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February  2019, 24(2): 511-528. doi: 10.3934/dcdsb.2018184

Bistable waves of a recursive system arising from seasonal age-structured population models

Yingli Pan 1, , Ying Su 1,, and  Junjie Wei 2,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
2. 

Department of Mathematics, Harbin Institute of Technology Weihai, Weihai, Shandong 264209, China

* Corresponding author

Received  December 2017 Revised  January 2018 Published  February 2019 Early access  June 2018

This paper is devoted to the existence, uniqueness and stability of bistable traveling waves for a recursive system, which is defined by the iterations of the Ponicaré map of a yearly periodic age-structured population model derived in the companion paper [8]. The existence of the wave is established by appealing to a monotone dynamical system theory, and the uniqueness and stability are obtained by employing a squeezing method.

Keywords: Bistable wave, exponentially asymptotical stability, recursive system.
Mathematics Subject Classification: Primary: 39A30; Secondary: 92D25, 45P05, 45M10.
Citation: Yingli Pan, Ying Su, Junjie Wei. Bistable waves of a recursive system arising from seasonal age-structured population models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 511-528. doi: 10.3934/dcdsb.2018184
References:
[1]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.   Google Scholar

[2]

R. Coutinho and B. Fernandez, Fronts in extended systems of bistable maps coupled via convolutions, Nonlinearity, 17 (2004), 23-27.  doi: 10.1088/0951-7715/17/1/002.  Google Scholar

[3]

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

[4]

J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556.  Google Scholar

[5]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[6]

R. Lui, A nonlinear integral operator arising from a model in population genetics, Ⅰ. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937.  doi: 10.1137/0513064.  Google Scholar

[7]

R. Lui, Existence and stability of traveling wave solutions of a nonlinear integral operator, J. Math. Biology, 16 (1983), 199-220.  doi: 10.1007/BF00276502.  Google Scholar

[8]

Y. Pan, J. Fang and J. Wei, Seasonal influence on stage-structured invasive species with yearly generation, SIAM J. Appl. Math., to appear, arXiv: 1712.06241. Google Scholar

[9]

H. L. Smith and X-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.  Google Scholar

[10]

Z. Wang, W. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, Journal of Differential Equations, 238 (2007), 152-200.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[11]

H. F. Weinberger, Long time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[12]

Y. Zhang and X-Q. Zhao, Bistable travelling waves in competitive recursion systems, Journal of Differential Equations, 252 (2012), 2630-2647.  doi: 10.1016/j.jde.2011.10.005.  Google Scholar

[13]

Y. Zhang and X.-Q. Zhao, Spatial dynamics of a reaction-diffusion model with distributed delay, Math. Model. Nat. Phenom., 8 (2013), 60-77.  doi: 10.1051/mmnp/20138306.  Google Scholar

[14]

Y. Zhang and X-Q. Zhao, Bistable travelling waves for a reaction and diffusion model with seasonal succession, Nonlinearity, 36 (2013), 691-709.  doi: 10.1088/0951-7715/26/3/691.  Google Scholar

[15]

X-Q. Zhao, Dynamical System in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.   Google Scholar

[2]

R. Coutinho and B. Fernandez, Fronts in extended systems of bistable maps coupled via convolutions, Nonlinearity, 17 (2004), 23-27.  doi: 10.1088/0951-7715/17/1/002.  Google Scholar

[3]

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

[4]

J. Fang and X.-Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Eur. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556.  Google Scholar

[5]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[6]

R. Lui, A nonlinear integral operator arising from a model in population genetics, Ⅰ. Monotone initial data, SIAM J. Math. Anal., 13 (1982), 913-937.  doi: 10.1137/0513064.  Google Scholar

[7]

R. Lui, Existence and stability of traveling wave solutions of a nonlinear integral operator, J. Math. Biology, 16 (1983), 199-220.  doi: 10.1007/BF00276502.  Google Scholar

[8]

Y. Pan, J. Fang and J. Wei, Seasonal influence on stage-structured invasive species with yearly generation, SIAM J. Appl. Math., to appear, arXiv: 1712.06241. Google Scholar

[9]

H. L. Smith and X-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.  Google Scholar

[10]

Z. Wang, W. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, Journal of Differential Equations, 238 (2007), 152-200.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[11]

H. F. Weinberger, Long time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar

[12]

Y. Zhang and X-Q. Zhao, Bistable travelling waves in competitive recursion systems, Journal of Differential Equations, 252 (2012), 2630-2647.  doi: 10.1016/j.jde.2011.10.005.  Google Scholar

[13]

Y. Zhang and X.-Q. Zhao, Spatial dynamics of a reaction-diffusion model with distributed delay, Math. Model. Nat. Phenom., 8 (2013), 60-77.  doi: 10.1051/mmnp/20138306.  Google Scholar

[14]

Y. Zhang and X-Q. Zhao, Bistable travelling waves for a reaction and diffusion model with seasonal succession, Nonlinearity, 36 (2013), 691-709.  doi: 10.1088/0951-7715/26/3/691.  Google Scholar

[15]

X-Q. Zhao, Dynamical System in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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