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Stability and Hopf bifurcation for a delay competition diffusion system. (English) Zbl 1038.35147

Summary: This paper investigates the stability and Hopf bifurcation of a delay competition diffusion system. Firstly we discuss the existence and stability of the corresponding steady state solutions. Secondly our purpose is to give more detail information about the Hopf bifurcation of this system. We derive the basis of the eigenfunction subspace and then convert the existence of periodic solutions to the study of the existence of the implicit function. Finally, we analyze the stability of the periodic solutions by reducing the original system on the center manifold.

MSC:

35R10 Partial functional-differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
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[1] Busenberg, S.; Mahaffy, J., Interaction of spatial diffusion and delay in models of genetic control by repression, J. Math. Biol., 22, 313-333 (1985) · Zbl 0593.92010
[2] Busenberg, S.; Huang, W. Z., Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 104, 80-107 (1996) · Zbl 0854.35120
[3] Hale, J., Theory of functional differential equations (1977), Springer: Springer Berlin · Zbl 0352.34001
[4] Pazy, A., Semigroup of linear operators and applications to partial differential equations (1983), Springer: Springer Berlin · Zbl 0516.47023
[5] Travis, C.; Webb, G., Existence and stability for partial functional differential equations, Trans. Am. Math. Soc., 204, 395-418 (1974) · Zbl 0299.35085
[6] Wu, J., Theory and applications of partial functional differential equations (1996), Springer: Springer Berlin · Zbl 0870.35116
[7] Zhou, L.; Pao, C. V., Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal. TMA, 6, 11, 1163-1184 (1982) · Zbl 0522.92017
[8] Zhou, L., Bifurcation method for analysis of coexistence of competition diffusion system, Acta Appl. Math. (Chinese), 14, 102-110 (1991) · Zbl 0728.35046
[9] Kanel, J. I.; Zhou, L., Existence of wave front solutions and estimates of wave speed for a competition-diffusin system, Nonlinear Anal. TMA, 27, 579-587 (1996) · Zbl 0857.35059
[10] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and applications of Hopf bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002
[11] Zhou, L.; Hussein, S., Stability and Hopf bifurcation for a competition delay model with diffusion, J. Math. (Chinese), 19, 441-446 (1999) · Zbl 0954.35025
[12] Pao, C. V., Couple nonlinear parabolic system with time delay, J. Math. Anal. Appl., 196, 237-265 (1995) · Zbl 0854.35122
[13] Zhou, L.; Hussein, S., The existence and stability of nontrivial steady state for a delay competition-diffusion model, Wuhan Univ. J. Natural Sci., 4, 4, 377-381 (1999) · Zbl 0955.35040
[14] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Math. Biosci., 111, 1-71 (1992) · Zbl 0783.92002
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