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Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield

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Abstract

In this paper, we consider a variable yield model of a single-species growth in a well-stirred tank containing fresh medium, assuming the instances of time as triggering factors in which the nutrient refilling process and the removal of microorganisms by the uptake of lethal external antibiotic are initiated. It is also assumed that the periodic nutrient refilling and the periodic antibiotic injection occur with the same periodicity, but not simultaneously. The model is then formulated in terms of autonomous differential equations subject to impulsive perturbations. It is observed that either the population of microorganisms essentially washes out, or more favorably, the system is permanent. To describe this dichotomy, some biologically significant integral conditions are introduced. Further, it is shown that in a certain critical situation, a nontrivial periodic solution emerges via a bifurcation phenomenon. Finally, the dynamics of the model is illustrated with numerical experiments and computer simulations.

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Authors and Affiliations

  1. Department of Mathematics, Jiangsu University, Zhenjiang, 212013, Jiangsu Province, P. R. China

    Hong Zhang  (张弘)

  2. Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, SE-901 87, Sweden

    Hong Zhang  (张弘)

  3. Department of Mathematics, “Gh. Asachi” Technical University of Iaşi, Bd. Copou 11, Iaşi, 700506, Romania

    Paul Georgescu

  4. Departamento de Análisis Matemático, Facultad de Matemáticas Universidad de Santiago de Compostela, Santiago de Compostela, 15782, Spain

    Juan J. Nieto

  5. Institute of Mathematics, Chinese Academy of Sciences, Beijing, 100080, P. R. China

    Lan-sun Chen  (陈兰荪)

Authors
  1. Hong Zhang  (张弘)
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  2. Paul Georgescu
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  3. Juan J. Nieto
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  4. Lan-sun Chen  (陈兰荪)
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Corresponding author

Correspondence to Hong Zhang  (张弘).

Additional information

Communicated by Li-qun CHEN

Project supported by the National Natural Science Foundation of China (No. 10771104)

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Zhang, H., Georgescu, P., Nieto, J.J. et al. Impulsive perturbation and bifurcation of solutions for a model of chemostat with variable yield. Appl. Math. Mech.-Engl. Ed. 30, 933–944 (2009). https://doi.org/10.1007/s10483-009-0712-x

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  • DOI: https://doi.org/10.1007/s10483-009-0712-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

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