Degenerate Hopf bifurcation in a Leslie–Gower predator–prey model with predator harvest

In this paper, we investigate the degenerate Hopf bifurcation of a Leslie–Gower predator–prey system with predator harvest. The known work discussed the Hopf bifurcation of this system when the first Lyapunov number does not vanish and gave an example of a stable weak focus with order 2. However, th...

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Bibliographic Details
Published in:Advances in difference equations 2020-05, Vol.2020 (1), p.1-19, Article 194
Main Author: Su, Juan
Format: Article
Language:English
Subjects:
Analysis
Computer simulation
Difference and Functional Equations
Focal value
Functional Analysis
Hopf bifurcation
Leslie–Gower predator–prey model
Mathematical models
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Parameters
Partial Differential Equations
Predator-prey simulation
Resultant elimination
Thermal energy
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Summary:In this paper, we investigate the degenerate Hopf bifurcation of a Leslie–Gower predator–prey system with predator harvest. The known work discussed the Hopf bifurcation of this system when the first Lyapunov number does not vanish and gave an example of a stable weak focus with order 2. However, the thorough discussion of center-type equilibrium for all possible parameters has not been completed. In this paper, by computing the first two focal values, we decompose the variety with resultant and prove that the center-type equilibrium is a weak focus with order at most 2 for all the possible parameter values. Moreover, numerical simulations are employed to show the appearance of two limit cycles from degenerate Hopf bifurcation. Our results finish the study of Hopf bifurcation in this system.
ISSN:1687-1847
1687-1839
1687-1847
DOI:10.1186/s13662-020-02656-3