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March  2015, 20(2): 423-443. doi: 10.3934/dcdsb.2015.20.423

Dynamical complexity of a prey-predator model with nonlinear predator harvesting

R. P. Gupta 1, , Peeyush Chandra 1, and  Malay Banerjee 2,

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

Received  May 2013 Revised  July 2014 Published  January 2015

The objective of this paper is to study systematically the dynamical properties of a predator-prey model with nonlinear predator harvesting. We show the different types of system behaviors for various parameter values. The results developed in this article reveal far richer dynamics compared to the model without harvesting. The occurrence of change of structure or bifurcation in a system with parameters is a way to predict global dynamics of the system. It has been observed that the model has at most two interior equilibria and can exhibit numerous kinds of bifurcations (e.g. saddle-node, transcritical, Hopf-Andronov and Bogdanov-Takens bifurcation). The stability (direction) of the Hopf-bifurcating periodic solutions has been obtained by computing the first Lyapunov number. The emergence of homoclinic loop has been shown through numerical simulation when the limit cycle arising though Hopf-bifurcation collides with a saddle point. Numerical simulations using MATLAB are carried out as supporting evidences of our analytical findings. The main purpose of the present work is to offer a complete mathematical analysis for the model.
Keywords: Prey-predator model, stability, bifurcation, harvesting, phase portraits..
Mathematics Subject Classification: Primary: 70K05, 34C23; Secondary: 34D2.
Citation: R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423
References:
[1]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1962.  Google Scholar

[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.  Google Scholar

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C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976.  Google Scholar

[4]

C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[14]

P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4 (2010), 791-803.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics (MPB-36), New York, Princeton University Press, 2003. doi: 10.1515/9781400847259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. doi: 10.1007/978-1-4684-0249-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Monographs in Population Biology, 35, Princeton Univ. Press, Princeton, NJ, 2003.  Google Scholar

[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.  Google Scholar

[27]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

G. Birkhoff and G. C. Rota, Ordinary Differential Equations, Ginn, Boston, 1962.  Google Scholar

[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.  Google Scholar

[3]

C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976.  Google Scholar

[4]

C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

P. Lenzini and J. Rebaza, Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci., 4 (2010), 791-803.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics (MPB-36), New York, Princeton University Press, 2003. doi: 10.1515/9781400847259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1996. doi: 10.1007/978-1-4684-0249-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Monographs in Population Biology, 35, Princeton Univ. Press, Princeton, NJ, 2003.  Google Scholar

[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.  Google Scholar

[27]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), 493-506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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