Directional migration and competition in fluid media: Global existence, uniqueness, and robust simulation of chemotaxis and tumor growth dynamics

Unlike classical isotropic models, this work establishes a mathematical framework for analyzing interspecies competition in fluid environments with nonlinear degenerate anisotropic diffusive fluxes, a setting that has not been rigorously treated before. The first major novelty is a rigorous proof of the global existence of weak solutions, despite the strong nonlinearity and degeneracy of the operators. Furthermore, we provide one of the first uniqueness results for weak solutions under Stokes coupling, obtained via a carefully tailored duality method. On the computational side, we generalize a convergent hybrid finite volume-finite element scheme that overcomes the traditional instability and mesh-dependence issues plaguing anisotropic systems. This scheme guarantees confinement properties consistent with biological admissibility and is implemented in a robust predictive solver. Numerical experiments conducted on heterogeneous domains reveal new classes of dynamical behaviors in two-species systems, including anisotropy-driven spatial segregation and complex domains. By integrating analytical rigor and advanced numerics, this study provides a novel benchmark for the well-posedness and simulation of nonlinear anisotropic ecological and biomedical systems, with particular relevance to tumor growth dynamics and multi-species chemotaxis competitive systems.