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

Changrong Zhu; Lei Kong

doi:10.3934/dcdss.2017065

Article Contents

2017, Volume 10, Issue 5: 1187-1206. Doi: 10.3934/dcdss.2017065

This issue Previous Article Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term Next Article

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

Changrong Zhu and
Lei Kong^,

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

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

Received: October 31, 2016

Revised: December 31, 2016

Published: October 2017

Acknowledgement: This work was supported by NSFC grant 11671058.

Abstract Full Text(HTML) Figure(12) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 70K05, 34D20; Secondary: 34C23.

Citation:

Full Text(HTML)

Figure 1. The number of interior equilibriums of system (4)

Download: Full-size image PowerPoint slide

Figure 2. There is no interior equilibrium

Download: Full-size image PowerPoint slide

Figure 3. A unique interior equilibrium $E_2$

Download: Full-size image PowerPoint slide

Figure 4. A unique interior equilibrium $E_3$

Download: Full-size image PowerPoint slide

Figure 5. The bi-stability occurred

Download: Full-size image PowerPoint slide

Figure 6. A stable limit cycle

Download: Full-size image PowerPoint slide

Figure 7. Two limit cycles

Download: Full-size image PowerPoint slide

Figure 8. An unstable limit cycle

Download: Full-size image PowerPoint slide

Figure 9. A cusp of codimension 2

Download: Full-size image PowerPoint slide

Figure 10. An unstable limit cycle

Download: Full-size image PowerPoint slide

Figure 11. An unstable homoclinic loop

Download: Full-size image PowerPoint slide

Figure 12. A saddle and a stable focus

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

	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.
	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.
	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.
	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.
	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.
	C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. doi: 10.1016/0025-5564(87)90046-0.
	C. W. Clark , Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979) , 317-337.
	C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Re-sources, John Wiley & Sons, Inc., New York-London-Sydney, 1976.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2 edition, Springer-verlag, New York, 1998. doi: 10.1003/978-1-4757-3978-7.
	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.
	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.
	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.
	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.
	P. Lenzini and J. Rebaza , Non-constant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci, 4 (2010) , 791-803.
	T. Lindstrom , Qualitative analysisnof a predator-prey system with limit cycles, J. Math. Biol., 31 (1993) , 541-561. doi: 10.1007/BF00161198.
	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.
	A. Lotka, Elements of Mathematical Biology, Dover, New York, 1958. doi: 10.1002/jps.3030471044.
	W. Murdoch, C. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, 2003.
	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.
	L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0.
	E. C. Pielou, An Introduction to Mathematical Ecology, 2 edition, John Wiley & Sons, New York, 1977. doi: 10.1002/bimj.19710130308.
	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.
	D. Xiao and S. Ruan , Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Inst. Commun., 21 (1999) , 493-506.
	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.