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October  2013, 18(8): 2101-2121. doi: 10.3934/dcdsb.2013.18.2101

## Bifurcation analysis 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 Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250

Received  April 2013 Revised  June 2013 Published  July 2013

In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.
Citation: Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101
##### References:
 [1] J. R. Beddington and J. G. Cooke, Harvesting from a prey-predator complex, Ecol. Modelling, 14 (1982), 155-177. doi: 10.1016/0304-3800(82)90016-3. [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] R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Trudy Sem. Petrovsk. Vyp., 2 (1976), 23-35. [4] R. I. Bogdanov, The versal deformations of a singular point on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk. Vyp., 2 (1976), 37-65. [5] 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. [6] 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. [7] 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. [8] L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215. doi: 10.1007/s00285-012-0546-5. [9] V. Christensen, Managing fisheries involving predator and prey species, Rev. Fish Biol. Fisher., 6 (1996), 417-442. doi: 10.1007/BF00164324. [10] C. W. Clark, "Mathematical Bioeconomics. The Optimal Management of Renewable Resources," Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, New York, 1990. [11] S.-N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields," Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639. [12] 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. [13] 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., 7 (1987), 375-413. doi: 10.1017/S0143385700004119. [14] 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. [15] O. Flaaten, On the bioeconomics of predator-prey fishing, Fish. Research, 37 (1998), 179-191. doi: 10.1016/S0165-7836(98)00135-0. [16] Y. Gong and J. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting,, Acta Math. Appl. Sinica Eng. Ser. (accepted)., (). [17] S. L. Hill, E. J. Murphy, K. Reid, P. N. Trathan and A. J. Constable, Modelling Southern Ocean ecosystems: Krill, the food-web, and the impacts of harvesting, Biol. Rev., 81 (2006), 581-608. [18] W. L. Hogarth, J. Norbury, I. Cunning and K. Sommers, Stability of a predator-prey model with harvesting, Ecol. Modelling, 62 (1992), 83-106. doi: 10.1016/0304-3800(92)90083-Q. [19] J. A. Hutchings, Collapse and recovery of marine fishes, Nature, 406 (2000), 882-885. [20] J. A. Hutchings and R. A. Myers, What can be learned from the collapse of a renewable resource? Atlantic code, Gadus morhua, of Newfoundland and Labrador, Can. J. Fish. Aquat. Sci., 51 (1994), 2126-2146. [21] 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. [22] 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. [23] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with nonconstant harvesting, Discrete Contin. Dynam. Syst. Ser. S, 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. [24] R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277. doi: 10.1126/science.205.4403.267. [25] R. A. Myers, J. A. Hutchings and N. J. Barrowman, Why do fish stocks collapse? The example of cod in Atlantic Canada, Ecol. Appl., 7 (1997), 91-106. [26] R. A. Myers and B. Worm, Rapid worldwide depletion of large predatory fish communities, Nature, 423 (2003), 280-283. [27] M. R. Myerscough, B. F. Gray, W. L. Hograth and J. Norbury, An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking, J. Math. Biol., 30 (1992), 389-411. doi: 10.1007/BF00173294. [28] D. Pauly, Theory and management of tropical multispecies stocks, ICLARM Stud. Rev., 1 (1979), 35 pp. [29] D. Pauly, et al., Towards sustainability in world fisheries, Nature, 418 (2002), 689-695. [30] L. Perko, "Differential Equations and Dynamical Systems," Second edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0. [31] F. Takens, Forced oscillations and bifurcation, in "Applications of Global Analysis, I" (Sympos., Utrecht State Univ., Utrecht, 1973), Comm. Math. Inst. Rijksuniversitat Utrecht., No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, (1974), 1-59. [32] Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: 10.1137/070700966. [33] 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. [34] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology" (Halifax, NS, 1997), Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, (1999), 493-506. [35] P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecol. Appl., 4 (1994), 51-58. doi: 10.2307/1942114. [36] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equation," Transl. Math. Monogr., 101, Amer. Math. Soc., Providence, RI, 1992. [37] 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. Dynam. Syst. Ser. B, 14 (2010), 289-306. doi: 10.3934/dcdsb.2010.14.289. [38] 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.

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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. doi: 10.1016/0304-3800(82)90016-3. [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] R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Trudy Sem. Petrovsk. Vyp., 2 (1976), 23-35. [4] R. I. Bogdanov, The versal deformations of a singular point on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk. Vyp., 2 (1976), 37-65. [5] 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. [6] 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. [7] 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. [8] L. Cai, G. Chen and D. Xiao, Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215. doi: 10.1007/s00285-012-0546-5. [9] V. Christensen, Managing fisheries involving predator and prey species, Rev. Fish Biol. Fisher., 6 (1996), 417-442. doi: 10.1007/BF00164324. [10] C. W. Clark, "Mathematical Bioeconomics. The Optimal Management of Renewable Resources," Second edition, Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, New York, 1990. [11] S.-N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields," Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639. [12] 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. [13] 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., 7 (1987), 375-413. doi: 10.1017/S0143385700004119. [14] 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. [15] O. Flaaten, On the bioeconomics of predator-prey fishing, Fish. Research, 37 (1998), 179-191. doi: 10.1016/S0165-7836(98)00135-0. [16] Y. Gong and J. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting,, Acta Math. Appl. Sinica Eng. Ser. (accepted)., (). [17] S. L. Hill, E. J. Murphy, K. Reid, P. N. Trathan and A. J. Constable, Modelling Southern Ocean ecosystems: Krill, the food-web, and the impacts of harvesting, Biol. Rev., 81 (2006), 581-608. [18] W. L. Hogarth, J. Norbury, I. Cunning and K. Sommers, Stability of a predator-prey model with harvesting, Ecol. Modelling, 62 (1992), 83-106. doi: 10.1016/0304-3800(92)90083-Q. [19] J. A. Hutchings, Collapse and recovery of marine fishes, Nature, 406 (2000), 882-885. [20] J. A. Hutchings and R. A. Myers, What can be learned from the collapse of a renewable resource? Atlantic code, Gadus morhua, of Newfoundland and Labrador, Can. J. Fish. Aquat. Sci., 51 (1994), 2126-2146. [21] 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. [22] 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. [23] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with nonconstant harvesting, Discrete Contin. Dynam. Syst. Ser. S, 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. [24] R. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277. doi: 10.1126/science.205.4403.267. [25] R. A. Myers, J. A. Hutchings and N. J. Barrowman, Why do fish stocks collapse? The example of cod in Atlantic Canada, Ecol. Appl., 7 (1997), 91-106. [26] R. A. Myers and B. Worm, Rapid worldwide depletion of large predatory fish communities, Nature, 423 (2003), 280-283. [27] M. R. Myerscough, B. F. Gray, W. L. Hograth and J. Norbury, An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking, J. Math. Biol., 30 (1992), 389-411. doi: 10.1007/BF00173294. [28] D. Pauly, Theory and management of tropical multispecies stocks, ICLARM Stud. Rev., 1 (1979), 35 pp. [29] D. Pauly, et al., Towards sustainability in world fisheries, Nature, 418 (2002), 689-695. [30] L. Perko, "Differential Equations and Dynamical Systems," Second edition, Texts in Applied Mathematics, 7, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0. [31] F. Takens, Forced oscillations and bifurcation, in "Applications of Global Analysis, I" (Sympos., Utrecht State Univ., Utrecht, 1973), Comm. Math. Inst. Rijksuniversitat Utrecht., No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, (1974), 1-59. [32] Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: 10.1137/070700966. [33] 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. [34] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, in "Differential Equations with Applications to Biology" (Halifax, NS, 1997), Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, (1999), 493-506. [35] P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecol. Appl., 4 (1994), 51-58. doi: 10.2307/1942114. [36] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equation," Transl. Math. Monogr., 101, Amer. Math. Soc., Providence, RI, 1992. [37] 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. Dynam. Syst. Ser. B, 14 (2010), 289-306. doi: 10.3934/dcdsb.2010.14.289. [38] 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.
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