Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-12T10:32:09.407Z Has data issue: false hasContentIssue false
  • English
  • Français

HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS

Part of: Local and nonlocal bifurcation theory Functional-differential and differential-difference equations

Published online by Cambridge University Press:  05 June 2014

E. KARAOGLU  and
H. MERDAN
E. KARAOGLU
Affiliation:
TOBB University of Economics and Technology, Faculty of Arts and Sciences, Department of Mathematics, Söğütözü 06530, Ankara, Turkey email [email protected], [email protected]
H. MERDAN*
Affiliation:
TOBB University of Economics and Technology, Faculty of Arts and Sciences, Department of Mathematics, Söğütözü 06530, Ankara, Turkey email [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

Keywords

Hopf bifurcationdelay differential equationtime delaystabilityperiodic solutionspopulation dynamics

MSC classification

Secondary: 34K18: Bifurcation theory 34K20: Stability theory 37G15: Bifurcations of limit cycles and periodic orbits 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 3 , January 2014 , pp. 214 - 231
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Akkocaoğlu, H., Merdan, H. and Çelik, C., “Hopf bifurcation analysis of a general non-linear differential equation with delay”, J. Comput. Appl. Math. 237 (2013) 565–575; doi:10.1016/j.cam.2012.06.029.CrossRefGoogle Scholar
Allen, L. J. S., An introduction to mathematical biology (Pearson/Prentice Hall, 2007).Google Scholar
Arditi, R. and Ginzburg, L. R., “Coupling in predator prey dynamics: ratio-dependence”, J. Theoret. Biol. 139 (1989) 311–326.CrossRefGoogle Scholar
Balachandran, B., Kalmar-Nagy, T. and Gilsinn, D. E., Delay differential equations: recent advances and new directions (Springer, Berlin, 2009).Google Scholar
Belair, J. and Campbell, S. A., “Stability and bifurcations of equilibrium in a multiple-delayed differential equation”, SIAM J. Appl. Math. 94 (1994) 1402–1424; doi:10.1137/S0036139993248853.CrossRefGoogle Scholar
Bellman, R. and Cooke, K. L., Differential-difference equations (Academic Press, New York, 1963).Google Scholar
Çelik, C., “The stability and Hopf bifurcation for a predator–prey system with time delay”, Chaos Solitons Fractals 37 (2008) 87–99; doi:10.1016/j.chaos.2007.10.045.CrossRefGoogle Scholar
Çelik, C., “Hopf bifurcation of a ratio-dependent predator–prey system with time delay”, Chaos Solitons Fractals 42 (2009) 1474–1484; doi:10.1016/j.chaos.2009.03.071.CrossRefGoogle Scholar
Çelik, C. and Merdan, H., “Hopf bifurcation analysis of a system of coupled delayed-differential equations”, Appl. Math. Comput. 219 (2013) 6605–6617; doi:10.1016/j.amc.2012.12.063.Google Scholar
Cooke, K. L. and van den Driessche, P., “On zeros of some transcendental equations”, Funkcial. Ekvac. 29 (1986) 77–90.Google Scholar
Cooke, K. L. and Grossman, Z., “Discrete delay, distributed delay and stability switches”, J. Math. Anal. Appl. 86 (1982) 592–627.CrossRefGoogle Scholar
Frisch, R. and Holme, H., “The characteristic solutions of a mixed difference and differential equation occurring in economic dynamics”, Econometrica 3 (1935) 225–239.CrossRefGoogle Scholar
Hale, J., Theory of functional differential equations (Springer, Berlin, 1977).CrossRefGoogle Scholar
Hassard, N. D. and Kazarinoff, Y. H., Theory and applications of Hopf bifurcation (Cambridge University Press, Cambridge, 1981).Google Scholar
Kuang, Y., Delay differential equations with application in population dynamics (Academic Press, New York, 1993).Google Scholar
Leslie, P. H., “Some further notes on the use of matrices in population mathematics”, Biometrika 35 (1948) 213–245.CrossRefGoogle Scholar
Li, X., Ruan, S. and Wei, J., “Stability and bifurcation in delay-differential equations with two delays”, J. Math. Anal. Appl. 236 (1999) 254–280; doi:10.1006/jmaa.1999.6418.CrossRefGoogle Scholar
Li, K. and Wei, J., “Stability and Hopf bifurcation analysis of a prey–predator system with two delays”, Chaos Solitons Fractals 42 (2009) 2606–2613; doi:10.1016/j.chaos.2009.04.001.CrossRefGoogle Scholar
Mackey, M. and Glass, L., “Oscillations and chaos in physiological control systems”, Science 197 (1977) 287–289.CrossRefGoogle ScholarPubMed
Murray, J. D., Mathematical biology (Springer-Verlag, Berlin, 2002).CrossRefGoogle Scholar
Ramussen, H., Wake, G. C. and Donaldson, J., “Analysis of a class of distributed delay logistic differential equations”, Math. Comput. Modelling 38 (2003) 123–132; doi:10.1016/S0895-7177(03)00196-1.CrossRefGoogle Scholar
Ruan, S. and Wei, J., “On the zero of some transcendental functions with applications to stability of delay differential equations with two delays”, Dyn. Contin. Discrete Impuls. Syst. Ser. 10 (2003) 863–874.Google Scholar
Wei, J. and Ruan, S., “Stability and bifurcation in a neural network model with two delays”, Physica D 30 (1999) 255–272.CrossRefGoogle Scholar
Wei, J. and Ruan, S., “Stability and global Hopf bifurcation for neutral differential equations”, Acta Math. Sin. 45 (2002) 93–104.Google Scholar
Yafia, R., “Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response”, Discrete Dyn. Nat. Soc. 2006 (2006) 1–9; doi:10.1155/DDNS/2006/58463.Google Scholar
Yan, X. P. and Chu, Y. D., “Stability and bifurcation analysis for a delayed Lotka–Volterra predator–prey system”, J. Comput. Appl. Math. 196 (2006) 198–210; doi:10.1016/j.cam.2005.09.001.CrossRefGoogle Scholar
Zhou, S. R., Liu, Y. F. and Wang, G., “The stability of predator–prey systems subject to the Allee effects”, Theor. Populat. Biol. 67 (2005) 23–31; doi: 10.1016j/j.tpb.2004.06.007.CrossRefGoogle Scholar