Stability and persistence analysis of a microorganism flocculation model with infinite delay

In this paper, we study the global stability and persistence of a microorganism flocculation model with infinite delay. First, we make a complete theoretical analysis on the local stability of the boundary equilibrium (microorganism-free equilibrium) and the positive equilibrium (microorganism co-existent equilibrium), and give a sufficient condition for the global stability of the boundary equilibrium (applicable to the forward bifurcation and the backward bifurcation). Then, for the persistence of the model, we present an explicit estimate of the eventual lower bound of any positive solution for which only the parameter threshold $ R_0 > 1 $ is required. The obtained results extend some of the conclusions of the existing literatures on the case of discrete time delay.