A PDE system modeling the competition and inhibition of harmful algae with seasonal variations

In this paper, we study a reaction-diffusion-advection system modeling the competition of harmful algae with seasonal variations in a flowing water habitat. We assume that harmful algae produce toxins, which have inhibitory effects on their algal competitors, that is, the produced toxins can inhibit the growth of its competitor. For the single population model, we prove that the algae will be washed out eventually if the trivial periodic state is locally asymptotically stable, while there exists a unique positive periodic state which is globally attractive if the trivial periodic state is unstable. When there is mutual invasibility of both semitrivial periodic solutions of the two-species model, we are able to prove the existence of periodic coexistence state.