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

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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 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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