Global dynamics and travelling waves for a periodic and diffusive chemostat model with two nutrients and one microorganism

We consider a classical chemostat model with two nutrients and one microorganism, which incorporates spatial diffusion, temporal heterogeneity, and spatial heterogeneity. We study the basic reproduction number R0 and the asymptotic behaviours, which provide us some new findings in chemostat models. Global dynamics in terms of R0 is investigated in a bounded spatial domain. In the general situation where the growth rate and the loss rate of microorganisms depend on the spatiotemporal heterogeneity, we observe that microorganisms will be persistent if either the domain is of fast-growth type or there exists at least one fast-growth site and the diffusion of microorganisms is sufficiently slow. Asymptotic behaviours of the microorganism-existent steady state are discussed for the large diffusion rates, and the existence of periodic travelling wave is established by analysing a fixed point problem of a nonlinear operator in an unbounded spatial domain.