American Institute of Mathematical Sciences

Advanced
March  2016, 21(2): 591-606. doi: 10.3934/dcdsb.2016.21.591

Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession

Manjun Ma 1, and  Xiao-Qiang Zhao 2,

1. 

Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China
2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7

Received  August 2014 Revised  November 2014 Published  November 2015

This paper is devoted to the study of propagation phenomena for a two-species competitive reaction-diffusion model with seasonal succession in the monostable case. By appealing to theory of traveling waves and spreading speeds for monotone semiflows, we establish the existence of the minimal wave speed for rightward traveling waves and its coincidence with the rightward spreading speed. We also obtain a set of sufficient conditions for the spreading speed to be linearly determinate.
Keywords: minimal wave speed, Reaction and diffusion, linear determinacy., periodic traveling waves, spreading speed, seasonal succession.
Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 37N25, 92D2.
Citation: Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591
References:
[1]

P. J. DuBowy, Waterfowl communities and seasonal environments: Temporal variabolity in interspecific competition,, Ecology, 69 (1988), 1439.  doi: 10.2307/1941641.  Google Scholar

[2]

J. Fang and X.-Q. Zhao, Travelling waves for monotone semiflows with weak compactness,, SIAM J. Math. Anal., 46 (2014), 3678.  doi: 10.1137/140953939.  Google Scholar

[3]

S. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay,, SIAM J. Appl. Dyn. Syst., 9 (2010), 589.  doi: 10.1137/090767261.  Google Scholar

[4]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession,, J. Math. Biol., 64 (2012), 109.  doi: 10.1007/s00285-011-0408-6.  Google Scholar

[5]

S. S. Hu and A. J. Tessier, Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage,, Ecology, 76 (1995), 2278.  doi: 10.2307/1941702.  Google Scholar

[6]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[7]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and travelling waves for periodic evolution systems,, J. Differential Equations, 231 (2006), 57.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[8]

X. Liang and X.-Q. Zhao, Asymptotic speeds of speed and travelling waves for monotone semiflows with application,, Comm. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[9]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light,, Amer. Naturalist, 157 (2001), 170.  doi: 10.1086/318628.  Google Scholar

[10]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete. Contin. Dyn. Syst. (Ser. A), 33 (2013), 2007.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[11]

C. F. Steiner, A. S. Schwaderer, V. Huber, C. A. Klausmeier and E. Litchman, Periodically forced food chain dynamics: Model predictions and experimental validation,, Ecology, 90 (2009), 3099.  doi: 10.1890/08-2377.1.  Google Scholar

[12]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183.  doi: 10.1007/s002850200145.  Google Scholar

[13]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[14]

Y. Zhang and X.-Q. Zhao, Bistable travelling waves for a reaction and diffusion model with seasonal succession,, Nonlinearity, 26 (2013), 691.  doi: 10.1088/0951-7715/26/3/691.  Google Scholar

show all references

References:
[1]

P. J. DuBowy, Waterfowl communities and seasonal environments: Temporal variabolity in interspecific competition,, Ecology, 69 (1988), 1439.  doi: 10.2307/1941641.  Google Scholar

[2]

J. Fang and X.-Q. Zhao, Travelling waves for monotone semiflows with weak compactness,, SIAM J. Math. Anal., 46 (2014), 3678.  doi: 10.1137/140953939.  Google Scholar

[3]

S. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay,, SIAM J. Appl. Dyn. Syst., 9 (2010), 589.  doi: 10.1137/090767261.  Google Scholar

[4]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession,, J. Math. Biol., 64 (2012), 109.  doi: 10.1007/s00285-011-0408-6.  Google Scholar

[5]

S. S. Hu and A. J. Tessier, Seasonal succession and the strength of intra- and interspecific competition in a Daphnia assemblage,, Ecology, 76 (1995), 2278.  doi: 10.2307/1941702.  Google Scholar

[6]

M. A. Lewis, B. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models,, J. Math. Biol., 45 (2002), 219.  doi: 10.1007/s002850200144.  Google Scholar

[7]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and travelling waves for periodic evolution systems,, J. Differential Equations, 231 (2006), 57.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar

[8]

X. Liang and X.-Q. Zhao, Asymptotic speeds of speed and travelling waves for monotone semiflows with application,, Comm. Pure Appl. Math., 60 (2007), 1.  doi: 10.1002/cpa.20154.  Google Scholar

[9]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light,, Amer. Naturalist, 157 (2001), 170.  doi: 10.1086/318628.  Google Scholar

[10]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession,, Discrete. Contin. Dyn. Syst. (Ser. A), 33 (2013), 2007.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[11]

C. F. Steiner, A. S. Schwaderer, V. Huber, C. A. Klausmeier and E. Litchman, Periodically forced food chain dynamics: Model predictions and experimental validation,, Ecology, 90 (2009), 3099.  doi: 10.1890/08-2377.1.  Google Scholar

[12]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183.  doi: 10.1007/s002850200145.  Google Scholar

[13]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[14]

Y. Zhang and X.-Q. Zhao, Bistable travelling waves for a reaction and diffusion model with seasonal succession,, Nonlinearity, 26 (2013), 691.  doi: 10.1088/0951-7715/26/3/691.  Google Scholar

[1]

Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126
[2]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441
[3]

Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067
[4]

Gregoire Nadin. How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1785-1803. doi: 10.3934/dcdsb.2015.20.1785
[5]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[6]

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707
[7]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
[8]

Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043
[9]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[10]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[11]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883
[12]

Juliette Bouhours, Grégroie Nadin. A variational approach to reaction-diffusion equations with forced speed in dimension 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1843-1872. doi: 10.3934/dcds.2015.35.1843
[13]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[14]

Hans Weinberger. On sufficient conditions for a linearly determinate spreading speed. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2267-2280. doi: 10.3934/dcdsb.2012.17.2267
[15]

Gary Bunting, Yihong Du, Krzysztof Krakowski. Spreading speed revisited: Analysis of a free boundary model. Networks & Heterogeneous Media, 2012, 7 (4) : 583-603. doi: 10.3934/nhm.2012.7.583
[16]

Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405
[17]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141
[18]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[19]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41
[20]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences