# American Institute of Mathematical Sciences

May  2016, 15(3): 1041-1055. doi: 10.3934/cpaa.2016.15.1041

## Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting

 1 School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China, China 2 School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning, Hubei, 437100, China 3 Department of Mathematics, University of Miami, Coral Gables, FL 33124

Received  June 2015 Revised  November 2015 Published  February 2016

Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.
Citation: Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
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##### References:
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