Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1775–1825 | Cite as

Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat

  • Sze-Bi Hsu
  • King-Yeung Lam
  • Feng-Bin Wang


This paper presents a PDE system modeling the growth of a single species population consuming inorganic carbon that is stored internally in a poorly mixed habitat. Inorganic carbon takes the forms of “CO2” (dissolved CO2 and carbonic acid) and “CARB” (bicarbonate and carbonate ions), which are substitutable in their effects on algal growth. We first establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear eigenvalue problem. If the habitat is the unstirred chemostat, we add biologically relevant assumptions on the uptake functions and prove the uniqueness and global attractivity of the positive steady state when the species persists.


Inorganic carbon Internal storage Extinction Persistence Global stability A nonlinear eigenvalue problem 

Mathematics Subject Classification

35B40 35K57 92D25 



We are grateful to three anonymous referees for their careful reading and helpful suggestions which led to improvements of our original manuscript. We also express our thanks to Prof. H. R. Thieme for suggesting us the key reference Mallet-Paret and Nussbaum (2010) in this paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsNational Tsing Hua UniversityHsinchuTaiwan
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA
  3. 3.Department of Natural Science in the Center for General EducationChang Gung UniversityGuishan, TaoyuanTaiwan
  4. 4.Community Medicine Research CenterChang Gung Memorial Hospital, KeelungKeelungTaiwan

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