• Previous Article
    Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system
  • DCDS-B Home
  • This Issue
  • Next Article
    The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition
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.

[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.

[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. 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.

[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.

[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. 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.

[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.

[1]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

[2]

Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659

[3]

Alberto d'Onofrio. On the interaction between the immune system and an exponentially replicating pathogen. Mathematical Biosciences & Engineering, 2010, 7 (3) : 579-602. doi: 10.3934/mbe.2010.7.579

[4]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255

[5]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601

[6]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329

[7]

Yuncheng You. Asymptotical dynamics of Selkov equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 193-219. doi: 10.3934/dcdss.2009.2.193

[8]

Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056

[9]

Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115

[10]

Yuncheng You. Asymptotical dynamics of the modified Schnackenberg equations. Conference Publications, 2009, 2009 (Special) : 857-868. doi: 10.3934/proc.2009.2009.857

[11]

Giuseppe Buttazzo, Filippo Santambrogio. Asymptotical compliance optimization for connected networks. Networks & Heterogeneous Media, 2007, 2 (4) : 761-777. doi: 10.3934/nhm.2007.2.761

[12]

Noam Presman, Simon Litsyn. Recursive descriptions of polar codes. Advances in Mathematics of Communications, 2017, 11 (1) : 1-65. doi: 10.3934/amc.2017001

[13]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

[14]

Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems & Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713

[15]

Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic & Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729

[16]

Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control & Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010

[17]

Shaolin Ji, Xiaomin Shi. Recursive utility optimization with concave coefficients. Mathematical Control & Related Fields, 2018, 8 (3&4) : 753-775. doi: 10.3934/mcrf.2018033

[18]

Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271

[19]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[20]

Claudia Valls. Stability of some waves in the Boussinesq system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 929-939. doi: 10.3934/cpaa.2006.5.929

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (61)
  • HTML views (293)
  • Cited by (0)

Other articles
by authors

[Back to Top]