Two scales hydrodynamic limit for a model of malignant tumor cells

We consider a model introduced in S. Luckhaus, L. Triolo, The continuum reaction-diffusion limit of a stochastic cellular growth model, Rend. Acc. Lincei (S.9) 15 (2004) 215-223] with two species (η and ξ) of particles, representing respectively malignant and normal cells. The basic motions of the η particles are independent random walks, scaled diffusively. The ξ particles move on a slower time scale and obey an exclusion rule among themselves and with the η particles. The competition between the two species is ruled by a coupled birth and death process. We prove convergence in the hydrodynamic limit to a system of two reaction-diffusion equations with measure valued initial data.