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doi: 10.3934/dcdsb.2021265
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The minimal wave speed of the Lotka-Volterra competition model with seasonal succession

Wentao Meng , Yuanxi Yue , and  Manjun Ma ,

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

* Corresponding authors

Received  February 2021 Revised  July 2021 Early access November 2021

Fund Project: The work of M. Ma was supported by National Natural Science Foundation of China (No. 12071434, No. 12011530398)

This paper focuses on the minimal wave speed of time-periodic traveling waves to a Lotka-Volterra competition model with seasonal succession. It is the first time the general conditions of linear selection and nonlinear selection have been derived by the comparison principle and the upper-lower solution method. Based on the decay characteristics of traveling waves, we obtain some explicit conditions for determining the selection mechanism of the minimal wave speed by constructing upper/lower solutions, which include the first explicit condition for the nonlinear selection and the explicit conditions for the linear selection that greatly improve the result in the reference.

Keywords: The minimal wave speed, periodic traveling waves, seasonal succession, linear and nonlinear selection.
Mathematics Subject Classification: 35K57, 35K40, 35C07, 92D25.
Citation: Wentao Meng, Yuanxi Yue, Manjun Ma. The minimal wave speed of the Lotka-Volterra competition model with seasonal succession. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021265
References:
[1]

N. Abrantes, S. C. Antunes and M. J. Pereira, Seasonal succession of cladocerans and phytoplankton and their interactions in a shallow eutrophic lake (Lake Vela, Portugal), Acta Oecologica, 29 (2006), 54-64.  doi: 10.1016/j.actao.2005.07.006.  Google Scholar

[2]

J. Gamier, G. Billen and M. Coste, Seasonal succession of diatoms and Chlorophyceae in the drainage network of the Seine River: Observation and modeling, Limnology and Oceanography, 40 (1995), 750-765.  doi: 10.4319/lo.1995.40.4.0750.  Google Scholar

[3]

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.  Google Scholar

[4]

J. Li and A. Zhao, Stability analysis of a non-autonomous Lotka-Volterra competition model with seasonal succession, Appl. Math. Model., 40 (2016), 763-781.  doi: 10.1016/j.apm.2015.10.035.  Google Scholar

[5]

M. Ma, Z. Huang and C. Ou, Speed of the traveling wave for the bistable Lotka-Volterra competition mode, Nonlinearity, 32 (2019), 3143-3162.  doi: 10.1088/1361-6544/ab231c.  Google Scholar

[6]

M. Ma and X.-Q. Zhao, Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 591-606.  doi: 10.3934/dcdsb.2016.21.591.  Google Scholar

[7]

H. Müller, A. Schöne and R. M. Pinto-Coelho, Seasonal succession of ciliates in lake constance, Microbial Ecology, 21 (1991), 119-138.   Google Scholar

[8]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[9]

J. Pinhassi and Å. Hagström, Seasonal succession in marine bacterioplankton, Aquatic Microbial Ecology, 21 (2000), 245-256.  doi: 10.3354/ame021245.  Google Scholar

[10]

D. E. Raitsos, Y. Pradhan and R. J. W. Brewin, et al, Remote sensing the phytoplankton seasonal succession of the red sea, PLoS ONE, 8 (2013). doi: 10.1371/journal.pone.0064909.  Google Scholar

[11]

S. K. Schmidt, E. K. Costello and D. R. Nemergut, Biogeochemical consequences of rapid microbial turnover and seasonal succession in soil, Ecology, 88 (2007), 1379-1385.  doi: 10.1890/06-0164.  Google Scholar

[12]

U. Sommer, Z. M. Gliwicz and W. I. Lampert, The PEG-model of seasonal succession of planktonic events in fresh waters, Archiv für Hydrobiologie, 106 (1986), 433-471.   Google Scholar

[13]

H. Y. Wang, H. L. Wang and C. H. Ou, Spreading dynamics of a Lotka-Volterra competition model in periodic habitats, J. Differential Equations, 270 (2021), 664-693.  doi: 10.1016/j.jde.2020.08.016.  Google Scholar

