## Bifurcations for a predator-prey system with two delays.(English)Zbl 1132.34053

Authors’ abstract: In this paper, a predator-prey system with two delays is investigated. By choosing the sum $$\tau$$ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as $$\tau$$ 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. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global bifurcation results of J. Wu [Trans. Am. Math. Soc. 350, 4799–4838 (1998; Zbl 0905.34034)], we may show the global existence of periodic solutions.

### MSC:

 34K18 Bifurcation theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations 34K60 Qualitative investigation and simulation of models involving functional-differential equations 92D25 Population dynamics (general)

Zbl 0905.34034
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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-Verlag New York [3] Cushing, J.M., Periodic time-dependent predator – prey systems, SIAM J. appl. math., 32, 82-95, (1977) · Zbl 0348.34031 [4] Faria, T.; Magalháes, 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] Faria, T.; Magalháes, L.T., Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations, 122, 181-200, (1995) · Zbl 0836.34068 [6] Faria, T., Stability and bifurcation for a delay predator – prey model and the effect of diffusion, J. math. anal. appl., 254, 433-463, (2001) · Zbl 0973.35034 [7] Giannakopoulos, F.; Zapp, A., Local and Hopf bifurcation in a scalar delay differential equation, J. math. anal. appl., 237, 425-450, (1999) · Zbl 1126.34371 [8] Hadelen, K.P.; Tomiuk, J., Periodic solutions of differential-difference equations, Arch. ration. mech. anal., 65, 87-95, (1977) · Zbl 0426.34058 [9] Hale, J.; Lunel, S.V., Introduction to functional differential equations, (1993), Springer-Verlag New York [10] He, X., Stability and delays in a predator – prey system, J. math. anal. appl., 198, 355-370, (1996) · Zbl 0873.34062 [11] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 [12] Leung, A., Periodic solutions for a prey – predator differential delay equation, J. differential equations, 26, 391-403, (1977) · Zbl 0365.34078 [13] Mallet-Paret, J.; Nussbaum, R.D., Global continuation and asymptotic behavior for periodic solutions of a differential – delay equation, Ann. math. pura appl., 145, 33-128, (1986) · Zbl 0617.34071 [14] Mallet-Paret, J.; Nussbaum, R.D., A differential – delay equation arising in optics and physiology, SIAM J. math. anal., 20, 249-292, (1989) · Zbl 0676.34043 [15] Ruan, S.; Wei, J., Periodic solutions of planar systems with two delays, Proc. roy. soc. Edinburgh sect. A, 129, 1017-1032, (1999) · Zbl 0946.34062 [16] Song, Y.; Wei, J., Local and global Hopf bifurcation in a delayed hematopoiesis model, Internat. J. bifur. chaos appl. sci. engrg., 14, 3909-3919, (2004) · Zbl 1090.37547 [17] Song, Y.; Wei, J., Local Hopf bifurcation and global periodic solutions in a delayed predator – prey system, J. math. anal. appl., 301, 1-21, (2005) · Zbl 1067.34076 [18] Taboas, P., Periodic solutions of a planar delay equation, Proc. roy. soc. Edinburgh sect. A, 116, 85-101, (1990) · Zbl 0719.34125 [19] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085 [20] Wei, J.; Li, M., Hopf bifurcation analysis in a delayed Nicholson blowflies equation, Nonlinear anal., 60, 1351-1367, (2005) · Zbl 1144.34373 [21] Wei, J.; Li, Y., Global existence of periodic solutions in a tri-neuron network model with delays, Phys. D, 198, 106-119, (2004) · Zbl 1062.34077 [22] Wu, J., Theory and applications of partial functional differential equations, (1996), Springer-Verlag New York [23] Wu, J., Symmetric functional differential equations and neural networks with memory, Trans. amer. math. soc., 350, 4799-4838, (1998) · Zbl 0905.34034 [24] 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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