Convergence of the finite difference scheme for a general class of the spatial segregation of reaction-diffusion systems

In this work we prove convergence of the finite difference scheme for equations of stationary states of a general class of the spatial segregation of reaction-diffusion systems with $m\ge 2$ components. More precisely, we show that the numerical solution $u_h^l$, given by the difference scheme, converges to the $l^{th}$ component $u_l,$ when the mesh size $h$ tends to zero, provided $u_l\in C^2(\mathrm{\Omega }),$ for every $l=1,2,\dots ,m\mathrm{.}$ In particular, our proof provides convergence of a difference scheme for the multi-phase obstacle problem.