Dynamics of a chemostat consumer-resource system with diffusion between multiple sink-source patches

This paper considers a chemostat system with a consumer and a resource. The consumer moves between multiple sink-source patches with both resource and toxicant. Using graph-theoretic method and Lyapunov theory, we show global stability of positive equilibria in the system. Rigorous analysis on the equilibria demonstrates that diffusion could make the consumer persist in sinks and even make the total population abundance larger than that without diffusion. Asymmetry in diffusion, toxicant distribution, and spatial pattern of patches play crucial roles in the persistence and increase, while two formulas are deduced for total abundances with/without diffusion. On the contrary, diffusion could also lead to decrease of the total abundance, even lead to extinction of the consumer in sources. Our results are consistent with empirical observations, and provide new insights. This work is important for both conservation of endangered species and control of pests.