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Study of Lotka-volterra food chain chemostat with periodically varying dilution rate

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In this paper, we introduce and study a model of Lotka-volterra chemostat food chain chemostat with periodically varying dilution rate, which contains with predator, prey, and substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system experiences following process: periodic solution → periodic doubling cascade → chaos.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Yulin Normal University, Yulin, Guangxi, 537000, People’s Republic of China

    Guoping Pang

  2. College of Science, Jimei University, Xiamen, Fujian, 361021, People’s Republic of China

    Fengyan Wang

  3. Department of Applied Mathematics, Dalian University of Technology, Dalian Liaoning, 116024, People’s Republic of China

    Lansun Chen

Authors
  1. Guoping Pang
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  2. Fengyan Wang
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  3. Lansun Chen
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Correspondence to Guoping Pang.

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Pang, G., Wang, F. & Chen, L. Study of Lotka-volterra food chain chemostat with periodically varying dilution rate. J Math Chem 43, 901–913 (2008). https://doi.org/10.1007/s10910-007-9263-5

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  • DOI: https://doi.org/10.1007/s10910-007-9263-5

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