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Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones
The minimal wave speed of the Lotka-Volterra competition model with seasonal succession
Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Müller, A. Schöne and R. M. Pinto-Coelho,
Seasonal succession of ciliates in lake constance, Microbial Ecology, 21 (1991), 119-138.
|
[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. |
[9] |
J. Pinhassi and Å. Hagström,
Seasonal succession in marine bacterioplankton, Aquatic Microbial Ecology, 21 (2000), 245-256.
doi: 10.3354/ame021245. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[17] |
X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer Nature, Switzerland, 2017. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Müller, A. Schöne and R. M. Pinto-Coelho,
Seasonal succession of ciliates in lake constance, Microbial Ecology, 21 (1991), 119-138.
|
[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. |
[9] |
J. Pinhassi and Å. Hagström,
Seasonal succession in marine bacterioplankton, Aquatic Microbial Ecology, 21 (2000), 245-256.
doi: 10.3354/ame021245. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[17] |
X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer Nature, Switzerland, 2017. |
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