• Previous Article
    A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions
  • DCDS-B Home
  • This Issue
  • Next Article
    Efficient resolution of metastatic tumor growth models by reformulation into integral equations
March  2015, 20(2): 423-443. doi: 10.3934/dcdsb.2015.20.423

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

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.
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, (1962). Google Scholar

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 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, (1976). Google Scholar

[4]

C. W. Clark, Bioeconomic Modelling and Fisheries Management,, Wiley, (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. 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. 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. 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. 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. 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. 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. doi: 10.1006/bulm.1997.0023. Google Scholar

[12]

A. Y. Kuznetsov, Elements of Applied Bifurcation Theory,, Appl. Math. Sci., (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. 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. 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. 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., 14 (1990), 955. 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, (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. 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. doi: 10.1007/s11071-010-9914-0. Google Scholar

[20]

L. Perko, Differential Equations and Dynamical Systems,, Springer, (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. Google Scholar

[22]

J. Rebaza, Dynamics of prey threshold harvesting and refuge,, J. Comput. Appl. Math., 236 (2012), 1743. 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. 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. 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 (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. 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. 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. 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. 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. doi: 10.1137/S0036139901397285. Google Scholar

show all references

References:
[1]

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

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 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, (1976). Google Scholar

[4]

C. W. Clark, Bioeconomic Modelling and Fisheries Management,, Wiley, (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. 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. 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. 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. 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. 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. 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. doi: 10.1006/bulm.1997.0023. Google Scholar

[12]

A. Y. Kuznetsov, Elements of Applied Bifurcation Theory,, Appl. Math. Sci., (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. 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. 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. 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., 14 (1990), 955. 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, (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. 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. doi: 10.1007/s11071-010-9914-0. Google Scholar

[20]

L. Perko, Differential Equations and Dynamical Systems,, Springer, (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. Google Scholar

[22]

J. Rebaza, Dynamics of prey threshold harvesting and refuge,, J. Comput. Appl. Math., 236 (2012), 1743. 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. 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. 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 (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. 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. 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. 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. 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. doi: 10.1137/S0036139901397285. Google Scholar

[1]

K. Q. Lan, C. R. Zhu. Phase portraits of predator--prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 901-933. doi: 10.3934/dcds.2012.32.901

[2]

C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289

[3]

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 & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[4]

Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101

[5]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[6]

Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128

[7]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[8]

Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737

[9]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[10]

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173

[11]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

[12]

Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061

[13]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[14]

J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059

[15]

Komi Messan, Yun Kang. A two patch prey-predator model with multiple foraging strategies in predator: Applications to insects. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 947-976. doi: 10.3934/dcdsb.2017048

[16]

Yun Kang, Sourav Kumar Sasmal, Komi Messan. A two-patch prey-predator model with predator dispersal driven by the predation strength. Mathematical Biosciences & Engineering, 2017, 14 (4) : 843-880. doi: 10.3934/mbe.2017046

[17]

Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172

[18]

Huiling Li, Peter Y. H. Pang, Mingxin Wang. Qualitative analysis of a diffusive prey-predator model with trophic interactions of three levels. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 127-152. doi: 10.3934/dcdsb.2012.17.127

[19]

Shanbing Li, Jianhua Wu. Effect of cross-diffusion in the diffusion prey-predator model with a protection zone. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1539-1558. doi: 10.3934/dcds.2017063

[20]

Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]