A periodic reaction-diffusion model with a quiescent stage

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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

Abstract
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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 and 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[1]	J. Cook, "Dispersive Variability and Invasion Wave Speeds," unpublished manuscript, 1993.
[2]	K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.
[3]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[4]	J. K. Hale, "Ordinary Differential Equations," Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.
[5]	P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Res. Notes Math. Series, 247, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
[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), 473-499.
[7]	J. Jiang, X. Liang and X.-Q. Zhao, Saddle-point behavior for monotone semiflows and reaction-diffusion models, J. Diff. Eqns., 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002.
[8]	X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution system, J. Diff. Eqns., 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.
[9]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018.
[10]	R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," New York, 1976.
[11]	R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. of AMS, 321 (1990), 1-44. doi: 10.2307/2001590.
[12]	J. D. Murray, "Mathematical Biology I, II," New York, 2003.
[13]	P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.
[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), 1029-1043. doi: 10.1098/rspa.2006.1806.
[15]	X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.

show all references

References:

[1]	J. Cook, "Dispersive Variability and Invasion Wave Speeds," unpublished manuscript, 1993.
[2]	K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.
[3]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[4]	J. K. Hale, "Ordinary Differential Equations," Second edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.
[5]	P. Hess, "Periodic-Parabolic Boundary Value Problems and Positivity," Pitman Res. Notes Math. Series, 247, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1991.
[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), 473-499.
[7]	J. Jiang, X. Liang and X.-Q. Zhao, Saddle-point behavior for monotone semiflows and reaction-diffusion models, J. Diff. Eqns., 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002.
[8]	X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution system, J. Diff. Eqns., 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.
[9]	X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), 857-903. doi: 10.1016/j.jfa.2010.04.018.
[10]	R. H. Martin, "Nonlinear Operators and Differential Equations in Banach Spaces," New York, 1976.
[11]	R. H. Martin and H. L. Smith, Abstract functional-differential equations and reaction-diffusion systems, Trans. of AMS, 321 (1990), 1-44. doi: 10.2307/2001590.
[12]	J. D. Murray, "Mathematical Biology I, II," New York, 2003.
[13]	P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.
[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), 1029-1043. doi: 10.1098/rspa.2006.1806.
[15]	X.-Q. Zhao, "Dynamical Systems in Population Biology," CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 16, Springer-Verlag, New York, 2003.

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