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Stability and bifurcation analysis for a delayed Lotka–Volterra predator–prey system. (English) Zbl 1095.92071

Summary: The present paper deals with a delayed Lotka-Volterra predator-prey system. By linearizing the equations and by analyzing the locations on the complex plane of the roots of the characteristic equation, we find necessary conditions that the parameters should verify in order to have oscillations in the system. In addition, the normal form of the Hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these Hopf bifurcations. To verify the obtained conditions, a special numerical example is also included.

MSC:

92D40 Ecology
34K18 Bifurcation theory of functional-differential equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
37N25 Dynamical systems in biology
34K13 Periodic solutions to functional-differential equations
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References:

[1] Beretta, E.; Kuang, Y., Convergence results in a well-known delayed predator – prey system, J. math. anal. appl., 204, 840-853, (1996) · Zbl 0876.92021
[2] Chow, S.-N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer New York
[3] 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
[4] Faria, T.; Magalh√£es, L.T., Normal form for retarded functional differential equations and applications to bogdanov – takens singularity, J. differential equations, 122, 201-224, (1995) · Zbl 0836.34069
[5] Freedman, H.I.; Rao, V.S.H., Stability for a system involving two time delays, SIAM. J. appl. math., 46, 552-560, (1986) · Zbl 0624.34066
[6] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064
[7] Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048
[8] Hale, J.K.; Infante, E.F.; Tsen, F.-S.P., Stability in linear delay equations, J. math. anal. appl., 105, 533-555, (1985) · Zbl 0569.34061
[9] He, X., Stability and delays in a predator – prey system, J. math. anal. appl., 198, 355-370, (1996) · Zbl 0873.34062
[10] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press New York
[11] Liu, Z.; Yuan, R., Stability and bifurcation in a harmonic oscillator with delays, Choas, solitons fractals, 23, 551-562, (2005) · Zbl 1078.34050
[12] Ma, W.; Takeuchi, Y., Stability analysis on a predator – prey system with distributed delays, J. comput. appl. math., 88, 79-94, (1998) · Zbl 0897.34062
[13] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
[14] Wei, J.; Li, M., Global existence of periodic solutions in a tri-neuron network model with delays, Physica D, 198, 106-119, (2004) · Zbl 1062.34077
[15] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511
[16] Zhao, T.; Kuang, Y.; Smith, H.L., Global existence of periodic solutions in a class of delayed gause-type predator – prey systems, Nonlinear anal., 28, 1373-1394, (1997) · Zbl 0872.34047
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