HOPF Bifurcation of a Predator-prey Model with Time Delay and Habitat Complexity
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 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.
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[1] J.M. Cushing, Integro-differential equations and delay models in population
dynamics, in: Lecture Notes in Biomathematics 20, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
[2] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations
of Population Dynamics, Kluwer Academic, Dordrecht, Norwell, MA 1992.
[3] Y. Kuang, Delay Differential Equations with Applications in Population
Dynamics, Academic Press, New York, 1993.
[4] M.S. Bartlett, On theoretical models for competitive and predatory
biological systems, Biometrika 44 (1957) 27-42.
[5] E. Beretta, Y. Kuang, Global analyses in some delayed ratio-dependent
predator-prey systems, Nonlinear Anal. TMA 32 (1998) 381-408.
[6] P.J. Wangersky, W.J. Cunningham, Time lag in prey-predator population
models, Ecology 38 (1957) 136-139.
[7] J.Wei, S. Ruan, Stability and bifurcation in a neural network model with
two delays, Physica D 130 (1999) 255-272.
[8] J.K. Hale, Theory of Functional Differential Equations, Springer, New
York, 1976.
[9] B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf
Bifurcation, Cambridge University Press, Cambridge, 1981.
[10] C.S. Holling, Some characteristics of simple types of predation and
parasitism, Can. Entomologist 91 (1959) 385-398.
[11] I.J. Winfield, The influence of simulated aquatic macrophytes on the
zooplankton consumption rate of juvenile roach, Rutilus rutilus, rudd,
Scardinius erythrophthalmus, and perch, Perca fluviatilis, J. Fish Biol. 29
(1986) 37-48.
[12] M. Kot, Elements of Mathematical Ecology, Cambridge Univ. Press,
Cambridge, 2001.
[13] R.V. Culshaw, S. Ruan, A Delay-differential equation model of HIV
infection of CD4+ T-cells, Math. Biosci. 165 (2000) 27-39.
[14] J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1960.
[15] L. Luckinbill, Coexistence in laboratory populations of Paramecium
aurelia and its predator Didinium nasutum, Ecology 54 (1973) 1320-1327.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59231", author = "Li Hongwei", title = "HOPF Bifurcation of a Predator-prey Model with Time Delay and Habitat Complexity", 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 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.
", keywords = "Predator-prey system, delay, habitat complexity, HOPF bifurcation.", volume = "5", number = "12", pages = "2020-5", }
