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January  2012, 17(1): 283-295. doi: 10.3934/dcdsb.2012.17.283

A periodic reaction-diffusion model with a quiescent stage

Feng-Bin Wang 1,

1. 

Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan

Received  June 2010 Revised  June 2011 Published  October 2011

In this paper, we investigate the asymptotic behaviour for a periodic reaction-diffusion model with a quiescent stage. By appealing to the theory of asymptotic speeds of spread and traveling waves for monotone periodic semiflow, we establish the existence of the spreading speed and show that it coincides with the minimal wave speed for monotone periodic traveling waves. Finally, we consider the case where the spatial domain is bounded. A threshold result on the global attractivity of either zero or a positive periodic solution are established.
Keywords: periodic coexistence state., Periodic traveling waves, Spreading speeds, Monotone systems.
Mathematics Subject Classification: Primary: 35K55, 37C65; Secondary: 92D2.
Citation: Feng-Bin Wang. A periodic reaction-diffusion model with a quiescent stage. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 283-295. doi: 10.3934/dcdsb.2012.17.283
References:
[1]

J. Cook, "Dispersive Variability and Invasion Wave Speeds,", unpublished manuscript, (1993).   Google Scholar

[2]

K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985).   Google Scholar

[3]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar

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J. K. Hale, "Ordinary Differential Equations,", Second edition, (1980).   Google Scholar

[5]

P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,", Pitman Res. Notes Math. Series, 247 (1991).   Google Scholar

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K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Q., 10 (2002), 473.   Google Scholar

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J. Jiang, X. Liang and X.-Q. Zhao, Saddle-point behavior for monotone semiflows and reaction-diffusion models,, J. Diff. Eqns., 203 (2004), 313.  doi: 10.1016/j.jde.2004.05.002.  Google Scholar

[8]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution system,, J. Diff. Eqns., 231 (2006), 57.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[9]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[10]

R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces,", New York, (1976).   Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. of AMS, 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar

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J. D. Murray, "Mathematical Biology I, II,", New York, (2003).   Google Scholar

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P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar

[14]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage,, Proc. R. Soc. A., 463 (2007), 1029.  doi: 10.1098/rspa.2006.1806.  Google Scholar

[15]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16 (2003).   Google Scholar

show all references

References:
[1]

J. Cook, "Dispersive Variability and Invasion Wave Speeds,", unpublished manuscript, (1993).   Google Scholar

[2]

K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1985).   Google Scholar

[3]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[4]

J. K. Hale, "Ordinary Differential Equations,", Second edition, (1980).   Google Scholar

[5]

P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity,", Pitman Res. Notes Math. Series, 247 (1991).   Google Scholar

[6]

K. P. Hadeler and M. A. Lewis, Spatial dynamics of the diffusive logistic equation with a sedentary compartment,, Can. Appl. Math. Q., 10 (2002), 473.   Google Scholar

[7]

J. Jiang, X. Liang and X.-Q. Zhao, Saddle-point behavior for monotone semiflows and reaction-diffusion models,, J. Diff. Eqns., 203 (2004), 313.  doi: 10.1016/j.jde.2004.05.002.  Google Scholar

[8]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution system,, J. Diff. Eqns., 231 (2006), 57.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[9]

X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems,, J. Funct. Anal., 259 (2010), 857.  doi: 10.1016/j.jfa.2010.04.018.  Google Scholar

[10]

R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces,", New York, (1976).   Google Scholar

[11]

R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems,, Trans. of AMS, 321 (1990), 1.  doi: 10.2307/2001590.  Google Scholar

[12]

J. D. Murray, "Mathematical Biology I, II,", New York, (2003).   Google Scholar

[13]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM. J. Math. Anal., 37 (2005), 251.  doi: 10.1137/S0036141003439173.  Google Scholar

[14]

K. F. Zhang and X.-Q. Zhao, Asymptotic behaviour of a reaction-diffusion model with a quiescent stage,, Proc. R. Soc. A., 463 (2007), 1029.  doi: 10.1098/rspa.2006.1806.  Google Scholar

[15]

X.-Q. Zhao, "Dynamical Systems in Population Biology,", CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16 (2003).   Google Scholar

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