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BIFURCATION ANALYSIS OF A LOGISTIC PREDATOR–PREY SYSTEM WITH DELAY

Published online by Cambridge University Press:  12 May 2016

CANAN ÇELİK Open the ORCID record for CANAN ÇELİK [Opens in a new window]  and
GÖKÇEN ÇEKİÇ Open the ORCID record for GÖKÇEN ÇEKİÇ [Opens in a new window]
CANAN ÇELİK
Affiliation:
Bahcesehir University, Faculty of Engineering and Natural Sciences, Ciragan Cad., Besiktas, 34353 Istanbul, Turkey email canan.celik@eng.bahcesehir.edu.tr
GÖKÇEN ÇEKİÇ*
Affiliation:
Istanbul University, Department of Mathematics, Vezneciler, 34134 Istanbul, Turkey email gokcen.cekic@gmail.com
*
✉gokcen.cekic@gmail.com
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Abstract

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We consider a coupled, logistic predator–prey system with delay. Mainly, by choosing the delay time ${\it\tau}$ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time ${\it\tau}$ passes some critical values. Based on the normal-form theory and the centre manifold theorem, we also derive formulae to obtain the direction, stability and the period of the bifurcating periodic solution at critical values of ${\it\tau}$. Finally, numerical simulations are investigated to support our theoretical results.

Keywords

predator–prey systemdelayed logistic differential equationHopf bifurcationstability

MSC classification

Primary: 92D25: Population dynamics (general)
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 4 , April 2016 , pp. 445 - 460
Copyright
© 2016 Australian Mathematical Society 

References

Chen, X., “Periodicity in a nonlinear discrete predator–prey system with state dependent delays”, Nonlinear Anal. Real World Appl. 8 (2007) 435–446; doi:10.1016/j.nonrwa.2005.12.005.Google Scholar
Evans, L. C., Partial differential equations (American Mathematical Society, Providence, RI, USA, 2010).Google Scholar
Faria, T., “Stability and bifurcation for a delayed predator–prey model and the effect of diffusion”, J. Math. Anal. Appl. 254 (2001) 433–463; doi:10.1006/jmaa.2000.7182.Google Scholar
Gopalsamy, K., “Time lags and global stability in two species competition”, Bull. Math. Biol. 42 (1980) 728–737; doi:10.1007/BF02460990.CrossRefGoogle Scholar
Hale, J. K., Theory of functional differential equations (Springer, New York, 1977).Google Scholar
Hassard, B. D., Kazarinoff, N. D. and Wan, Y.-H., Theory and applications of Hopf bifurcation (Cambridge University Press, Cambridge, 1981).Google Scholar
Krise, S. and Choudhury, S. R., “Bifurcations and chaos in a predator–prey model with delay and a laser-diode system with self-sustained pulsations”, Chaos Solitons Fractals 16 (2003) 59–77; doi:10.1016/S0960-0779(02)00199-6.Google Scholar
Kuang, Y., Delay differential equations with applications in population dynamics (Academic Press, Boston, 1993).Google Scholar
Kuznetsov, Yu. A., Elements of applied bifurcation theory (Springer, New York, 2004).CrossRefGoogle Scholar
Leung, A., “Periodic solutions for a prey–predator differential delay equation”, J. Differential Equations 26 (1977) 391–403; doi:10.1016/0022-0396(77)90087-0.CrossRefGoogle Scholar
Liu, B., Teng, Z. and Chen, L., “Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy”, J. Comput. Appl. Math. 193 (2006) 347–362; doi:10.1016/j.cam.2005.06.023.Google Scholar
Liu, X. and Xiao, D., “Complex dynamic behaviors of a discrete-time predator–prey system”, Chaos Solitons Fractals 32 (2007) 80–94; doi:10.1016/j.chaos.2005.10.081.Google Scholar
Liu, Z. and Yuan, R., “Stability and bifurcation in a harvested one-predator–two-prey model with delays”, Chaos Solitons Fractals 27 (2006) 1395–1407; doi:10.1016/j.chaos.2005.05.014.Google Scholar
Ma, W. and Takeuchi, Y., “Stability analysis on a predator–prey system with distributed delays”, J. Comput. Appl. Math. 88 (1998) 79–94; doi:10.1016/S0377-0427(97)00203-3.CrossRefGoogle Scholar
May, R. M., “Time delay versus stability in population models with two and three trophic levels”, Ecology 4 (1973) 315–325; doi:10.2307/1934339.CrossRefGoogle Scholar
Murray, J. D., Mathematical biology (Springer, New York, 1993).CrossRefGoogle Scholar
Ruan, S., “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator–prey systems with discrete delays”, Quart. Appl. Math. 59 (2001) 159–173; http://www.math.miami.edu/∼ruan/MyPapers/Ruan-QAM2001.pdf.Google Scholar
Ruan, S. and Wei, J., “Periodic solutions of planar systems with two delays”, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 1017–1032; doi:10.1017/S0308210500031061.Google Scholar
Song, Y. L. and Wei, J. J., “Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system”, J. Math. Anal. Appl. 301 (2005) 1–21; doi:10.1016/j.jmaa.2004.06.056.Google Scholar
Yan, X. P. and Li, W. T., “Hopf bifurcation and global periodic solutions in a delayed predator–prey system”, Appl. Math. Comput. 177 (2006) 427–445; doi:10.1016/j.amc.2005.11.020.Google Scholar
Yan, X. P. and Zhang, C. H., “Hopf bifurcation in a delayed Lotka–Volterra predator–prey system”, Nonlinear Anal. Real World Appl. 9 (2008) 114–127; doi:10.1016/j.nonrwa.2006.09.007.Google Scholar
Zhou, L. and Tang, Y., “Stability and Hopf bifurcation for a delay competition diffusion system”, Chaos Solitons Fractals 14 (2002) 1201–1225; doi:10.1016/S0960-0779(02)00068-1.Google Scholar