Finite Volume Analysis of the Two Competing-species Chemotaxis Models with General Diffusive Functions

This paper aims to see how different spatial and environmental factors affect the coexistence or the exclusion of two species, while chemotaxis draws them towards a higher concentration of nutrients. For that, we analyze a robust numerical scheme applied for competitive two-species chemotaxis models with heterogeneous and potentially discontinuous diffusive coefficients. This extension is essential because diffusion can lead to discontinuities when the conductivities of the medium’s components differ. In this work, we examine a generalized finite volume scheme on admissible meshes, where the line joining the circumcenters of two neighboring volumes is orthogonal to their common interface, and the discontinuities coincide with the mesh interfaces. Finite volume methods are well-suited for problems involving conservation laws and can naturally handle discontinuities, making them an ideal candidate. To achieve the convergence, we first derive the discrete problem and then we show that the discrete solution converges to a weak solution of the continuous model. Finally, many simulations were performed using Fortran software, with the introduction of a reliable computational algorithm. The efficiency of our numerical approach for finding the discrete solutions is then carefully evaluated with many test cases focusing on the heterogeneity and the discontinuity of the diffusive coefficients.