ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Hopf Bifurcation and Stability Analysis of a Predator-Prey System with Holling Type IV Functional Response

Journal of Environmental Accounting and Management 7(4) (2018) 337--348 | DOI:10.5890/JAND.2018.12.002

Z. Lajmiri$^{1}$, R. Khoshsiar Ghaziani$^{2}$, I. Orak$^{1}$

$^{1}$ Sama technical and vocational training college, Islamic Azad University , Izeh branch, Izeh Iran

$^{2}$ Department of Applied Mathematics, Shahrekord University, P.O.Box. 115, Shahrekord, Iran

Abstract

In this paper, we investigate the dynamical complexities of a predator-prey model with Holling type IV functional response, which describes interaction between two populations of prey and predator. We perform a bifurcation analysis of this model analytically and numerically. Our bifurcation analysis indicates that the system exhibits numerous types of codimension one and two bifurcations including fold, subcritical and supercritical Hopf, cusp and Bogdanov-Takens. By numerical continuation method, we also compute several curves of equilibria and bifurcations. Further, by numerical simulations we reveal more complex dynamics of the model.

References

1.  [1] Gao, Y. (2013), Dynamics of a ratio-dependent predator-prey system with a strong Allee effect, Discrete and continuous dynamical systems series B, 18(9), 2283-2313.
2.  [2] Kuznetsov, Y. (1998), Elements of applied bifurcation theory, 112, Springer Verlag.
3.  [3] Agarwal, M. and Pathak, R. (2012), Harvesting and Hopf bifurcation in a prey-predator model with Holling type IV functional response, International Journal of Mathematics and Soft Computing, 2(1), 83-92.
4.  [4] Aziz-Alaoui, M.A. and Daher Okiye, M. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075.
5.  [5] Camara, B.I. and Aziz-Alaoui, M.A. (2008), Complexity in a prey predator model, International conference in honor of Claude Lobry, 9, 109-122.
6.  [6] Chen, S., shi, J., and wei, J. (2013), The effect of delay on a diffusive predator-prey system with Holling type-II predator functional response, Communications on pure and applied analysis, 12(1), 481-501.
7.  [7] Lu, H. and Wang, W. (2000), Application of fractional calculus in physics, world scientific.
8.  [8] Mukherjee, D., das, P., and kesh, D., Dynamics of a plant-herbivore model with Holling type II functional response, J. Math.
9.  [9] Zhang, L., Wang, W., Xue, Y., and Jin, Z. (2008), Complex dynamics of a Holling-type IV predator-prey model, 1-23.
10.  [10] Zhang, Z.Z and Yang H.Z. (2013), Hopf bifurcation in a delayed predator-prey system with modified Leslie- Gower and Holling type III schemes, Acta Automatica Sinica, 39(5), 610-616.
11.  [11] Banerjee, M. and Banerjee, S. (2012), Turing instabilities and spatiotemporal chaos in ratio-dependent Holling-Tanner model, Math. Biosci, 236, 64-76.
12.  [12] Holling, C.S. (1965), The functional response of predator to prey density and its role in mimicry and popu- lation regulation, Mem. Entomol. Sec. Can., 45, 1-60.
13.  [13] Murray, J.D. (1989), Mathematical Biology, Springer, Berlin.
14.  [14] Skalski, G. and Gilliam, J.F. (2001), Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82, 3083-3092.
15.  [15] Ruan, S. and Xiao, D. (2001), Global analysis in a predator-prey system with nonmonotonic functional response, Society for industrial and applied mathematics, 61(4), 1445-1472.
16.  [16] Vijaya Lakshmi, G.M., Vijaya, S., and Gunasekaran, M. (2014), Bifurcation and stability analysis in dynamics of prey-predator model with holling type IV functional response and intra-specific competition, International Journal Of Engineering And Science, 4, 52-61.
17.  [17] Perko, L. (2000), Differential Equations and and Dynamical Systems, New York, Springer-Verlag.
18.  [18] Wiggins, S. (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Verlag New York Berlin Heidelberg.
19.  [19] Chow, S., Li, C., andWang, D. (2011), Dynamics of a delayed discrete semi-ratio dependent predator-prey system with Holling type IV functional response, Advances in Difference Equations, 3-19.
20.  [20] Agiza, H.N., Elabbasy, E.M., Metwally, E.K., and Elsadany, A.A. (2003), Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075.
21.  [21] Allgower, E.L. and Georg, K. (1990), Numerical Continuation Methods: An Introduction, Springer- Verlag, Berlin.
22.  [22] Wang, Y.H. (2009), Numerical algorithm based on Adomian decomposition for fractional differential equa- tions, Computer and Mathematics with Applications, 57, 1627-1681.