Open AccessFeedback-mediated coexistence and oscillations in the chemostat
Author(s) -
Willard S. Keeran,
Patrick De Leenheer,
Sergei S. Pilyugin
Publication year - 2008
Publication title -
discrete and continuous dynamical systems - b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2008.9.321
Subject(s) - chemostat , hopf bifurcation , mathematics , bifurcation , pitchfork bifurcation , control theory (sociology) , biological applications of bifurcation theory , dilution , stability (learning theory) , transcritical bifurcation , affine transformation , physics , pure mathematics , biology , thermodynamics , computer science , control (management) , nonlinear system , genetics , quantum mechanics , artificial intelligence , machine learning , bacteria
We consider a mathematical model that describes the competition of three species for a single nutrient in a chemostat in which the dilution rate is assumed to be controllable by means of state dependent feedback. We consider feedback schedules that are affine functions of the species concentrations. In case of two species, we show that the system may undergo a Hopf bifurcation and oscillatory behavior may be induced by appropriately choosing the coefficients of the feedback function. When the growth of the species obeys Michaelis-Menten kinetics, we show that the Hopf bifurcation is supercritical in the relevant parameter region, and the bifurcating periodic solutions for two species are always stable. Finally, we show that by adding a third species to the system, the two-species stable periodic solutions may bifurcate into the coexistence region via a transcritical bifurcation. We give conditions under which the bifurcating orbit is locally asymptotically stable.
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