Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case

Journal of Differential Equations

Volumn 245, Issue 11, 2008, Pages 3376-3388

  Yi, Taishan ^a   Zou, Xingfu ^b  

^a HUNAN UNIVERSITY   (China)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

Diffusive Nicholson's blowflies equation; Global attractivity; Maximum principle; Neumann boundary value; Non monotone dynamical system

Indexed keywords

EID: 53349160548     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.03.007     Document Type: Article

References (28)

1. Cooke K., van den Driessche P., and Zou X. Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39 (1999) 332-352
2. Faria T. Asymptotic stability for delayed logistic type equations. Math. Comput. Modelling 43 (2006) 433-445
3. Fitzgibbon W.E. Semilinear functional differential equations in Banach space. J. Differential Equations 29 (1978) 1-14
4. Gourley S.A. Traveling fronts in the diffusive Nicholson's blowflies equation with distributed delays. Math. Comput. Modelling 32 (2000) 843-853
5. Gourley S.A., and Ruan S. Dynamics of the diffusive Nicholson's blowflies equation with distributed delay. Proc. Roy. Soc. Edinburgh Sect. A 130 (2000) 1275-1291
6. Gurney W.S.C., Blythe S.P., and Nisbet R.M. Nicholson's blowflies revisited. Nature 287 (1980) 17-21
7. ′ (t) = - δ x (t) + p f (x (t - τ)). Dynam. Systems Appl. 8 (1999) 197-210
8. Hale J.K. Asymptotic Behavior of Dissipative Systems (1988), Amer. Math. Soc., Providence, RI
9. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional-Differential Equations (1993), Springer-Verlag, New York
10. Karakostas G., Philos Ch.G., and Sficas Y.C. Stable steady state of some population models. J. Dynam. Differential Equations 4 (1992) 161-190
11. Kuang Y. Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology. Japan J. Indust. Appl. Math. 9 (1992) 205-238
12. Kulenovic N.M.R.S., and Ladas G. Linearized oscillations in population dynamics. Bull. Math. Biol. 49 (1987) 615-627
13. Liang D., So J.W.-H., Zhang F., and Zou X. Population dynamic models with nonlocal delay on bounded fields and their numeric computations. Differential Equations Dynam. Systems 11 (2003) 117-139
14. Liang D., and Wu J. Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects. J. Nonlinear Sci. 13 (2003) 289-310
15. Liang D., Wu J., and Zhang F. Modelling population growth with delayed nonlocal reaction in 2-dimensions. Math. Biosci. Eng. 2 (2005) 111-132
16. Martin R., and Smith H.L. Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc. 321 (1990) 1-44
17. Martin R., and Smith H.L. Reaction-diffusion systems with time delay: Monotonicity, invariance, comparison and convergence. J. Reine Angew. Math. 413 (1991) 1-35
18. Mei M., So J.W.-H., Li M.Y., and Shen S. Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594
19. Nicholson A.J. An outline of the dynamics of animal populations. Aust. J. Zool. 2 (1954) 9-65
20. Protter M.H., and Weinberger H.F. Maximum Principles in Differential Equations (1984), Springer-Verlag, Berlin, New York
21. Smith H.L. Monotone Dynamical Systems. Math. Surveys Monogr. (1995), Amer. Math. Soc., Providence, RI
22. So J.W.-H., and Yang Y. Dirichlet problem for the diffusive Nicholson's blowflies equation. J. Differential Equations 150 (1998) 317-348
23. So J.W.-H., and Yu J.S. Global attractivity and uniform persistence in Nicholson's blowflies. Differential Equations Dynam. Systems 2 (1994) 11-18
24. So J.W.-H., and Zou X. Traveling waves for the diffusive Nicholson's blowflies equation. Appl. Math. Comput. 122 (2001) 385-392
25. So J.W.-H., Wu J., and Zou X. A reaction diffusion model for a single species with age structure, I. Traveling wave fronts on unbounded domains. Proc. R. Soc. Lond. Ser. A 457 (2001) 1841-1854
26. Travis C.C., and Webb F.F. Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc. 200 (1974) 395-418
27. Wu J. Theory and Applications of Partial Functional Differential Equations. Appl. Math. Sci. vol. 119 (1996), Springer-Verlag, New York
28. Y. Yang, J.W.-H. So, Dynamics for the diffusive Nicholson blowflies equation, in: Proceedings of the International Conference on Dynamical Systems and Differential Equations, held in Springfield, Missouri, USA, May 29-June 1, 1996