New exact solutions of a nonlinear cross-diffusion system

A system of two reaction-diffusion equations with cross-diffusion and quadratic nonlinearities is considered. The system is a particular case, the well-known model proposed by Shigesada et al (1979 J. Theor. Biol. 79 83). New non-Lie ansaetze (special-type substitutions) reducing the system to systems of first-order ordinary differential equations are obtained and applied to find its exact solutions. Several families of exact solutions are constructed, their asymptotic behavior is investigated and some biological interpretation of the solutions is provided. Lie symmetry of this system is also discussed