{"title":"Bifurcations of a Delayed Prototype Model","authors":"Changjin Xu","volume":68,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1123,"pagesEnd":1128,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/1950","abstract":"<p>In this paper, a delayed prototype model is studied. Regarding the delay as a bifurcation parameter, we prove that a sequence of Hopf bifurcations will occur at the positive equilibrium when the delay increases. Using the normal form method and center manifold theory, some explicit formulae are worked out for determining the stability and the direction of the bifurcated periodic solutions. Finally, Computer simulations are carried out to explain some mathematical conclusions.<\/p>\r\n","references":"[1] A. U\u00e7ar, A prototype model for chaos studies. Int. J. Eng. Sci. 40 (2002)\r\n251-258.\r\n[2] A. U\u00e7ar, On the chaotic behavior of the prototype delayed dynamical\r\nsystem. Chaos, Solitons and Fractals 16( 2003) 187-194.\r\n[3] M.S. Peng,Bifurcation and chaotic behavior in the Euler method for a\r\nU\u00e7ar prototype delay model. Chaos, Solitons and Fractals 22( 2004) 483-493.\r\n[4] C.G. LI, X.F. Liao and J.B. Yu. Hopf bifurcation in a prototype delayed\r\nsystem. Chaos, Solitons and Fractals 19 (2004) 779-787.\r\n[5] X.P. Yan and W.T. Li. Hopf bifurcation and global periodic solutions\r\nin a delayed predator-prey system. Appl. Math. Comput. 177(1) (2006) 427-445.\r\n[6] X.P. Yan and Y.D. Chu. Stability and bifurcation analysis for a delayed\r\nLotka-Volterra predator-prey system. J. Comput. Appl. Math. 196(1)\r\n(2006) 198-210.\r\n[7] X.P. Yan and C.H. Zhang. Hopf bifurcation in a delayed Lokta-Volterra\r\npredator-prey system. Nonlinear Anal.: Real World App. 9(1) (2008) 114-127.\r\n[8] X.P. Yan and C.H. Zhang. Direction of Hopf bifurcation in a delayed Lotka-Volterra competition diffusion system. Nonlinear Anal.: Real World\r\nAppl. 10(5) (2009) 2758-2773.\r\n[9] X.P. Yan and W.T. Li. Bifurcation and global periodic solutions in a\r\ndelayed facultative mutualism system. Physica D 227( 2007) 51-69.\r\n[10] Y.L. Song and J.J. Wei. Local Hopf bifurcation and global periodic\r\nsolutions in a delayed predator-prey system. J. Math. Anal. Appl. 301(1)\r\n(2005) 1-21.\r\n[11] Y.L. Song, Y.H. Peng and J.J. Wei. Bifurcations for a predator-prey\r\nsystem with two delays. J. Math. Anal. Appl. 337(1) (2008) 466-479.\r\n[12] Y.L. Song, S.L. Yuan and J.M. Zhang. Bifurcation analysis in the delayed\r\nLeslie-Gower predator-prey system. Appl. Math. Modelling 33(11) (2009)\r\n4049-4061.\r\n[13] Y.L. Song and S.L. Yuan. Bifurcation analysis in a predator-prey system\r\nwith time delay. Nonlinear Anal.: Real World Appl.7(2) (2006) 265-284.\r\n[14] S.L. Yuan and F.Q. Zhang. Stability and global Hopf bifurcation in a\r\ndelayed predator-prey system. Nonlinear Anal.: Real World Appl. 11(2)\r\n(2010) 959-977.\r\n[15] X.Y. Zhou, X.Y. Shi and X.Y. Song. Analysis of non-autonomous\r\npredator-prey model with nonlinear diffusion and time delay. Appl. Math.\r\nComput. 196 (2008) 129-136.\r\n[16] C.J. XU, X.H. Tang and M.X. Liao. Stability and bifurcation analysis\r\nof a delayed predator-prey model of prey dispersal in two-patch environments.\r\nAppl. Math. Comput. 216(10) (2010) 2920-2936.\r\n[17] C.J. XU, X.H. Tang, M.X. Liao. and X.F. He. Bifurcation analysis in a\r\ndelayed Lokta-Volterra predator-prey model with two delays. Nonlinear\r\nDynam., doi: 10.1007\/s11071-010-9919-8\r\n[18] C.J. XU, M.X. Liao. and X.F. He. Stability and Hopf bifurcation analysis\r\nfor a Lokta-Volterra predator-prey model with two delays. Int. J. Appl.\r\nMath. Comput. Sci. 21( 2011) 97-107.\r\n[19] C.J., Xu and Y.F. Shao, Bifurcations in a predator-prey model with discrete\r\nand distributed time delay. Nonlinear Dynam., doi: 10.1007\/s11071-\r\n011-0140-1\r\n[20] Y. Kuang, Delay Differential Equations with Applications in Population\r\nDynamics. INC: Academic Press, 1993.\r\n[21] J. Hale, Theory of Functional Differential Equations. Springer-Verlag,\r\n1977.\r\n[22] B. Hassard, D. Kazatina and Y. Wan, Theory and applications of Hopf\r\nbifurcation. Cambridge: Cambridge University Press, 1981.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 68, 2012"}