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

  • Wei Liu (a1) (a2) and Yaolin Jiang (a1)
Abstract
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.

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*Corresponding author. Email addresses: wliu2015@163.com (W. Liu), yljiang@mail.xjtu.edu.cn (Y. Jiang)
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[1] F. Brauer and A.C. Soudack , Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol. 8, 5571 (1979).

[2] F. Brauer and A.C. Soudack , Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol. 12, 101114 (1981).

[4] L.S. Chen , Mathematical Models and Methods in Ecology (in Chinese), Science Press, Beijing (1988).

[5] G.F. Gause , The Struggle for Existence, Hafner Publishing Co. Inc., New York (1934).

[6] H.S. Gordon , Economic theory of a common property resource: The fishery, J. Polit. Econ. 62, 124142 (1954).

[7] J. Guckenheimer and P. Holmes , Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983).

[10] T.K. Kar and U.K. Pahari , Non-selective harvesting in prey-predator models with delay, Commun. Nonlinear Sci. Numer. Simul. 11, 499509 (2006).

[12] X.Y. Wu and B.S. Chen , Bifurcations and stability of a discrete singular bioeconomic system, Nonlinear Dyn. 73, 18131828 (2013).

[13] S. Liu , L. Chen and G. Luo , Extinction and permanence in competitive stage structured system with time-delays, Nonlinear Anal. Th. Meth. Appl. 51, 13471361 (2002).

[16] M.L. Rosenzweig , Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science 171, 385387 (1971).

[19] X. Yan and W. Li , Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Appl. Math. Comput. 177, 427445 (2006).

[20] B.S. Chen and J.J. Chen , Bifurcation and chaotic behavior of a discrete singular biological economic system, Appl. Math. Comp. 219, 23712386 (2012).

[21] G.D. Zhang , Y. Shen and B.S. Chen , Hopf bifurcation of a predator-prey system with predator harvesting and two delays, Nonlinear Dyn. 73, 21192131 (2013).

[22] G.D. Zhang , Y. Shen and B.S. Chen , Bifurcation analysis in a discrete differential-algebraic predator-prey system, Applied Math. Modelling 38, 48354848 (2014).

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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