{"title":"HOPF Bifurcation of a Predator-prey Model with Time Delay and Habitat Complexity","authors":"Li Hongwei","volume":60,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":2150,"pagesEnd":2155,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/10173","abstract":"
In this paper, a predator-prey model with time delay and habitat complexity is investigated. By analyzing the characteristic equations, the local stability of each feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By choosing the sum of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as \u001c crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main theoretical results.<\/p>\r\n","references":"[1] J.M. Cushing, Integro-differential equations and delay models in population\r\ndynamics, in: Lecture Notes in Biomathematics 20, Springer-Verlag, Berlin, Heidelberg, New York, 1977.\r\n[2] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations\r\nof Population Dynamics, Kluwer Academic, Dordrecht, Norwell, MA 1992.\r\n[3] Y. Kuang, Delay Differential Equations with Applications in Population\r\nDynamics, Academic Press, New York, 1993.\r\n[4] M.S. Bartlett, On theoretical models for competitive and predatory\r\nbiological systems, Biometrika 44 (1957) 27-42.\r\n[5] E. Beretta, Y. Kuang, Global analyses in some delayed ratio-dependent\r\npredator-prey systems, Nonlinear Anal. TMA 32 (1998) 381-408.\r\n[6] P.J. Wangersky, W.J. Cunningham, Time lag in prey-predator population\r\nmodels, Ecology 38 (1957) 136-139.\r\n[7] J.Wei, S. Ruan, Stability and bifurcation in a neural network model with\r\ntwo delays, Physica D 130 (1999) 255-272.\r\n[8] J.K. Hale, Theory of Functional Differential Equations, Springer, New\r\nYork, 1976.\r\n[9] B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf\r\nBifurcation, Cambridge University Press, Cambridge, 1981.\r\n[10] C.S. Holling, Some characteristics of simple types of predation and\r\nparasitism, Can. Entomologist 91 (1959) 385-398.\r\n[11] I.J. Winfield, The influence of simulated aquatic macrophytes on the\r\nzooplankton consumption rate of juvenile roach, Rutilus rutilus, rudd,\r\nScardinius erythrophthalmus, and perch, Perca fluviatilis, J. Fish Biol. 29\r\n(1986) 37-48.\r\n[12] M. Kot, Elements of Mathematical Ecology, Cambridge Univ. Press,\r\nCambridge, 2001.\r\n[13] R.V. Culshaw, S. Ruan, A Delay-differential equation model of HIV\r\ninfection of CD4+ T-cells, Math. Biosci. 165 (2000) 27-39.\r\n[14] J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1960.\r\n[15] L. Luckinbill, Coexistence in laboratory populations of Paramecium\r\naurelia and its predator Didinium nasutum, Ecology 54 (1973) 1320-1327.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 60, 2011"}