Global dynamics in a chemostat and an epidemic model

Two models are studied in this work; a periodically forced Droop model for phytoplankton growth with two competing species in a chemostat and a time-delayed SIR epidemic model with dispersal. -- For the competition model, both uniform persistence and the existence of periodic coexistence state are established for a periodically forced Droop model on two phytoplankton species competition in a chemostat under some appropriate conditions. Numerical simulations using biological data are presented as well to illustrate the main result. -- The global dynamics of a time-delayed model with population dispersal between two patches is also investigated. For a general class of birth functions, persistence theory is applied to prove that a disease is persistent when the basic reproduction number is greater than one. It is also shown that the disease will die out if the basic reproduction number is less than one, provided that the invasion intensity is not strong. Numerical simulations are presented using some typical birth functions from biological literature to illustrate the main ideas and the relevance of dispersal