Global Dynamics of a Time-Delayed Microorganism Flocculation Model with Saturated Functional Responses

Abstract

In this paper, a time-delayed model of microorganism flocculation with saturated functional responses is presented. We first analyse the local dynamics of this model with bifurcations in parameter fields, and then prove the collection of microorganisms is sustainable as well as obtain an explicit eventual lower bound of microorganism concentration when threshold parameter \(R_{0}>1\). This model has a backward bifurcation if \(w<R_{0}<1\) under an additional condition, which implies that the microorganism-free equilibrium coexists with a microorganism equilibrium. In these cases, we establish some sufficient conditions for the global stability by using a variant of the Lyapunov–LaSalle theorem.