Stability and Bifurcation Analysis in a Food Chain Model with Delay

Abstract:

In this paper, we investigate a delayed three-species food chain model. The existence and stability of equilibria are obtained. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory.

• Periodical: Applied Mechanics and Materials (Volumes 110-116)
• Pages: 3382-3388
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.110-116.3382
• Online since: October 2011
• Authors: Zhang Li
• Keywords: Delay, Food Chain Model, Hopf Bifurcation, Stability

References

[1] D. Strong, Are trophic cascades all wet? Differentiation and donor control in species ecosystems, Ecology, vol. 73, pp.747-754 , June (1992).

DOI: 10.2307/1940154

[2] G. Polis and D. Strong, Food web complexity and community dynamics, Am. Nat., vol. 147, pp.813-846 , May (1996).

[3] M. Arim and P. Marquet, Intraguild predation: a widespread interaction related to species biology, Ecol. Lett., vol. 7, pp.557-564 , June (2004).

DOI: 10.1111/j.1461-0248.2004.00613.x

[4] I. Perfecto, J. Vandermeer, G. L_opez Bautista, G. Ibarra Nu~nez, R. Greenberg, P. Bichier, and S. Langridge, Greater predation in shaded coffee farms: the role of resident neotropical birds, Ecology, vol. 85, pp.2677-2681, October (2004).

DOI: 10.1890/03-3145

[5] W. Fagan, Omnivory as a stabilizing feature of natural communities, Am. Nat., vol. 150, pp.554-567 , November (1997).

DOI: 10.1086/286081

[6] R. Arditi, N. Perrin and H. Saiah, Functional responses and heterogeneities: an experimental test with cladocerans, OIKOS, vol. 60, pp.69-75 , Febrary (1991).

DOI: 10.2307/3544994

[7] A. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholsons blowflies as an example, Ecology, vol. 73, pp.1552-1563, October (1992).

DOI: 10.2307/1940008

[8] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., vol. 36, pp.389-406, July (1998).

DOI: 10.1007/s002850050105

[9] C. Holling, The functional response of predator to prey density and its role in mimcry and population regulation, Mem Ent Sec Can, vol. 45, pp.1-60 , May (1965).

[10] J. Andrews, A mathematical model for the continuous culture of microorganism utilizing inhibitory sunbstrates, Biotechnol Bioeng, vol. 10, pp.707-723, March (1968).

DOI: 10.1002/bit.260100602

[11] S. Zhang, F. Wang and L. Chen, A food chain model with impulsive perturbations and Holling IV functional response, vol. 26, pp.855-866 , November (2005).

DOI: 10.1016/j.chaos.2005.01.053

[12] R. Kara and M. Can, Ratio-dependent food chain models with three trophic levels, International journal of computer science, vol. 1, pp.85-91 , August (2006).

[13] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependant arameters, SIAM. J. Math. Anal., vol. 33, pp.1144-1165 , April (2002).

DOI: 10.1137/s0036141000376086

[14] S. Ruan and J. Wei, On the zeros of a thirds degree exponential polynomial with applications to a delayed model for control of testosterone sectetion, Journal of Mathematics Applied inMedicine and Biology, vol. 18, pp.41-52 , March (2001).

DOI: 10.1093/imammb/18.1.41

[15] B. Hassard, N. Kazarinoff, and Y. Wan, Theory and Applications of Hopf Bifurcation, CambridgeCambridge Univ. Press, (1981).