• Previous Article
    Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay
  • DCDS-B Home
  • This Issue
  • Next Article
    A simple regulatory circuit that can simultaneously generate excitability of two different mechanisms
January  2012, 17(1): 283-295. doi: 10.3934/dcdsb.2012.17.283

A periodic reaction-diffusion model with a quiescent stage

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.
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J. Cook, "Dispersive Variability and Invasion Wave Speeds," unpublished manuscript, 1993.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.

[1]

Xiongxiong Bao, Wenxian Shen, Zhongwei Shen. Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems. Communications on Pure and Applied Analysis, 2019, 18 (1) : 361-396. doi: 10.3934/cpaa.2019019

[2]

Nar Rawal, Wenxian Shen, Aijun Zhang. Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1609-1640. doi: 10.3934/dcds.2015.35.1609

[3]

Haiyan Wang, Carlos Castillo-Chavez. Spreading speeds and traveling waves for non-cooperative integro-difference systems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2243-2266. doi: 10.3934/dcdsb.2012.17.2243

[4]

Liang Kong, Tung Nguyen, Wenxian Shen. Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1613-1636. doi: 10.3934/cpaa.2019077

[5]

Shuang-Ming Wang, Zhaosheng Feng, Zhi-Cheng Wang, Liang Zhang. Spreading speed and periodic traveling waves of a time periodic and diffusive SI epidemic model with demographic structure. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2005-2034. doi: 10.3934/cpaa.2021145

[6]

Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663

[7]

Weiwei Ding, Xing Liang, Bin Xu. Spreading speeds of $N$-season spatially periodic integro-difference models. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3443-3472. doi: 10.3934/dcds.2013.33.3443

[8]

Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268

[9]

Isabel Coelho, Carlota Rebelo, Elisa Sovrano. Extinction or coexistence in periodic Kolmogorov systems of competitive type. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5743-5764. doi: 10.3934/dcds.2021094

[10]

Wei-Jian Bo, Guo Lin. Asymptotic spreading of time periodic competition diffusion systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3901-3914. doi: 10.3934/dcdsb.2018116

[11]

Guo Lin, Shuxia Pan. Periodic traveling wave solutions of periodic integrodifference systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3005-3031. doi: 10.3934/dcdsb.2020049

[12]

Guangyu Zhao. Multidimensional periodic traveling waves in infinite cylinders. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 1025-1045. doi: 10.3934/dcds.2009.24.1025

[13]

Guy Métivier, Kevin Zumbrun. Large-amplitude modulation of periodic traveling waves. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022070

[14]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043

[15]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601

[16]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397

[17]

Bendong Lou. Periodic traveling waves of a mean curvature flow in heterogeneous media. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 231-249. doi: 10.3934/dcds.2009.25.231

[18]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993

[19]

Thuc Manh Le, Nguyen Van Minh. Monotone traveling waves in a general discrete model for populations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3221-3234. doi: 10.3934/dcdsb.2017171

[20]

Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]