# American Institute of Mathematical Sciences

October  2017, 10(5): 1187-1206. doi: 10.3934/dcdss.2017065

## Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting

 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: klcqu20132016@163.com (L. Kong)

Received  November 2016 Revised  January 2017 Published  June 2017

Fund Project: Acknowledgement: This work was supported by NSFC grant 11671058.

In the present paper the dynamics of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting is studied. We give out all the possible ranges of parameters for which the model has up to five equilibria. We prove that these equilibria can be topological saddles, nodes, foci, centers, saddle-nodes, cusps of codimension 2 or 3. Numerous kinds of bifurcations also occur, such as the transcritical bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Several numerical simulations are carried out to illustrate the validity of our results.

Citation: Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065
##### References:
 [1] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program for Scientific Translations, Jerusalem-London, 1973. doi: 10.1016/0022-0396(77)90136-X. [2] R. I. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421. [3] 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. [4] F. D. Chen, On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33-49.  doi: 10.1016/j.cam.2004.10.001. [5] S. N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511665639. [6] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985.  doi: 10.1016/0025-5564(87)90046-0. [7] C. W. Clark, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317-337. [8] C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Re-sources, John Wiley & Sons, Inc., New York-London-Sydney, 1976. [9] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity withnilponent linear part. The cusp case of codimension 3, Ergodic Theor. Dyn. Syst., 7 (1987), 375-413.  doi: 10.1017/S0143385700004119. [10] T. C. Gard, Persistence in food webs: Holling-type food chains, Math. Biosci., 49 (1980), 61-67.  doi: 10.1016/0025-5564(80)90110-8. [11] Y. Gong and J. Huang, Bogdanove-Takens bifurcations in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Apple. Sinica Eng. Ser., 30 (2014), 239-244.  doi: 10.1007/s10255-014-0279-x. [12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.  doi: 10.1007/2F978-1-4612-1140-2. [13] R. P. Gupta, M. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6. [14] R. P. Gupta, P. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear harvesting, Disc. Cont. Dyna. Sys. Ser. B., 20 (2015), 423-443.  doi: 10.3934/dcdsb.2015.20.423. [15] 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. [16] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5. [17] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201. [18] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Disc. Cont. Dyna. Sys. Ser. B., 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101. [19] C. Jost, O. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey model, J. Comput. Bull. Math. Biol., 61 (1999), 19-32.  doi: 10.1006/bulm.1998.0072. [20] S. V. Krishna, P. D. N. Srinivasu and B. Kaymackcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.  doi: 10.1006/bulm.1997.0023. [21] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105. [22] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2 edition, Springer-verlag, New York, 1998.  doi: 10.1003/978-1-4757-3978-7. [23] Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type Ⅲ functional response, J. Dynam. Diff. Equ., 20 (2008), 535-571.  doi: 10.1007/s10884-008-9102-9. [24] K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Analysis: RAW., 12 (2011), 1961-1973.  doi: 10.1016/j.nonrwa.2010.12.012. [25] K. Q. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys., 32 (2012), 901-933.  doi: 10.3934/dcds.2012.32.901. [26] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with non-constant harvesting, Disc. Cont. Dyna. Sys. Ser. S., 1 (2008), 303-315.  doi: 10.3934/dcdss.2008.1.303. [27] P. Lenzini and J. Rebaza, Non-constant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci, 4 (2010), 791-803. [28] T. Lindstrom, Qualitative analysisnof a predator-prey system with limit cycles, J. Math. Biol., 31 (1993), 541-561.  doi: 10.1007/BF00161198. [29] J. M. Lorca, E. G. Olivares and B. G. Yanez, The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, In: Mondaini, R. (ed. ) Proceedings of the International Symposium on Mathematical and Computational Biology: BIOMAT 2006, E-papers Servicos Editoriais Ltda. , R'io de Janeiro, (2007), 105–132. [30] A. Lotka, Elements of Mathematical Biology, Dover, New York, 1958.  doi: 10.1002/jps.3030471044. [31] W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, 2003. [32] P. J. Pal, S. Sarwardi, T. Saha and P. K. Mandal, Mean square stability in a modified Leslie-Gower and holling-type Ⅱ predator-prey model, J. Appl. Math. Inform., 29 (2011), 781-802. [33] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996.  doi: 10.1007/978-1-4684-0249-0. [34] E. C. Pielou, An Introduction to Mathematical Ecology, 2 edition, John Wiley & Sons, New York, 1977.  doi: 10.1002/bimj.19710130308. [35] D. Xiao and L. S. 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. [36] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Inst. Commun., 21 (1999), 493-506. [37] C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys. Ser. B., 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.

