Qualitative Theory of Dynamical Systems

, Volume 18, Issue 3, pp 1201–1224 | Cite as

Hopf Bifurcations in a Predator–Prey Model with an Omnivore

  • Yongjun Li
  • Valery G. RomanovskiEmail author


We study limit cycle bifurcations in a three-dimensional Lotka–Volterra system which models a food web of three species, one of which is an omnivore. First, using a new approach based on the elimination theory of the computational algebra we find necessary and sufficient conditions for existence of a pair of pure imaginary eigenvalues for the Jacobian of the system at the stationary point with positive coordinates. Then it is shown that the system can have two small limit cycles bifurcating from the singular point.


Lotka–Volterra systems Center manifold Limit cycle Bifurcation 

Mathematics Subject Classification

34C23 37G15 34C60 



The first author acknowledges the support by the National Natural Science Foundation of China (11761044, 11661048) and the key constructive discipline of Lanzhou City University (LZCU-ZDJSXK-201706). The second author acknowledges the support by the Slovenian Research Agency (Program P1-0306, project N1-0063) and the grant of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) related to the Workshop on “Advances in Chemical Reaction Network Theory”. We also thank the reviewers for careful reading and valuable suggestions which helped to improve the manuscript.


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Authors and Affiliations

  1. 1.School of MathematicsLanzhou City UniversityLanzhouChina
  2. 2.Faculty of Natural Science and MathematicsUniversity of MariborMariborSlovenia
  3. 3.Faculty of Electrical Engineering and Computer ScienceUniversity of MariborMariborSlovenia
  4. 4.Center for Applied Mathematics and Theoretical PhysicsUniversity of MariborMariborSlovenia

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