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Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting
- Source :
- Mathematical Methods in the Applied Sciences. 40:6715-6731
- Publication Year :
- 2017
- Publisher :
- Wiley, 2017.
-
Abstract
- The Bogdanov-Takens bifurcations of a Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339-366,” Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.
- Subjects :
- Cusp (singularity)
Phase portrait
General Mathematics
010102 general mathematics
General Engineering
Codimension
Type (model theory)
01 natural sciences
Nonlinear Sciences::Chaotic Dynamics
010101 applied mathematics
Control theory
Limit cycle
Quantitative Biology::Populations and Evolution
Applied mathematics
Homoclinic bifurcation
Limit (mathematics)
Homoclinic orbit
0101 mathematics
Nonlinear Sciences::Pattern Formation and Solitons
Mathematics
Subjects
Details
- ISSN :
- 01704214
- Volume :
- 40
- Database :
- OpenAIRE
- Journal :
- Mathematical Methods in the Applied Sciences
- Accession number :
- edsair.doi...........eda892a0003c36020ccc5aa5c2fb984b