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Dynamics of a Modified Predator-Prey System to allow for a Functional Response and Time Delay

Published online by Cambridge University Press:  19 October 2016

Wei Liu  and
Yaolin Jiang
Wei Liu*
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China School of Mathematics and Computer Science, Xinyu University, Xinyu 338004, China
Yaolin Jiang*
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
*
*Corresponding author. Email addresses:[email protected] (W. Liu), [email protected] (Y. Jiang)
*Corresponding author. Email addresses:[email protected] (W. Liu), [email protected] (Y. Jiang)
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Abstract

A modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.

Keywords

Predator-prey systemharvestingstabilitytime delayperiodic solutions

MSC classification

Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 4 , November 2016 , pp. 384 - 399
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Brauer, F. and Soudack, A.C., Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol. 8, 55–71 (1979).Google Scholar
[2] Brauer, F. and Soudack, A.C., Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol. 12, 101–114 (1981).CrossRefGoogle Scholar
[3] Chen, B.S., Liao, X.X. and Liu, Y.Q., Normal forms and bifurcations for the differential-algebraic systems (in Chinese), Acta Math. Appl. Sin. 23, 429–433 (2000).Google Scholar
[4] Chen, L.S., Mathematical Models and Methods in Ecology (in Chinese), Science Press, Beijing (1988).Google Scholar
[5] Gause, G.F., The Struggle for Existence, Hafner Publishing Co. Inc., New York (1934).Google Scholar
[6] Gordon, H.S., Economic theory of a common property resource: The fishery, J. Polit. Econ. 62, 124–142 (1954).Google Scholar
[7] Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983).Google Scholar
[8] Hassard, B., Kazarinoff, D. and Wan, Y., Theory and Applications of Hopf Bifurcation, Cambridge University Press (1981).Google Scholar
[9] Jost, C., Comparing predator-prey models qualitatively and quantitatively with ecological time-series data, PhD Thesis, Institut National Agronomique Paris-Grignon (1998).Google Scholar
[10] Kar, T.K. and Pahari, U.K., Non-selective harvesting in prey-predator models with delay, Commun. Nonlinear Sci. Numer. Simul. 11, 499–509 (2006).CrossRefGoogle Scholar
[11] Kot, M., Elements of Mathematical Biology, Cambridge University Press (2001).Google Scholar
[12] Wu, X.Y. and Chen, B.S., Bifurcations and stability of a discrete singular bioeconomic system, Nonlinear Dyn. 73, 1813–1828 (2013).Google Scholar
[13] Liu, S., Chen, L. and Luo, G., Extinction and permanence in competitive stage structured system with time-delays, Nonlinear Anal. Th. Meth. Appl. 51, 1347–1361 (2002).Google Scholar
[14] Lucas, W.F., Modules in Applied Mathematics: Differential Equation Models, Springer, New York (1983).Google Scholar
[15] Ou, Y.J. and Yang, D.Q., On qualitative analysis of functional reaction function prey-predacious system (in Chinese), J. Biomath. 13, 329–333 (1998).Google Scholar
[16] Rosenzweig, M.L., Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science 171, 385–387 (1971).CrossRefGoogle ScholarPubMed
[17] Ruan, S. and Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10. 863–874 (2003).Google Scholar
[18] Wang, W., Global dynamics of a populationmodel with stage structure a predator-prey system with stage structure for predator, in: Chen, L., Ruan, S., Zhu, J. (Eds.), Advanced Topics in Biomathematics, pp. 253–257, World Scientific (1998).Google Scholar
[19] Yan, X. and Li, W., Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Appl. Math. Comput. 177, 427–445 (2006).Google Scholar
[20] Chen, B.S. and Chen, J.J., Bifurcation and chaotic behavior of a discrete singular biological economic system, Appl. Math. Comp. 219, 2371–2386 (2012).CrossRefGoogle Scholar
[21] Zhang, G.D., Shen, Y. and Chen, B.S., Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn. 73, 2119–2131 (2013).CrossRefGoogle Scholar
[22] Zhang, G.D., Shen, Y. and Chen, B.S., Bifurcation analysis in a discrete differential-algebraic predator-prey system, Applied Math. Modelling 38, 4835–4848 (2014).Google Scholar