Hopf bifurcation in three-species food chain models with group defense

H. I. Freedman; Shigui Ruan

doi:10.1016/0025-5564(92)90079-C

Hopf bifurcation in three-species food chain models with group defense

H. I. Freedman, Shigui Ruan

Research output: Contribution to journal › Article › peer-review

46 Scopus citations

Abstract

Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.

Original language	English (US)
Pages (from-to)	73-87
Number of pages	15
Journal	Mathematical Biosciences
Volume	111
Issue number	1
DOIs	https://doi.org/10.1016/0025-5564(92)90079-C
State	Published - Sep 1992
Externally published	Yes

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Biochemistry, Genetics and Molecular Biology(all)
Immunology and Microbiology(all)
Agricultural and Biological Sciences(all)
Applied Mathematics

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title = "Hopf bifurcation in three-species food chain models with group defense",

abstract = "Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.",

author = "Freedman, {H. I.} and Shigui Ruan",

note = "Funding Information: *Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. NSERC A 4823. **Research partially supported by a University of Alberta Ph.D. Scholarship. Copyright: Copyright 2014 Elsevier B.V., All rights reserved.",

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AB - Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.

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Hopf bifurcation in three-species food chain models with group defense

Abstract

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