Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1247–1271 | Cite as

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

  • Songbai Guo
  • Wanbiao MaEmail author
  • Xiao-Qiang Zhao


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.


Microorganism flocculation model Delay differential equations Global stability Lyapunov–LaSalle theorem Permanence 

Mathematics Subject Classification

37N25 37L15 34D23 34K20 



We are grateful to the editor and anonymous reviewers for their careful reading and valuable comments. This work was supported in part by the China Scholarship Council, the Fundamental Research Funds for the Central Universities (FRF-BY-14-036), the National Natural Science Foundation of China (11471034), and the Natural Science and Engineering Research Council of Canada.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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