-conditional symmetry operators) for a multi-component non-linear reaction-diffusion system are constructed for the first time. An example of non-Lie symmetry reduction for solving a biologically motivated problem is presented.

💡 Research Summary

The paper conducts a comprehensive symmetry analysis of the classical three‑component diffusive Lotka‑Volterra (LV) system in one spatial dimension. The authors first formulate the system as a set of coupled reaction‑diffusion equations for the densities (u(t,x), v(t,x), w(t,x)) with diffusion coefficients (d_i) and interaction parameters (a_i, b_{ij}). They then perform a full Lie‑symmetry classification. By constructing the most general infinitesimal generator
(X = \tau(t,x,u,v,w)\partial_t + \xi(t,x,u,v,w)\partial_x + \phi^1\partial_u + \phi^2\partial_v + \phi^3\partial_w)
and substituting into the prolonged invariance conditions, they derive the determining equations. Solving these equations shows that, for arbitrary parameters, the only continuous Lie symmetries are the obvious translations in time and space and a uniform scaling of the dependent variables. No non‑trivial Lie symmetries exist unless the parameters satisfy very restrictive relations, confirming the typical rigidity of multi‑species reaction‑diffusion models.

The second, and more original, part of the work deals with Q‑conditional (also called non‑classical) symmetries of the first type. A Q‑conditional symmetry operator (Q) must satisfy not only the invariance of the differential equations but also the invariant surface conditions (Q