Single species growth consuming inorganic carbon with internal storage in a poorly mixed habitat
- 338 Downloads
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.
KeywordsInorganic carbon Internal storage Extinction Persistence Global stability A nonlinear eigenvalue problem
Mathematics Subject Classification35B40 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.
- Cunningham A, Nisbet RM (1983) Transient and oscillation in continuous culture. In: Bazin MJ (ed) Mathematics in microbiology. Academic, New York, pp 77–103Google Scholar
- Diekmann O, Metz JAJ (1986) The dynamics of physiologically structured populations. Lecture Notes in Biomath, vol 68. Springer, BerlinGoogle Scholar
- Droop M (1973) Some thoughts on nutrient limitation in algae. J Phycol 9:264–272Google Scholar
- Henry D (1981) Geometric theory of semilinear parabolic equations. Lecture Notes in Math, vol 840. Springer, BerlinGoogle Scholar
- De Leenheer P, Levin SA, Sontag ED, Klausmeier CA (2006) Global stability in a chemostat with multiple nutrients. J Math Biol 52:419–438Google Scholar
- Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Math. Surveys Monogr, vol 41. American Mathematical Society, ProvidenceGoogle Scholar
- Smith HL, Thieme HR (2011) Dynamical systems and population persistence, Graduate Studies in Mathematics, vol 118. American Mathematical Society, ProvidenceGoogle Scholar
- Van de Waal DB, Verspagen JMH, Finke JF, Vournazou V, Immers AK, Kardinaal W, Edwin A, Tonk L, Becker S, Van Donk E, Visser PM, Huisman J (2011) Reversal in competitive dominance of a toxic versus non-toxic cyanobacterium in response to rising CO2, I.S.M.E. J. 5 (2011), pp 1438–1450Google Scholar