Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities

Q -conditional symmetries (nonclassical symmetries) for the general class of two-component reaction–diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction–diffusion systems admitting such symmetry is derived. The results obtained for the reaction–diffusion systems are compared with those for the scalar reaction–diffusion equations. The symmetries found for reducing reaction–diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction–diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.