A Class of Competitive Systems on Graphs

A class of Lotka-Volterra competitive systems on a finite graph (network) is investigated, where both diffusive and advective movements are considered. By the theory of graphs, the theory of monotone dynamical systems and the theory of principal eigenvalues, the global dynamics is determined in terms of critical competition coefficients, where the key is to establish the linear stability of all possible positive equilibria. This work can be seen as a further development of Slavík (2020) and Chen et al. (2022), where there is diffusion only (no advection).