Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation. (English) Zbl 1191.35046
Summary: The dynamics of a diffusive Nicholson’s blowflies equation with a finite delay and Dirichlet boundary condition have been investigated in this paper. The occurrence of steady state bifurcation with the changes of parameter is proved by applying phase plane ideas. The existence of Hopf bifurcation at the positive equilibrium with the changes of specify parameters is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions is found by analyzing the distribution of the eigenvalues. By the theory of normal form and center manifold, an explicit algorithm for determining the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are derived.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35R10 | Partial functional-differential equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35K58 | Semilinear parabolic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
steady state bifurcation; Hopf bifurcation; Dirichlet boundary condition; bifurcation direction; normal form; center manifoldReferences:
| [1] | Gurney, M. S.; Blythe, S. P.; Nisbet, R. M., Nicholson’s bowflies revisited, Nature, 287, 17-21 (1980) |
| [2] | Györi, I.; Ladas, G., Oscillation Theory of Delay Differential Equations with Applications (1991), Oxford University Press: Oxford University Press Oxford · Zbl 0780.34048 |
| [3] | Kulenovic, M. R.S.; Ladas, G., Linearized oscillations in population dynamics, Bull. Math. Bio., 49, 615-627 (1987) · Zbl 0634.92013 |
| [4] | Kulenovic, M. R.S.; Ladas, G.; Sficas, Y. G., Global attractivity in Nicholson’s blowflies, Appl. Anal., 43, 109-124 (1992) · Zbl 0754.34078 |
| [5] | Feng, Q. X.; Yan, J. R., Global attractivity and oscillation in a kind of Nicholson’s blowflies, J. Biomath., 17, 21-26 (2002) |
| [6] | Györi, I.; Trofimchuk, S. I., On the existence of rapidly oscillatory solutions in the Nicholson’s blowflies equation, Nonlinear Anal. TMA, 48, 1033-1042 (2002) · Zbl 1007.34063 |
| [7] | Li, J., Global attractivity in Nicholson’s blowflies, Appl. Math. Ser. B, 11, 425-434 (1996) · Zbl 0867.34042 |
| [8] | Sakar, S. H.; Agarwal, S., Oscillation and global attractivity in a periodic Niholson’s blowflies model, Math. Comput. Modelling, 35, 719-731 (2002) · Zbl 1012.34067 |
| [9] | Weng, P.; Liang, M., Existence and global attractivity of periodic solution of a model in population dynamics, Acta Math. Appl. Sinica (English Ser.), 12, 427-434 (1996) · Zbl 0886.45005 |
| [10] | So, J. W.-H.; Yu, J. S., On the stability and uniform persistence of a discrete model of Nicholson’s blowflies, J. Math. Anal. Appl., 193, 233-244 (1995) · Zbl 0834.39009 |
| [11] | Li, J., Global attractivity in a discrete model of Nicholson’s blowflies, Ann. Differential Equations, 12, 173-182 (1996) · Zbl 0855.34092 |
| [12] | Kocić, V. Lj.; Ladas, G., Oscillation and global attractivity in a discrete model of Nicholson’s blowflies, Appl. Anal., 38, 21-31 (1990) · Zbl 0715.39003 |
| [13] | Karakostas, G.; Philos, Ch. G.; Sficas, Y. G., The dynamics of some discrete population models, Nonlinear Anal. TMA, 17, 1069-1084 (1991) · Zbl 0760.92019 |
| [14] | Zhang, B. G.; Xu, H. X., A note on the global attractivity of a discrete model of Nicholson’s blowflies, Discrete Dyn. Nat. Soc., 3, 51-55 (1999) · Zbl 0962.39011 |
| [15] | So, J. W.-H.; Yu, J. S., Global attractivity and uniform persistence in Nichlson’s blowflies, Differ. Equ. Dyn. Syst., 2, 11-18 (1994) · Zbl 0869.34056 |
| [16] | Yang, Y.; So, J. W.-H., Dynamics of the diffusive Nicholson’s blowflies equation, (Chen, Wenxiong; Hu., Shouchuan, Proceedings of the International Conference on Dynamical Systems and Differential Equations, Vol. II (1996), Springfield: Springfield Missouri, U.S.A), An added volume to Discrete Contin. Dyn. Syst., (1998) 333-352 |
| [17] | So, J. W.-H.; Yang, Y., Dirichlet problem for the diffusive Nicholson’s blowflies equation, J. Differential Equations, 150, 317-348 (1998) · Zbl 0923.35195 |
| [18] | So, J. W.-H.; Wu, J.; Yang, Y., Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson’s blowflies equation, Appl. Math. Cmput., 111, 33-51 (2000) · Zbl 1028.65138 |
| [19] | Gourley, S. A.; Ruan, S., Dynamics of the diffusive Nicholson’s blowflies with distributed delay, Proc. Roy. Soc. Edinburgh Sect. A, 130, 1275-1291 (2000) · Zbl 0973.34064 |
| [20] | So, J. W.-H.; Wu, J.; Zou, X., A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains, Proc. R. Soc. Lond. A, 457, 1841-1853 (2001) · Zbl 0999.92029 |
| [21] | Faria, T.; Trofimchuk, S., Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228, 357-376 (2006) · Zbl 1217.35102 |
| [22] | Li, W. T.; Ruan, S.; Wang, Z. C., On the diffusive Nicholson’s blowflies equation with nonlocal delay, J. Nonlinear Sciences, 17, 505-525 (2007) · Zbl 1134.35064 |
| [23] | Robinson, J. C., Infinite-Dimensional Dynamical Systems—An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0980.35001 |
| [24] | Wu, J., Theory and Applications of Partial Functional-Differential Equations (1996), Springer: Springer New York · Zbl 0870.35116 |
| [25] | Lin, X.; So, J. W.-H.; Wu, J., Center manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh Sect. A, 122, 237-254 (1992) · Zbl 0801.35062 |
| [26] | Faria, T., Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352, 2217-2238 (2000) · Zbl 0955.35008 |
| [27] | 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 |
| [28] | 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 |
| [29] | Hale, J. K., Theory of Functional Differential Equations (1977), Springer: Springer New York · Zbl 0352.34001 |
| [30] | Kulenovic, M. R.S.; Ladas, G.; Sficas, Y. G., Global attractivity in population dynamics, Comput. Math. Appl., 18, 925-928 (1989) · Zbl 0686.92019 |
| [31] | Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam. Contin., Discrete Impulsive Syst. Ser. A Math. Anal., 10, 863-874 (2003) · Zbl 1068.34072 |
| [32] | Wei, J.; Li, M. Y., Hopf bifurcation analysis in a delayde Nicholson’s blowflies equation, Nonlinear Anal. TMA, 60, 1351-1367 (2005) · Zbl 1144.34373 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