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##### References:
 [1] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program for Scientific Translations, Jerusalem-London, 1973. doi: 10.1016/0022-0396(77)90136-X. [2] R. I. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421. [3] 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. [4] F. D. Chen, On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33-49.  doi: 10.1016/j.cam.2004.10.001. [5] S. N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511665639. [6] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985.  doi: 10.1016/0025-5564(87)90046-0. [7] C. W. Clark, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317-337. [8] C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Re-sources, John Wiley & Sons, Inc., New York-London-Sydney, 1976. [9] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity withnilponent linear part. The cusp case of codimension 3, Ergodic Theor. Dyn. Syst., 7 (1987), 375-413.  doi: 10.1017/S0143385700004119. [10] T. C. Gard, Persistence in food webs: Holling-type food chains, Math. Biosci., 49 (1980), 61-67.  doi: 10.1016/0025-5564(80)90110-8. [11] Y. Gong and J. Huang, Bogdanove-Takens bifurcations in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Apple. Sinica Eng. Ser., 30 (2014), 239-244.  doi: 10.1007/s10255-014-0279-x. [12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.  doi: 10.1007/2F978-1-4612-1140-2. [13] R. P. Gupta, M. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6. [14] R. P. Gupta, P. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear harvesting, Disc. Cont. Dyna. Sys. Ser. B., 20 (2015), 423-443.  doi: 10.3934/dcdsb.2015.20.423. [15] 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. [16] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320.  doi: 10.4039/Ent91293-5. [17] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201. [18] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Disc. Cont. Dyna. Sys. Ser. B., 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101. [19] C. Jost, O. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey model, J. Comput. Bull. Math. Biol., 61 (1999), 19-32.  doi: 10.1006/bulm.1998.0072. [20] S. V. Krishna, P. D. N. Srinivasu and B. Kaymackcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.  doi: 10.1006/bulm.1997.0023. [21] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105. [22] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2 edition, Springer-verlag, New York, 1998.  doi: 10.1003/978-1-4757-3978-7. [23] Y. Lamontagne, C. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type Ⅲ functional response, J. Dynam. Diff. Equ., 20 (2008), 535-571.  doi: 10.1007/s10884-008-9102-9. [24] K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Analysis: RAW., 12 (2011), 1961-1973.  doi: 10.1016/j.nonrwa.2010.12.012. [25] K. Q. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys., 32 (2012), 901-933.  doi: 10.3934/dcds.2012.32.901. [26] B. Leard, C. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with non-constant harvesting, Disc. Cont. Dyna. Sys. Ser. S., 1 (2008), 303-315.  doi: 10.3934/dcdss.2008.1.303. [27] P. Lenzini and J. Rebaza, Non-constant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci, 4 (2010), 791-803. [28] T. Lindstrom, Qualitative analysisnof a predator-prey system with limit cycles, J. Math. Biol., 31 (1993), 541-561.  doi: 10.1007/BF00161198. [29] J. M. Lorca, E. G. Olivares and B. G. Yanez, The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, In: Mondaini, R. (ed. ) Proceedings of the International Symposium on Mathematical and Computational Biology: BIOMAT 2006, E-papers Servicos Editoriais Ltda. , R'io de Janeiro, (2007), 105–132. [30] A. Lotka, Elements of Mathematical Biology, Dover, New York, 1958.  doi: 10.1002/jps.3030471044. [31] W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, 2003. [32] P. J. Pal, S. Sarwardi, T. Saha and P. K. Mandal, Mean square stability in a modified Leslie-Gower and holling-type Ⅱ predator-prey model, J. Appl. Math. Inform., 29 (2011), 781-802. [33] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996.  doi: 10.1007/978-1-4684-0249-0. [34] E. C. Pielou, An Introduction to Mathematical Ecology, 2 edition, John Wiley & Sons, New York, 1977.  doi: 10.1002/bimj.19710130308. [35] D. Xiao and L. S. 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. [36] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Inst. Commun., 21 (1999), 493-506. [37] C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys. Ser. B., 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.
The number of interior equilibriums of system (4)
There is no interior equilibrium
A unique interior equilibrium $E_2$
A unique interior equilibrium $E_3$
The bi-stability occurred
A stable limit cycle
Two limit cycles
An unstable limit cycle
A cusp of codimension 2
An unstable limit cycle
An unstable homoclinic loop
A saddle and a stable focus
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