Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30319
HOPF Bifurcation of a Predator-prey Model with Time Delay and Habitat Complexity

Authors: Li Hongwei

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: Hopf Bifurcation, delay, habitat complexity, Predator-prey system

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1074735

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1489

References:


[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.