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 nonlinear reaction–diffusion system are constructed for the first time. The results are compared with those derived for the two-component diffusive Lotka–Volterra system. The conditional symmetry obtained for the non-Lie reduction of the three-component system used for modeling competition between three species in population dynamics is applied and the relevant exact solutions are found. Particularly, the exact solution describing different scenarios of competition between three species is constructed. (paper)