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Stability and bifurcation for a delayed predator-prey model and the effect of diffusion. (English) Zbl 0973.35034
The author studies the dynamics of a predator-prey system with one or two delays in terms of the local stability of the unique positive equilibrium \(E_*\). Comparison between the dynamics of the system with and without diffusion terms is made in a neighbourhood of \(E_*\).

MSC:
35B32 Bifurcations in context of PDEs
35R10 Functional partial differential equations
35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
92D25 Population dynamics (general)
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[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] Faria, T., Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. amer. math. soc., 352, 2217-2238, (2000) · Zbl 0955.35008
[4] Faria, T., Bifurcations aspects for some delayed population models with diffusion, Fields inst. commun., 21, 143-158, (1999) · Zbl 0922.35016
[5] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations and applications to bogdanov – takens singularity, J. differential equations, 122, 201-224, (1995) · Zbl 0836.34069
[6] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations, 122, 181-200, (1995) · Zbl 0836.34068
[7] 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
[8] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064
[9] Hale, J.K.; Verduyn-Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002
[10] He, X.-Z., Stability and delays in a predator – prey system, J. math. anal. appl., 198, 335-370, (1996)
[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 delay equation, J. differential equations, 26, 391-403, (1977) · Zbl 0365.34078
[13] Lin, X.; So, J.W.-H.; Wu, J., Centre manifolds for partial differential equations with delays, Proc. roy. soc. Edinburgh sect. A, 122, 237-254, (1992) · Zbl 0801.35062
[14] 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
[15] Memory, M.C., Stable and unstable manifolds for partial functional differential equations, Nonlinear anal., 16, 131-142, (1991) · Zbl 0729.35138
[16] Táboas, P., Periodic solutions of a planar delay equation, Proc. roy. soc. Edinburgh sect. A, 116, 85-101, (1990) · Zbl 0719.34125
[17] Travis, C.C.; Webb, G.F., Existence and stability for partial functional differential equations, Trans. amer. math. soc., 200, 395-418, (1974) · Zbl 0299.35085
[18] Wu, J., Theory and applications of partial functional differential equations, (1996), Springer-Verlag New York
[19] 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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