Abstract: We considered the coexistence solution problem of a class of Lotka-Volterra competitive systems with nonlinear cross-diffusion under homogeneous Dirichlet boundary conditions. Firstly, the stability of trivial solutions and semi-trivial solutions of the problem was analyzed by using spectral theory of linear operators. Secondly,  sufficient conditions for the existence of coexistence solutions of the problem were given by using the fixed point index theory on positive cone.

Key words: competition model, cross-diffusion, stability, fixed point index, coexistence solution