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Transient oscillations induced by delayed growth response in the chemostat

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Abstract.

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.

When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

    Huaxing Xia, Gail S.K. Wolkowicz & Lin Wang

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  1. Huaxing Xia
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  2. Gail S.K. Wolkowicz
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  3. Lin Wang
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Additional information

Mathematics Subject Classification: 34D20, 34K20, 92D25

Research was partially supported by NSERC of Canada.

This work was partly done while this author was a postdoc at McMaster.

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Xia, H., Wolkowicz, G. & Wang, L. Transient oscillations induced by delayed growth response in the chemostat. J. Math. Biol. 50, 489–530 (2005). https://doi.org/10.1007/s00285-004-0311-5

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  • DOI: https://doi.org/10.1007/s00285-004-0311-5

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