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Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system
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 |
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 [
References:
[1] |
X. Chen,
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2 (1997), 125-160.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
X-Q. Zhao, Dynamical System in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
X-Q. Zhao, Dynamical System in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
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