Lie symmetries of the Shigesada–Kawasaki–Teramoto system

Abstract

The Shigesada–Kawasaki–Teramoto system, which consists of two reaction-diffusion equations with variable cross-diffusion and quadratic nonlinearities, is considered. The system is the most important case of the biologically motivated model proposed by Shigesada et al. (J. Theor. Biol. 79(1979) 83–99). A complete description of Lie symmetries for this system is derived. It is proved that the Shigesada–Kawasaki–Teramoto system admits a wide range of different Lie symmetries depending on coefficient values. In particular, the Lie symmetry operators with highly unusual structure are unveiled and applied for finding exact solutions of the relevant nonlinear system with cross-diffusion.