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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

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  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.

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. WangW. 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. WangW. 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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