Hopf bifurcation for a class of predator-prey system with small immigration

Maurıicio F. S. Lima; Jaume Llibre; Maurıicio F. S. Lima; Jaume Llibre

doi:10.3934/era.2024209

Electronic Research Archive

2024, Volume 32, Issue 7: 4604-4613. doi: 10.3934/era.2024209

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Hopf bifurcation for a class of predator-prey system with small immigration

Maurıicio F. S. Lima ^{1
,
,},
Jaume Llibre ²

1.
Centro de Matematica Computação e Cognição, Universidade Federal do ABC, 09210-170, Santo André, SP, Brazil
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

Received: 10 April 2024 Revised: 28 June 2024 Accepted: 09 July 2024 Published: 26 July 2024

The subject of this paper concerns with the bifurcation of limit cycles for a predator-prey model with small immigration. Since, in general, the biological systems are not isolated, taking into account immigration in the model becomes more realistic. In this context, we deal with a model with a Holling type Ⅰ function response and study, using averaging theory of second order, the Hopf bifurcation that can emerge under small perturbation of the biological parameters.
- predator-prey system,
- periodic orbit,
- limit cycle,
- Hopf bifurcation,
- averaging equation
Citation: Maurıicio F. S. Lima, Jaume Llibre. Hopf bifurcation for a class of predator-prey system with small immigration[J]. Electronic Research Archive, 2024, 32(7): 4604-4613. doi: 10.3934/era.2024209

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Abstract

The subject of this paper concerns with the bifurcation of limit cycles for a predator-prey model with small immigration. Since, in general, the biological systems are not isolated, taking into account immigration in the model becomes more realistic. In this context, we deal with a model with a Holling type Ⅰ function response and study, using averaging theory of second order, the Hopf bifurcation that can emerge under small perturbation of the biological parameters.

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