American Institute of Mathematical Sciences

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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 and 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), 1439-1453. doi: 10.2307/1941641.

[2]

J. Fang and X.-Q. Zhao, Travelling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.1137/140953939.

[3]

S. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay, SIAM J. Appl. Dyn. Syst., 9 (2010), 589-610. doi: 10.1137/090767261.

[4]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130. doi: 10.1007/s00285-011-0408-6.

[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), 2278-2294. doi: 10.2307/1941702.

[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), 219-233. doi: 10.1007/s002850200144.

[7]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and travelling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.

[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), 1-40. doi: 10.1002/cpa.20154.

[9]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light, Amer. Naturalist, 157 (2001), 170-187. doi: 10.1086/318628.

[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), 2007-2031. doi: 10.3934/dcds.2013.33.2007.

[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), 3099-3107. doi: 10.1890/08-2377.1.

[12]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[13]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.

[14]

Y. Zhang and X.-Q. Zhao, Bistable travelling waves for a reaction and diffusion model with seasonal succession, Nonlinearity, 26 (2013), 691-709. doi: 10.1088/0951-7715/26/3/691.

show all references

References:
[1]

P. J. DuBowy, Waterfowl communities and seasonal environments: Temporal variabolity in interspecific competition, Ecology, 69 (1988), 1439-1453. doi: 10.2307/1941641.

[2]

J. Fang and X.-Q. Zhao, Travelling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46 (2014), 3678-3704. doi: 10.1137/140953939.

[3]

S. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay, SIAM J. Appl. Dyn. Syst., 9 (2010), 589-610. doi: 10.1137/090767261.

[4]

S.-B. Hsu and X.-Q. Zhao, A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), 109-130. doi: 10.1007/s00285-011-0408-6.

[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), 2278-2294. doi: 10.2307/1941702.

[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), 219-233. doi: 10.1007/s002850200144.

[7]

X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and travelling waves for periodic evolution systems, J. Differential Equations, 231 (2006), 57-77. doi: 10.1016/j.jde.2006.04.010.

[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), 1-40. doi: 10.1002/cpa.20154.

[9]

E. Litchman and C. A. Klausmeier, Competition of phytoplankton under fluctuating light, Amer. Naturalist, 157 (2001), 170-187. doi: 10.1086/318628.

[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), 2007-2031. doi: 10.3934/dcds.2013.33.2007.

[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), 3099-3107. doi: 10.1890/08-2377.1.

[12]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145.

[13]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.

[14]

Y. Zhang and X.-Q. Zhao, Bistable travelling waves for a reaction and diffusion model with seasonal succession, Nonlinearity, 26 (2013), 691-709. doi: 10.1088/0951-7715/26/3/691.

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