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Traveling wavefronts for time-delayed reaction-diffusion equation. I: Local nonlinearity. (English) Zbl 1173.35071
A class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate is studied. For all wavefronts with the speed strictly greater than the critical wave speed, it is proved that these wavefronts are asymptotically stable when the initial perturbation around the traveling waves decays exponentially as $x\rightarrow - \infty $, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results in previous studies. The approach adopted in this paper is the technical weighted energy method and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any speed strictly greater than the critical wave speed and for an arbitrary time-delay. [For part II see ibid., 511--529 (2009; Zbl 1173.35072).]

35K57Reaction-diffusion equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
35R10Partial functional-differential equations
Full Text: DOI
[1] Al-Omari, J.; Gourley, S. A.: Monotone travelling fronts in an age-structured reaction -- diffusion model of a single species, J. math. Biol. 45, 294-312 (2002) · Zbl 1013.92032 · doi:10.1007/s002850200159
[2] Al-Omari, J.; Gourley, S. A.: Monotone wave-fronts in a structured population model with distributed maturation delay, IMA J. Appl. math. 70, 858-879 (2005) · Zbl 1086.92043 · doi:10.1093/imamat/hxh073
[3] Fang, J.; Wei, J.; Zhao, X. -Q.: Spatial dynamics of a nonlocal and time-delayed reaction -- diffusion system, J. differential equations 245, 2749-2770 (2008) · Zbl 1180.35536 · doi:10.1016/j.jde.2008.09.001
[4] Fisher, R. A.: The wave of advance of advantageous genes, Ann. eugen. 7, 353-369 (1937) · Zbl 63.1111.04
[5] Gourley, S. A.: Linear stability of travelling fronts in an age-structured reaction -- diffusion population model, Quart. J. Mech. appl. Math. 58, 257-268 (2005) · Zbl 1069.92018 · doi:10.1093/qjmamj/hbi012
[6] Gourley, S. A.; Kuang, Y.: Wavefronts and global stability in a time-delayed population model with stage structure, Proc. R. Soc. lond. Ser. A 459, 1563-1579 (2003) · Zbl 1047.92037 · doi:10.1098/rspa.2002.1094
[7] Gourley, S. A.; So, J. W. -H.; Wu, J.: Non-locality of reaction -- diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. math. Sci. 124, 84-120 (2003) · Zbl 1128.35360 · doi:10.1023/B:JOTH.0000047249.39572.6d
[8] Gourley, S. A.; Wu, J.: Delayed nonlocal diffusive system in biological invasion and disease spread, Fields inst. Commun. 48, 137-200 (2006) · Zbl 1130.35127
[9] Gurney, W. S. C.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowflies revisited, Nature 287, 17-21 (1980)
[10] Kolmogorov, A.; Petrovsky, I.; Piskounov, N.: Study of the diffusion equation with growth of the quantity of matter and its application to biological problems, Bull. univ. Etat moscou 6, 1-25 (1937)
[11] Li, G.; Mei, M.; Wong, Y. S.: Nonlinear stability of travelling wavefronts in an age-structured reaction -- diffusion population model, Math. biosci. Eng. 5, 85-100 (2008) · Zbl 1148.35038 · doi:10.3934/mbe.2008.5.85
[12] Li, W. -T.; Ruan, S.; Wang, Z. -C.: On the diffusive Nicholson’s blowflies equation with nonlocal delays, J. nonlinear sci. 17, 505-525 (2007) · Zbl 1134.35064 · doi:10.1007/s00332-007-9003-9
[13] Li, W. -T.; Wang, Z. -C.; Wu, J.: Entire solutions in monostable reaction -- diffusion equations with delayed nonlinearity, J. differential equations 245, 102-129 (2008) · Zbl 1185.35314 · doi:10.1016/j.jde.2008.03.023
[14] Liang, D.; Wu, J.: Traveling waves and numerical approximations in a reaction -- diffusion equation with nonlocal delayed effect, J. nonlinear sci. 13, 289-310 (2003) · Zbl 1017.92024 · doi:10.1007/s00332-003-0524-6
[15] C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson’s blowflies equations with diffusion, submitted for publication