[14]

Y. X. Yue, Y. Z. Han, J. C. Tao and M. Ma, The minimal wave speed to the Lotka-Volterra competition model, J. Math. Anal. Appl., 488 (2020), 124106, 11pp. doi: 10.1016/j.jmaa.2020.124106.  Google Scholar

[15]

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.  Google Scholar

[16]

G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[17]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer Nature, Switzerland, 2017. Google Scholar

show all references

References:
[1]

N. Abrantes, S. C. Antunes and M. J. Pereira, Seasonal succession of cladocerans and phytoplankton and their interactions in a shallow eutrophic lake (Lake Vela, Portugal), Acta Oecologica, 29 (2006), 54-64.  doi: 10.1016/j.actao.2005.07.006.  Google Scholar

[2]

J. Gamier, G. Billen and M. Coste, Seasonal succession of diatoms and Chlorophyceae in the drainage network of the Seine River: Observation and modeling, Limnology and Oceanography, 40 (1995), 750-765.  doi: 10.4319/lo.1995.40.4.0750.  Google Scholar

[3]

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.  Google Scholar

[4]

J. Li and A. Zhao, Stability analysis of a non-autonomous Lotka-Volterra competition model with seasonal succession, Appl. Math. Model., 40 (2016), 763-781.  doi: 10.1016/j.apm.2015.10.035.  Google Scholar

[5]

M. Ma, Z. Huang and C. Ou, Speed of the traveling wave for the bistable Lotka-Volterra competition mode, Nonlinearity, 32 (2019), 3143-3162.  doi: 10.1088/1361-6544/ab231c.  Google Scholar

[6]

M. Ma and X.-Q. Zhao, Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 591-606.  doi: 10.3934/dcdsb.2016.21.591.  Google Scholar

[7]

H. Müller, A. Schöne and R. M. Pinto-Coelho, Seasonal succession of ciliates in lake constance, Microbial Ecology, 21 (1991), 119-138.   Google Scholar

[8]

R. Peng and X.-Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007.  Google Scholar

[9]

J. Pinhassi and Å. Hagström, Seasonal succession in marine bacterioplankton, Aquatic Microbial Ecology, 21 (2000), 245-256.  doi: 10.3354/ame021245.  Google Scholar

[10]

D. E. Raitsos, Y. Pradhan and R. J. W. Brewin, et al, Remote sensing the phytoplankton seasonal succession of the red sea, PLoS ONE, 8 (2013). doi: 10.1371/journal.pone.0064909.  Google Scholar

[11]

S. K. Schmidt, E. K. Costello and D. R. Nemergut, Biogeochemical consequences of rapid microbial turnover and seasonal succession in soil, Ecology, 88 (2007), 1379-1385.  doi: 10.1890/06-0164.  Google Scholar

[12]

U. Sommer, Z. M. Gliwicz and W. I. Lampert, The PEG-model of seasonal succession of planktonic events in fresh waters, Archiv für Hydrobiologie, 106 (1986), 433-471.   Google Scholar

[13]

H. Y. Wang, H. L. Wang and C. H. Ou, Spreading dynamics of a Lotka-Volterra competition model in periodic habitats, J. Differential Equations, 270 (2021), 664-693.  doi: 10.1016/j.jde.2020.08.016.  Google Scholar

[14]

Y. X. Yue, Y. Z. Han, J. C. Tao and M. Ma, The minimal wave speed to the Lotka-Volterra competition model, J. Math. Anal. Appl., 488 (2020), 124106, 11pp. doi: 10.1016/j.jmaa.2020.124106.  Google Scholar

[15]

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.  Google Scholar

[16]

G. Zhao and S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011), 627-671.  doi: 10.1016/j.matpur.2010.11.005.  Google Scholar

[17]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer Nature, Switzerland, 2017. Google Scholar

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