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Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case. (English) Zbl 1152.35511
Summary: We establish the global attractivity of the positive steady state of the diffusive Nicholson equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities.
MSC:
35R10Partial functional-differential equations
35B35Stability of solutions of PDE
35B40Asymptotic behavior of solutions of PDE
35B41Attractors (PDE)
References:
[1]Cooke, K.; Den Driessche, P. Van; Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models, J. math. Biol. 39, 332-352 (1999) · Zbl 0945.92016 · doi:10.1007/s002850050194
[2]Faria, T.: Asymptotic stability for delayed logistic type equations, Math. comput. Modelling 43, 433-445 (2006) · Zbl 1145.34043 · doi:10.1016/j.mcm.2005.11.006
[3]Fitzgibbon, W. E.: Semilinear functional differential equations in Banach space, J. differential equations 29, 1-14 (1978) · Zbl 0392.34041 · doi:10.1016/0022-0396(78)90037-2
[4]Gourley, S. A.: Traveling fronts in the diffusive Nicholson’s blowflies equation with distributed delays, Math. comput. Modelling 32, 843-853 (2000) · Zbl 0969.35133 · doi:10.1016/S0895-7177(00)00175-8
[5]Gourley, S. A.; Ruan, S.: Dynamics of the diffusive Nicholson’s blowflies equation with distributed delay, Proc. roy. Soc. Edinburgh sect. A 130, 1275-1291 (2000) · Zbl 0973.34064 · doi:10.1017/S0308210500000688
[6]Gurney, W. S. C.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowflies revisited, Nature 287, 17-21 (1980)
[7]Györi, I.; Trofimchuk, S.: Global attractivity in x ' (t)=-δx(t)+pf(x(t-τ)), Dynam. systems appl. 8, 197-210 (1999) · Zbl 0965.34064
[8]Hale, J. K.: Asymptotic behavior of dissipative systems, (1988)
[9]Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional-differential equations, (1993)
[10]Karakostas, G.; Philos, Ch.G.; Sficas, Y. C.: Stable steady state of some population models, J. dynam. Differential equations 4, 161-190 (1992) · Zbl 0744.34071 · doi:10.1007/BF01048159
[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, 205-238 (1992) · Zbl 0758.34065 · doi:10.1007/BF03167566
[12]Kulenovic, N. M. R.S.; Ladas, G.: Linearized oscillations in population dynamics, Bull. math. Biol. 49, 615-627 (1987) · Zbl 0634.92013 · doi:10.1007/BF02460139
[13]Liang, D.; So, J. W. -H.; Zhang, F.; Zou, X.: Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Differential equations dynam. Systems 11, 117-139 (2003) · Zbl 1231.35287
[14]Liang, D.; Wu, J.: Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. nonlinear sci. 13, 289-310 (2003) · Zbl 1017.92024 · doi:10.1007/s00332-003-0524-6
[15]Liang, D.; Wu, J.; Zhang, F.: Modelling population growth with delayed nonlocal reaction in 2-dimensions, Math. biosci. Eng. 2, 111-132 (2005) · Zbl 1061.92048
[16]Martin, R.; Smith, H. L.: Abstract functional differential equations and reaction – diffusion systems, Trans. amer. Math. soc. 321, 1-44 (1990) · Zbl 0722.35046 · doi:10.2307/2001590
[17]Martin, R.; Smith, H. L.: Reaction – diffusion systems with time delay: monotonicity, invariance, comparison and convergence, J. reine angew. Math. 413, 1-35 (1991) · Zbl 0709.35059 · doi:crelle:GDZPPN002208148
[18]Mei, M.; So, J. W. -H.; Li, M. Y.; Shen, S.: Asymptotic stability of travelling waves for Nicholson’s blowflies equation with diffusion, Proc. roy. Soc. Edinburgh sect. A 134, 579-594 (2004) · Zbl 1059.34019 · doi:10.1017/S0308210500003358
[19]Nicholson, A. J.: An outline of the dynamics of animal populations, Aust. J. Zool. 2, 9-65 (1954)
[20]Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations, (1984) · Zbl 0549.35002
[21]Smith, H. L.: Monotone dynamical systems, Math. surveys monogr. (1995)
[22]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 · doi:10.1006/jdeq.1998.3489
[23]So, J. W. -H.; Yu, J. S.: Global attractivity and uniform persistence in Nicholson’s blowflies, Differential equations dynam. Systems 2, 11-18 (1994) · Zbl 0869.34056
[24]So, J. W. -H.; Zou, X.: Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. math. Comput. 122, 385-392 (2001) · Zbl 1027.35051 · doi:10.1016/S0096-3003(00)00055-2
[25]So, J. W. -H.; Wu, J.; 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, 1841-1854 (2001) · Zbl 0999.92029 · doi:10.1098/rspa.2001.0789
[26]Travis, C. C.; Webb, F. F.: Existence and stability for partial functional differential equations, Trans. amer. Math. soc. 200, 395-418 (1974) · Zbl 0299.35085 · doi:10.2307/1997265
[27]Wu, J.: Theory and applications of partial functional differential equations, Appl. math. Sci. 119 (1996) · Zbl 0870.35116
[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