A reaction–advection–diffusion system modeling the competition for two complementary resources with seasonality in a flowing habitat

In this current paper, we investigate a periodic bio-reactor model where microorganisms compete for two essential nutrient resource(s) needed for growth. For the single population growth model: we show that when the trivial solution is (locally) asymptotically stable, then the single population will be washed out; when the trivial solution is unstable, there is a unique periodic positive solution which attracts all solutions with nonzero initial data. For the two species model, we prove that the existence of a periodic coexistence state is possible if each species can invade the semi-trivial periodic state established by the other species. More precisely, if the semi-trivial periodic solutions are both unstable, there exists at least one periodic coexistence solution. Numerical work indicates conditions for persistence depend on the flow characteristics (advection and diffusivity). From our numerical simulations, competitive exclusion, bistability, and coexistence are all observed.