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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 and 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.

[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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