Stability and Hopf bifurcations in a competitive Lotka–Volterra system with two delays. (English) Zbl 1067.34075

Summary: We consider a Lotka-Volterra competition system with two delays. We first investigate the stability of the positive equilibrium and the existence of Hopf bifurcations, and then using the normal form theory and center manifold argument, derive explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions.


34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
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[1] Tang, X.; Zou, X., Global attractivity of non-autonomous lotka – volterra competition system without instantaneous negative feedback, J. differ. equat., 19, 2502-2535, (2003) · Zbl 1035.34085
[2] Gopalsamy, K.; Weng, P., Global attractivity in a competition system with feedback controls, Comput. math. appl., 45, 665-676, (2003) · Zbl 1059.93111
[3] Tang, X.; Zou, X., 3/2-type criteria for global attractivity of lotka – volterra competition system without instantaneous negative feedbacks, J. differ. equat., 186, 420-439, (2002) · Zbl 1028.34070
[4] Saito, Y., The necessary and sufficient condition for global stability of a lotka – volterra cooperative or competition system with delays, J. math. anal. appl., 268, 109-124, (2002) · Zbl 1012.34072
[5] Jin, Z.; Ma, Z., Uniform persistence of n-dimensional lotka – volterra competition systems with finite delay, Adv. top. biomath., 91-95, (1998) · Zbl 0984.92037
[6] Gopalsamy, K.; He, X., Oscillations and convergence in an almost periodic competition system, Acta appl. math., 46, 247-266, (1997) · Zbl 0872.34050
[7] Lu, Z.; Takeuchi, Y., Permanence and global attractivity for competitive lotka – volterra system with delay, Nonlinear anal. TMA, 22, 847-856, (1994) · Zbl 0809.92025
[8] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Boston · Zbl 0752.34039
[9] Hale, J.; Lunel, S., Introduction to functional differential equations, (1993), Spring-Verlag New York · Zbl 0787.34002
[10] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002
[11] Zhou, L.; Tang, Y.; Hussein, S., Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, solitons & fractals, 14, 1201-1225, (2002) · Zbl 1038.35147
[12] Hassard, B.; Kazarinoff, D.; Wan, Y., Theory and applications of Hopf bifurcation, (1981), Cambridge University Press Cambridge · Zbl 0474.34002
[13] Li, X.; Ruan, S.; Wei, J., Stability and bifurcation in delay-differential equations with two delays, J. math. anal. appl., 236, 254-280, (1999) · Zbl 0946.34066
[14] Faria, T., Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. math. anal. appl., 254, 433-463, (2001) · Zbl 0973.35034
[15] Song Y, Wei J. Local and global Hopf bifurcation in a predator-prey system with two delays, preprint
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