# 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 and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
##### References:
 [1] J. R. Beddington and J. G. Cooke, Harvesting from a prey-predator complex, Ecol. Modelling, 14 (1982), 155-177. [2] J. R. Beddington and R. M. May, Maximum sustainable yields in systems subject to harvesting at more than one trophic level, Math. Biosci., 51 (1980), 261-281. doi: 10.1016/0025-5564(80)90103-0. [3] F. Brauer and A. C. Soudack, Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 7 (1979), 319-337. doi: 10.1007/BF00275152. [4] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8 (1979), 55-71. doi: 10.1007/BF00280586. [5] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12 (1981), 101-114. doi: 10.1007/BF00275206. [6] J. Chen, J. Huang, S. Ruan and J. Wang, Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting, SIAM J Appl Math, 73 (2013), 1876-1905. doi: 10.1137/120895858. [7] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, New York, 1994. doi: 10.1017/CBO9780511665639. [8] C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, 2nd ed., John Wily and Sons, New York, Toronto, 1990. [9] G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), 193-210. doi: 10.1137/S0036139994275799. [10] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theor. Dyn. Syst., 3 (1987), 375-413. doi: 10.1017/S0143385700004119. [11] F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields Nilpotent Singularities and Abelian Integrals, Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. [12] R. M. Etoua and C. Rousseau, Bifurcation analysis of a Generalized Gause model with prey harvesting and a generalized Holling response function of type III, J. Differential Equations, 249 (2010), 2316-2356. doi: 10.1016/j.jde.2010.06.021. [13] Y. Gong and J. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sinica Eng. Ser., 30 (2014), 239-244. doi: 10.1007/s10255-014-0279-x. [14] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201. [15] J. Huang, J. Chen, Y. Gong and W. Zhang, Complex dynamics in predator-prey models with nonmonotonic functional response and harvesting, Math. Model. Nat. Phenom, 8 (2013), 95-118. doi: 10.1051/mmnp/20138507. [16] J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos, 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642. [17] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121. doi: 10.3934/dcdsb.2013.18.2101. [18] Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type III functional response, J. Dynam. Differential Equations, 20 (2008), 535-571. doi: 10.1007/s10884-008-9102-9. [19] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with nonconstant harvesting, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. [20] C. Li, J. Li and Z. Ma, Codimension 3 B-T bifurcation in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116. doi: 10.3934/dcdsb.2015.20.1107. [21] R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277. [22] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. doi: 10.1007/978-1-4684-0249-0. [23] D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753. doi: 10.1137/S0036139903428719. [24] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. [25] H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682. doi: 10.1137/S0036139901397285. [26] C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306. doi: 10.3934/dcdsb.2010.14.289.

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##### References:
 [1] J. R. Beddington and J. G. Cooke, Harvesting from a prey-predator complex, Ecol. Modelling, 14 (1982), 155-177. [2] J. R. Beddington and R. M. May, Maximum sustainable yields in systems subject to harvesting at more than one trophic level, Math. Biosci., 51 (1980), 261-281. doi: 10.1016/0025-5564(80)90103-0. [3] F. Brauer and A. C. Soudack, Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 7 (1979), 319-337. doi: 10.1007/BF00275152. [4] F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8 (1979), 55-71. doi: 10.1007/BF00280586. [5] F. Brauer and A. C. Soudack, Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12 (1981), 101-114. doi: 10.1007/BF00275206. [6] J. Chen, J. Huang, S. Ruan and J. Wang, Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting, SIAM J Appl Math, 73 (2013), 1876-1905. doi: 10.1137/120895858. [7] S. N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, New York, 1994. doi: 10.1017/CBO9780511665639. [8] C. W. Clark, Mathematical Bioeconomics, The Optimal Management of Renewable Resources, 2nd ed., John Wily and Sons, New York, Toronto, 1990. [9] G. Dai and M. Tang, Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), 193-210. doi: 10.1137/S0036139994275799. [10] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theor. Dyn. Syst., 3 (1987), 375-413. doi: 10.1017/S0143385700004119. [11] F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields Nilpotent Singularities and Abelian Integrals, Lecture Notes in Mathematics, 1480, Springer-Verlag, Berlin, 1991. [12] R. M. Etoua and C. Rousseau, Bifurcation analysis of a Generalized Gause model with prey harvesting and a generalized Holling response function of type III, J. Differential Equations, 249 (2010), 2316-2356. doi: 10.1016/j.jde.2010.06.021. [13] Y. Gong and J. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sinica Eng. Ser., 30 (2014), 239-244. doi: 10.1007/s10255-014-0279-x. [14] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201. [15] J. Huang, J. Chen, Y. Gong and W. Zhang, Complex dynamics in predator-prey models with nonmonotonic functional response and harvesting, Math. Model. Nat. Phenom, 8 (2013), 95-118. doi: 10.1051/mmnp/20138507. [16] J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos, 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642. [17] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121. doi: 10.3934/dcdsb.2013.18.2101. [18] Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type III functional response, J. Dynam. Differential Equations, 20 (2008), 535-571. doi: 10.1007/s10884-008-9102-9. [19] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with nonconstant harvesting, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. [20] C. Li, J. Li and Z. Ma, Codimension 3 B-T bifurcation in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116. doi: 10.3934/dcdsb.2015.20.1107. [21] R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277. [22] L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. doi: 10.1007/978-1-4684-0249-0. [23] D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753. doi: 10.1137/S0036139903428719. [24] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. [25] H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682. doi: 10.1137/S0036139901397285. [26] C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306. doi: 10.3934/dcdsb.2010.14.289.
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