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

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.

[1]  S. Chow,et al.  Normal Forms and Bifurcation of Planar Vector Fields , 1994 .

[2]  Zhien Ma,et al.  Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence , 2015 .

[3]  Benjamin Leard,et al.  Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting , 2008 .

[4]  Robert M. May,et al.  Maximum sustainable yields in systems subject to harvesting at more than one trophic level , 1980 .

[5]  J. Chen,et al.  Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Harvesting , 2013 .

[6]  Moxun Tang,et al.  Coexistence Region and Global Dynamics of a Harvested Predator-Prey System , 1998, SIAM J. Appl. Math..

[7]  A. C. Soudack,et al.  Stability regions in predator-prey systems with constant-rate prey harvesting , 1979 .

[8]  Colin W. Clark,et al.  Mathematical Bioeconomics: The Optimal Management of Renewable Resources. , 1993 .

[9]  Sze-Bi Hsu,et al.  Global Stability for a Class of Predator-Prey Systems , 1995, SIAM J. Appl. Math..

[10]  Huaiping Zhu,et al.  Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response , 2003, SIAM J. Appl. Math..

[11]  Dongmei Xiao,et al.  Bifurcations of a Ratio-Dependent Predator-Prey System with Constant Rate Harvesting , 2005, SIAM J. Appl. Math..

[12]  K. Lan,et al.  Phase portraits, Hopf bifurcations and limit cyclesof Leslie-Gower predator-prey systems with harvesting rates , 2010 .

[13]  Freddy Dumortier,et al.  Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3 , 1987, Ergodic Theory and Dynamical Systems.

[14]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[15]  Christiane Rousseau,et al.  Bifurcation Analysis of a Predator–Prey System with Generalised Holling Type III Functional Response , 2008 .

[16]  A. C. Soudack,et al.  Coexistence properties of some predator-prey systems under constant rate harvesting and stocking , 1982 .

[17]  J. Beddington,et al.  Harvesting from a prey-predator complex , 1982 .

[18]  Colin W. Clark,et al.  Management of Multispecies Fisheries , 1979, Science.

[19]  Shigui Ruan,et al.  Bifurcation analysis in a predator-prey model with constant-yield predator harvesting , 2013 .

[20]  Jing Chen,et al.  Bifurcations of Invariant Tori in Predator-Prey Models with Seasonal Prey Harvesting , 2013, SIAM J. Appl. Math..

[21]  Christiane Rousseau,et al.  Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III , 2010 .

[22]  A. C. Soudack,et al.  Stability regions and transition phenomena for harvested predator-prey systems , 1979 .

[23]  Jicai Huang,et al.  Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting , 2014 .

[24]  Jicai Huang,et al.  Multiple bifurcations in a predator-prey System of Holling and Leslie Type with Constant-Yield prey harvesting , 2013, Int. J. Bifurc. Chaos.