[16] Ma, S.; Zou, X.: Existence, uniqueness and stability of travelling waves in a discrete reaction -- diffusion monostable equation with delay, J. differential equations 217, 54-87 (2005) · Zbl 1085.34050 · doi:10.1016/j.jde.2005.05.004
[17] Martin, R. H.; 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
[18] Mei, M.; Lin, C. -K.; Lin, C. -T.; So, J. W. -H.: Traveling wavefronts for time-delayed reaction -- diffusion equation: (II) nonlocal nonlinearity, J. differential equations 247, No. 2, 511-529 (2009) · Zbl 1173.35072 · doi:10.1016/j.jde.2008.12.020
[19] Mei, M.; Nishihara, K.: Nonlinear stability of travelling waves for one-dimensional viscoelastic materials with non-convex nonlinearity, Tokyo J. Math. 20, 241-264 (1997) · Zbl 0880.35018 · doi:10.3836/tjm/1270042411
[20] Mei, M.; So, J. W. -H.: Stability of strong traveling waves for a nonlocal time-delayed reaction -- diffusion equation, Proc. roy. Soc. Edinburgh sect. A 138, 551-568 (2008) · Zbl 1148.35093 · doi:10.1017/S0308210506000333
[21] Mei, M.; So, J. W. -H.; Li, M. Y.; Shen, S. S. P.: Asymptotic stability of traveling waves for the Nicholson’s blowflies equation with diffusion, Proc. roy. Soc. Edinburgh sect. A 134, 579-594 (2004) · Zbl 1059.34019 · doi:10.1017/S0308210500003358
[22] Metz, J. A. J.; Diekmann, O.: The dynamics of physiologically structured populations, (1986) · Zbl 0614.92014
[23] Ou, C.; Wu, J.: Persistence of wavefronts in delayed non-local reaction -- diffusion equations, J. differential equations 235, 219-261 (2007) · Zbl 1117.35037 · doi:10.1016/j.jde.2006.12.010
[24] Schaaf, K. W.: Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. amer. Math. soc. 302, 587-615 (1987) · Zbl 0637.35082 · doi:10.2307/2000859
[25] Smith, H. L.; Zhao, X. -Q.: Global asymptotic stability of traveling waves in delayed reaction -- diffusion equations, SIAM J. Math. anal. 31, 514-534 (2000) · Zbl 0941.35125 · doi:10.1137/S0036141098346785
[26] So, J. W. -H.; Wu, J.; Zou, X.: A reaction -- diffusion model for a single species with age structure: (I) traveling wavefronts on unbounded domains, Proc. R. Soc. lond. Ser. A 457, 1841-1853 (2001) · Zbl 0999.92029 · doi:10.1098/rspa.2001.0789
[27] So, J. W. -H.; Wu, J.; Yang, Y.: Numerical Hopf bifurcation analysis on the diffusive Nicholson’s blowflies equation, Appl. math. Comput. 111, 53-69 (2000) · Zbl 1028.65138 · doi:10.1016/S0096-3003(99)00063-6
[28] 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
[29] So, J. W. -H.; Zou, X.: Traveling waves for the diffusive Nicholson’s blowflies equation, Appl. math. Comput. 22, 385-392 (2001) · Zbl 1027.35051 · doi:10.1016/S0096-3003(00)00055-2
[30] Thieme, H. R.: Mathematics in population biology, (2003) · Zbl 1054.92042
[31] Thieme, H.; Zhao, X. -Q.: Asymptotic speeds of spread and traveling waves for integral equation and delayed reaction -- diffusion models, J. differential equations 195, 430-470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X
[32] Volpert, A.; Volpert, Vi.; Volpert, Vl.: Traveling wave solutions of parabolic systems, Transl. math. Monogr. 140 (1994) · Zbl 1001.35060 · http://www.ams.org/online_bks/mmono140/
[33] D. Wei, J.-Y. Wu, M. Mei, Remark on critical speed of traveling wavefronts for Nicholson’s blowflies equation with diffusion, Acta Math. Sci., in press · Zbl 1240.35565
[34] Wu, J. -Y.; Wei, D.; Mei, M.: Analysis on critical speed of traveling waves, Appl. math. Lett. 20, 712-718 (2007) · Zbl 1113.35097 · doi:10.1016/j.aml.2006.08.006
[35] Wu, J. -H.: Theory and applications of partial functional-diffusion equations, Appl. math. Sci. 119 (1996) · Zbl 0870.35116
[36] Zhao, X. -Q.: Dynamical systems in population biology, CMS books math. 16 (2003)
[37] Yang, Y.; So, J. W. -H.: Dynamics for the diffusive Nicholson’s blowflies equation, Dynamical systems and differential equations, vol. II, 333-352 (1998)