Co-existence of a Period Annulus and a Limit Cycle in a Class of Predator-Prey Models with Group Defense

Robert E. Kooij; André Zegeling

doi:10.1142/S0218127421501546

Co-existence of a Period Annulus and a Limit Cycle in a Class of Predator-Prey Models with Group Defense

Robert E. Kooij^*, André Zegeling

^*Corresponding author for this work

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Abstract

For a family of two-dimensional predator-prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and ehibits group defense. We prove the eistence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.

Original language	English
Article number	2150154
Number of pages	1
Journal	International Journal of Bifurcation and Chaos
Volume	31
Issue number	10
DOIs	https://doi.org/10.1142/S0218127421501546
Publication status	Published - 2021

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

bifurcation
functional response
Generalized Gause model
limit cycle

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title = "Co-existence of a Period Annulus and a Limit Cycle in a Class of Predator-Prey Models with Group Defense",

abstract = "For a family of two-dimensional predator-prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and ehibits group defense. We prove the eistence and uniqueness of the limit cycle using a new theorem for Li{\'e}nard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle. ",

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N2 - For a family of two-dimensional predator-prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and ehibits group defense. We prove the eistence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.

AB - For a family of two-dimensional predator-prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and ehibits group defense. We prove the eistence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.

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Co-existence of a Period Annulus and a Limit Cycle in a Class of Predator-Prey Models with Group Defense

Abstract

Bibliographical note

Keywords

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