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Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay†

Published online by Cambridge University Press:  10 August 2012

RUI HU  and
YUAN YUAN
RUI HU
Affiliation:
Department of Mathematical and Statistical Sciences, University of AlbertaEdmonton AB T6G 2G1Canada email: ([email protected])
YUAN YUAN
Affiliation:
Department of Mathematics and Statistics, Memorial University of NewfoundlandSt. John's NL A1C 5S7Canada
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Abstract

We consider a diffusive Nicholson's blowflies equation with non-local delay and study the stability of the uniform steady states and the possible Hopf bifurcation. By using the upper- and lower solutions method, the global stability of constant steady states is obtained. We also discuss the local stability via analysis of the characteristic equation. Moreover, for a special kernel, the occurrence of Hopf bifurcation near the steady state solution and the stability of bifurcated periodic solutions are given via the centre manifold theory. Based on laboratory data and our theoretical results, we address the influence of various types of vaccinations in controlling the outbreak of blowflies.

Keywords

Nicholson's blowflies equationDiffusionNon-local delayHopf bifurcationHomogeneous Neumann boundary condition
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 6 , December 2012 , pp. 777 - 796
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

†

Project supported in part by Natural Science and Engineering Research Council of Canada.

References

[1]Arundel, J. H. & Sutherland, A. K. (1988) Blowflies of sheep. In: Arundel, J. J. and Sutherland, A. K. (editors), Animal Health in Australia, Vol. 10, Ectoparasitic Diseases of Sheep, Cattle, Goats and Horses, Australian Government Publishing Service, Canberra, Australia, pp. 33–60.Google Scholar
[2]Britton, N. F. (1986) Reaction-Diffusion Equations and Their Applications to Biology, Academic, London.Google Scholar
[3]East, I. J. & Eisemann, C. H. (1993) Vaccination against Lucilia cuprina: The causative agent of sheep blowfly strike, J Immunol. Cell Biol. 71, 453–462.Google Scholar
[4]East, I. J., Fitzgerald, C. J., Pearson, R. D., Donaldson, R. A., Vuocolo, T., Cadogan, L. C., Tellam, R. L. & Eisemann, C. H. (1993) Lucilia cuprina inhibition of larval growth induced by immunisation of host sheep with extracts of larval peritrophic membrane. Int. J. Parasitol. 23, 221–229.Google Scholar
[5]Fry, K. L., Farrell, J. M. & Sandeman, R. M. (1993) Characterisation of potential vaccine antigens in blowfly larvae using monoconal antibodies. In: Sandeman, R. M., Arundel, J. H. and Scheurmann, E. A. (editors), Immunological Control of Blowfly Strike, Australian Wool Corporation, Melbourne, Australia, pp. 82–89.Google Scholar
[6]Gourley, S. A. & Britton, N. F. (1996) A predator prey reaction diffusion system with nonlocal effects. J. Math. Biol. 34 297–333.Google Scholar
[7]Gourley, S. A. & Ruan, S. (2000) Dynamics of the diffusive Nicholson's blowflies equation with distributed delay. Proc. Roy. Soc. Edin. A. 130, 1275–1291.Google Scholar
[8]Gourley, S. A. & So, J. W.-H. (2002) Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain. J. Math. Biol. 44, 49–78.Google Scholar
[9]Gourley, S. A.So, J. W.-H. & Wu, J. (2004) Nonlocality of reaction-diffusion equations induced by delay: Biological modeling and nonlinear dynamics. J. Math. Sci. 124, 5119–5153.Google Scholar
[10]Gurney, W. S. C., Blythe, S. P. & Nisbet, R. M. (1980) Nicholson's blowflies revisited. Nature 287, 17–21.Google Scholar
[11]Hassard, B. D., Kazarinoff, N. D. & Wan, Y.-H. (1981) Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, UK.Google Scholar
[12]Hu, R. & Yuan, Y. (2011) Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay. J. Differ. Equ. 250, 2779–2806.Google Scholar
[13]Li, W. T., Ruan, S. & Wang, Z. C. (2007) On the diffusive Nicholson's blowflies equation with nonlocal delay. J. Nonlinear Sci. 17, 505–525.Google Scholar
[14]Madras, N., Wu, J. & Zou, X. (1996) Local-nonlocal interaction and spatial-temporal patterns in single species population over a patchy environment. Canad. Appl. Math. Quar. 4, 109–134.Google Scholar
[15]Nicholson, A. J. (1954) An outline of the dynamics of animal populations. Austral. J. Zoo. 2, 9–65.Google Scholar
[16]Pao, C. V. (2002) Convergence of solutions of reaction-diffusion systems with time delays. Nonlinear Anal. 48, 349–362.Google Scholar
[17]Redlinger, R. (1984) Existence theorems for semilinaer parabolic system with functionals. Nonlinear Anal. 8, 667–682.Google Scholar
[18]So, J. W.-H. & Yang, Y. (1998) Dirichlet problem for the diffusive Nicholson's blowflies equation. J. Differ. Equ. 150, 317–348.Google Scholar
[19]Thieme, H. R. & Zhao, X. (2001) A non-local delayed and diffusive predator-prey model. Nonlinear Anal. Real World Appl. 2, 145–160.Google Scholar
[20]Wu, J. (1996) Theory and Applications of Partial Functional Differential Equations Springer-Verlag, New York.Google Scholar
[21]Wu, S. L., Li, W. T. & Liu, S. Y. (2009) Oscillatory waves in reaction-diffusion equations with nonlocal delay and crossing monostability. Nonlinear Anal. Real World Appl. 10, 3141–3151.Google Scholar
[22]Xu, D. & Zhao, X. (2003) A nonlocal reaction-diffusion population model with stage structure. Canad. Appl. Math. Q. 11, 303–320.Google Scholar
[23]Yamada, Y. (1982) On a certain class of semilinear Volterra diffusion equations. J. Math. Anal. Appl. 88, 433–451.Google Scholar
[24]Yang, Y. & So, J. W.-H. (1996) Dynamics of the diffusive Nicholson's blowflies equation. In: Chen, Wenxiong and Hu, Shouchuan (editors), Proceedings of the International Conference on Dynamical Systems and Differential Equations, Springfield, MO, May 29–Jun 1, Vol. II An Added Volume to Discrete and Continuous Dynamical Systems. . , pp. 333–352.Google Scholar
[25]Zied, E. M. A., Gabre, R. M. & Chi, H. (2003) Life table of the Australian sheep blowfly Lucilia cuprian (wiedemann) (Diptera: Calliphoridae). Egypt. J. Zool. 41, 29–45.Google Scholar