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Lajmiri, Z.; Khoshsiar Ghaziani, R.; Orak, Iman

Bifurcation and stability analysis of a ratio-dependent predator-prey model with predator harvesting rate. (English) Zbl 1392.93040

Chaos Solitons Fractals 106, 193-200 (2018).
Summary: In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov-Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate.

Cited in 18 Documents

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34C23 Bifurcation theory for ordinary differential equations
37G10 Bifurcations of singular points in dynamical systems
37N25 Dynamical systems in biology

Keywords:

Hopf bifurcation; Bogdanov-Takens bifurcation; cusp bifurcation; dynamical behavior; limit cycle

Software:

MATCONT
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Full Text: DOI

References:

[1] Allgower, E. L.; Georg, K., Numerical continuation methods: an introduction, (1990), Springer- Verlag Berlin · Zbl 0717.65030
[2] Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. G., Qualitative theory of second-order dynamical systems, (1973), John Wiley and Sons New York · Zbl 0282.34022
[3] Beyn, W. J.; Champneys, A.; Doedel, E.; Govaerts, W.; Kuznetsov, Y. A.; Sandstede, B., Numerical continuation and computation of normal forms, (Fiedler, B.; Iooss, G.; Kopell, N., Handbook of Dynamical Systems, vol. 2, (2002), Elsevier), 149-219 · Zbl 1034.37048
[4] Chakraborty, S.; Pal, S.; Bairagi, N., Predator-prey interaction with harvesting: mathematical study with biological ramifications, Appl Math Model, 36, (2012) · Zbl 1252.34038
[5] Dhooge, A.; Govaerts, W.; Kuznetsov, Y. A., MATCONT: a MATLAB package for numerical bifurcation analysis of odes, ACM Trans Math Software, 29, 2, 141-164, (2003) · Zbl 1070.65574
[6] W. Govaerts W., Y. A.Kuznetsov A., Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM J Sci Comput, 27, 1, 231-252.; W. Govaerts W., Y. A.Kuznetsov A., Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM J Sci Comput, 27, 1, 231-252. · Zbl 1087.65118
[7] Govaerts, W., Numerical methods for bifurcations of dynamical equilibria, (2000), SIAM Philadelphia · Zbl 0935.37054
[8] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J Appl Math, 42, 489-506, (2001) · Zbl 0984.92035
[9] Heggerad, C. M.; Lan, K., A Lyapunov function for lesliegower predator prey model, Appl Math Lett, 14, 697-699, (2001) · Zbl 0999.92036
[10] Huang, J.; Liu, S., Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting, Commun Pure Appl Anal, 15, 3, 1041-1055, (2016) · Zbl 1347.34060
[11] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator prey system, J Math Biol, 36, 389-406, (1998) · Zbl 0895.92032
[12] Lamontagne, Y.; Coutu, C.; Rousseau, C., Bifurcation analysis of a predator-prey system with generalized Holling type III functional response, J Dynam Differ Equ, 20, 3, 535-571, (2008) · Zbl 1160.34047
[13] Lan, K. Q.; Zhu, C. R., Phase portraits of predator-prey systems with harvesting rates, Discret Contin Dyn Syst Ser-A, 32, 3, 901-933, (2012) · Zbl 1252.34052
[14] Lan, K. Q.; Zhu, C. R., Phase portraits, Hopf bifurcations and limit cycles of the Holling-tanner models for predator prey interactions, nonlinear anal, Signal Process Real World Appl, 12, 1961-1973, (2011) · Zbl 1220.34068
[15] Lenzini, P.; Rebaza, J., Nonconstant predator harvesting on ratio-dependent predator-prey models, Appl Math Sci, 4, 16, 791-803, (2010) · Zbl 1189.37098
[16] Liang, Z.; Pan, H., Qualitative analysis of a ratio-dependent Holling-tanner model, J Math Anal Appl, 334, 954-964, (2007) · Zbl 1124.34030
[17] Li, B.; Kuang, Y., Heteroclinic bifurcations in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J Appl Math, 67, 5, 1453-1464, (2007) · Zbl 1132.34320
[18] Luo, G. P.; Lan, K. Q.; Zhu, C. R., Bifurcations of ratio-dependent predator prey Holling type III systems with harvesting rates, J Nonlinear Funct Anal, 16, (2014)
[19] Murray, J. D., Differential equations and dynamical systems, (1996), Springer-Verlag New York · Zbl 0854.34001
[20] Perko, L., Mathematical biology, (1989), Springer Berlin · Zbl 0682.92001
[21] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J Appl Math, 61, 4, 1445-1472, (2001) · Zbl 0986.34045
[22] Scheffer, M.; Carpenter, S.; Foley, J. A.; Folke, C.; Walker, B., Catastrophic shifts in ecosystems, Nature, 413, 591-596, (2001)
[23] Tang, Y.; Zhang, W.; Foley, J. A.; Folke, C.; Walker, B., Heteroclinic bifurcations in a ratio-dependent predator-prey system, J Math Biol, 50, 699-712, (2005) · Zbl 1067.92050
[24] Xiao, D., Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting, Discret Contin Dyn Syst - Ser B, 21, 2, 699-719, (2015) · Zbl 1331.34101
[25] Xiao, Q.; Dai, B.; Xu, B.; Bao, L., Homoclinic bifurcation for a general state-dependent Kolmogorov type predator prey model with harvesting, Nonlinear Anal Real World Appl, 26, 263-273, (2015) · Zbl 1331.34102
[26] Xiao D., Jennings L.S., SIAM J.A.M.. Bifurcations of a ratio-dependent predator prey system with constant rate harvesting. 2005. 65, 737-753.; Xiao D., Jennings L.S., SIAM J.A.M.. Bifurcations of a ratio-dependent predator prey system with constant rate harvesting. 2005. 65, 737-753. · Zbl 1094.34024
[27] Xiao, D. M.; Li, W. X.; Han, M., Dynamics in ratio-dependent predator-prey model with predator harvesting, J Math Anal Appl, 324, 1, 14-29, (2006) · Zbl 1122.34035
[28] Zhu, C. R.; Lan, K. Q., Phase portraits, Hopf bifurcations and limit cycles of Leslie-gower predator-prey systems with harvesting rates, Discrete Contin Dyn Syst Ser-B, 14, 289-306, (2010) · Zbl 1201.34065
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