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Multiphase mechanical models are now commonly used to describe living tissuesincluding tumour growth. The specific model we study here consists of twoequations of mixed parabolic and hyperbolic type which extend the standardcompressible porous medium equation, including cross-reaction terms. We studythe incompressible limit, when the pressure becomes stiff, which generates afree boundary problem. We establish the complementarity relation and also asegregation result. Several major mathematical difficulties arise in the twospecies case. Firstly, the system structure makes comparison principles fail.Secondly, segregation and internal layers limit the regularity available onsome quantities to BV. Thirdly, the Aronson-B{\'e}nilan estimates cannot beestablished in our context. We are lead, as it is classical, to add correctionterms. This procedure requires technical manipulations based on BV estimatesonly valid in one space dimension. Another novelty is to establish an L1version in place of the standard upper bound.

Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues