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Dynamical complexity of a prey-predator model with nonlinear predator harvesting
1. | Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India, India |
2. | Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh |
References:
[1] |
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1962. |
[2] |
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York, Berlin, 251 1982.
doi: 10.1007/978-1-4613-8159-4. |
[3] |
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976. |
[4] |
C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. |
[5] |
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. |
[6] |
T. Das, R. N. Mukherjee and K. S. Chaudhari, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462.
doi: 10.1080/17513750802560346. |
[7] |
H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508.
doi: 10.1007/BF02462320. |
[8] |
R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
doi: 10.1016/j.jmaa.2012.08.057. |
[9] |
M. Haque, A detailed study of the Beddington-DeAngelis predator-prey model, Math. Biosci., 234 (2011), 1-16.
doi: 10.1016/j.mbs.2011.07.003. |
[10] |
S. B. Hsu, T. W. Hwang and Y. Kuang, Global dynamics of a predator-prey model with Hassell-Varley type functional response, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 857-871.
doi: 10.3934/dcdsb.2008.10.857. |
[11] |
S. V. Krishna, P. D. N. Srinivasu and B. Prasad Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.
doi: 10.1006/bulm.1997.0023. |
[12] |
A. Y. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[13] |
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. |
[14] |
P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4 (2010), 791-803. |
[15] |
Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34 (2007), 606-620.
doi: 10.1016/j.chaos.2006.03.068. |
[16] |
K. Mischaikow and G. S. K. Wolkowicz, A predator-prey system involving group defense: A connection matrix approach, Nonlin. Anal., Theory, Methods & Applications, 14 (1990), 955-969.
doi: 10.1016/0362-546X(90)90112-T. |
[17] |
W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics (MPB-36), New York, Princeton University Press, 2003.
doi: 10.1515/9781400847259. |
[18] |
G. J. Peng, Y. L. Jiang and C. P. Li, Bifurcations of a Holling-type II predator-prey system with constant rate harvesting, Int. J. Bifurc. Chaos Appl. Sci. Eng., 19 (2009), 2499-2514.
doi: 10.1142/S021812740902427X. |
[19] |
G. J. Peng and Y. L. Jiang, Practical computation of normal forms of the Bogdonov-Takens bifurcation, Nonlinear Dyn., 66 (2011), 99-132.
doi: 10.1007/s11071-010-9914-0. |
[20] |
L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996.
doi: 10.1007/978-1-4684-0249-0. |
[21] |
T. Pradhan and K. S. Chaudhuri, Bioeconomic harvesting of a schooling fish species: A dynamic reaction model, Korean J. Comput. Appl. Math., 6 (1999), 127-141. |
[22] |
J. Rebaza, Dynamics of prey threshold harvesting and refuge, J. Comput. Appl. Math., 236 (2012), 1743-1752.
doi: 10.1016/j.cam.2011.10.005. |
[23] |
S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.
doi: 10.1137/S0036139999361896. |
[24] |
P. D. N. Srinivasu, Bioeconomics of a renewable resource in presence of a predator, Nonlin. Anal. Real World Appl., 2 (2001), 497-506.
doi: 10.1016/S1468-1218(01)00006-2. |
[25] |
P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Monographs in Population Biology, 35, Princeton Univ. Press, Princeton, NJ, 2003. |
[26] |
G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606.
doi: 10.1137/0148033. |
[27] |
D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. |
[28] |
D. Xiao and L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.
doi: 10.1137/S0036139903428719. |
[29] |
D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.
doi: 10.1016/j.jmaa.2005.11.048. |
[30] |
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. |
show all references
References:
[1] |
G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1962. |
[2] |
S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York, Berlin, 251 1982.
doi: 10.1007/978-1-4613-8159-4. |
[3] |
C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976. |
[4] |
C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. |
[5] |
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. |
[6] |
T. Das, R. N. Mukherjee and K. S. Chaudhari, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462.
doi: 10.1080/17513750802560346. |
[7] |
H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508.
doi: 10.1007/BF02462320. |
[8] |
R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.
doi: 10.1016/j.jmaa.2012.08.057. |
[9] |
M. Haque, A detailed study of the Beddington-DeAngelis predator-prey model, Math. Biosci., 234 (2011), 1-16.
doi: 10.1016/j.mbs.2011.07.003. |
[10] |
S. B. Hsu, T. W. Hwang and Y. Kuang, Global dynamics of a predator-prey model with Hassell-Varley type functional response, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 857-871.
doi: 10.3934/dcdsb.2008.10.857. |
[11] |
S. V. Krishna, P. D. N. Srinivasu and B. Prasad Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.
doi: 10.1006/bulm.1997.0023. |
[12] |
A. Y. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112. Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[13] |
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. |
[14] |
P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4 (2010), 791-803. |
[15] |
Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34 (2007), 606-620.
doi: 10.1016/j.chaos.2006.03.068. |
[16] |
K. Mischaikow and G. S. K. Wolkowicz, A predator-prey system involving group defense: A connection matrix approach, Nonlin. Anal., Theory, Methods & Applications, 14 (1990), 955-969.
doi: 10.1016/0362-546X(90)90112-T. |
[17] |
W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics (MPB-36), New York, Princeton University Press, 2003.
doi: 10.1515/9781400847259. |
[18] |
G. J. Peng, Y. L. Jiang and C. P. Li, Bifurcations of a Holling-type II predator-prey system with constant rate harvesting, Int. J. Bifurc. Chaos Appl. Sci. Eng., 19 (2009), 2499-2514.
doi: 10.1142/S021812740902427X. |
[19] |
G. J. Peng and Y. L. Jiang, Practical computation of normal forms of the Bogdonov-Takens bifurcation, Nonlinear Dyn., 66 (2011), 99-132.
doi: 10.1007/s11071-010-9914-0. |
[20] |
L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996.
doi: 10.1007/978-1-4684-0249-0. |
[21] |
T. Pradhan and K. S. Chaudhuri, Bioeconomic harvesting of a schooling fish species: A dynamic reaction model, Korean J. Comput. Appl. Math., 6 (1999), 127-141. |
[22] |
J. Rebaza, Dynamics of prey threshold harvesting and refuge, J. Comput. Appl. Math., 236 (2012), 1743-1752.
doi: 10.1016/j.cam.2011.10.005. |
[23] |
S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472.
doi: 10.1137/S0036139999361896. |
[24] |
P. D. N. Srinivasu, Bioeconomics of a renewable resource in presence of a predator, Nonlin. Anal. Real World Appl., 2 (2001), 497-506.
doi: 10.1016/S1468-1218(01)00006-2. |
[25] |
P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Monographs in Population Biology, 35, Princeton Univ. Press, Princeton, NJ, 2003. |
[26] |
G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606.
doi: 10.1137/0148033. |
[27] |
D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506. |
[28] |
D. Xiao and L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.
doi: 10.1137/S0036139903428719. |
[29] |
D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.
doi: 10.1016/j.jmaa.2005.11.048. |
[30] |
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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