Search Advanced

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4151-4181. doi: 10.3934/mbe.2019207
Previous Article Next Article
Research article Special Issues

Existence and stability of traveling wavefronts for discrete three species competitive-cooperative systems

  • 1.
    Department of Mathematics, National Central University, Chungli 32001, Taiwan
  • 2.
    General Education Center, National Taipei University of Technology, Taipei 10608, Taiwan
  • 3.
    School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi, 710071, P.R. China
  • Received: 18 January 2019 Accepted: 28 April 2019 Published: 13 May 2019

  • The purpose of this work is to investigate the existence and stability of traveling wavefronts for competitive-cooperative systems with three species. The existence result can be derived by using the technique of monotone method with the help of a pair of explicit supersolution and subsolution. Moreover, some su cient conditions ensure the linear determinacy for the minimal speed is given. Then, applying the weighted energy method, we prove that the traveling wavefronts are asymptotically stable in the weighted Banach spaces provided that the initial perturbations of the traveling wavefronts also belong to the same spaces.

    Citation: Cheng-Hsiung Hsu, Jian-Jhong Lin, Shi-Liang Wu. Existence and stability of traveling wavefronts for discrete three species competitive-cooperative systems[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4151-4181. doi: 10.3934/mbe.2019207

    Related Papers:

  • The purpose of this work is to investigate the existence and stability of traveling wavefronts for competitive-cooperative systems with three species. The existence result can be derived by using the technique of monotone method with the help of a pair of explicit supersolution and subsolution. Moreover, some su cient conditions ensure the linear determinacy for the minimal speed is given. Then, applying the weighted energy method, we prove that the traveling wavefronts are asymptotically stable in the weighted Banach spaces provided that the initial perturbations of the traveling wavefronts also belong to the same spaces.


    加载中


    [1] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355–369.
    [2] C. C. Chen, L. C. Hung, M. Mimura, et al., Exact travelling wave solutions of three-species competition-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653–2669.
    [3] C. C. Chen, L, C, Hung, M. Mimura, et al., Semi-exact equilibrium solutions for three-species competition-diffusion systems, Hiroshima Math. J., 43 (2013), 179–206.
    [4] M. Mimura and M. Tohma, Dynamic coexistence in a three-species competition-diffusion system, Ecol. Complex., 21 (2015), 215–232.
    [5] H. Ikeda, Travelling wave solutions of three-component systems with competition and diffusion, Toyama Math. J., 24 (2001), 37–66.
    [6] H. Ikeda, Dynamics of weakly interacting front and back waves in three-component systems, Toyama Math. J., 30 (2007), 1–34.
    [7] Y. Kan-on and M. Mimura, Singular perturbation approach to a 3-component reaction-diffusion system arising in population dynamics, SIAM J. Math. Anal., 29 (1998), 1519–1536.
    [8] P. D. Miller, Nonmonotone waves in a three species reaction-diffusion model, Methods and Applications of Analysis, 4 (1997), 261–282.
    [9] M. Mimura and P. C. Fife, A 3-component system of competition and diffusion, Hiroshima Math. J., 16 (1986), 189–207.
    [10] J.-S. Guo, Y.Wang, C.-H.Wu, et al., The minimal speed of traveling wave solutions for a diffusive three species competition system, Taiwan. J. Math., 19 (2015), 1805–1829.
    [11] L.-C. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Anal. Real World Appl., 12 (2011), 3691–3700.
    [12] C.-H. Chang, Existence and stability of traveling wave solutions for a competitive-cooperative system of three species, preprint, (2018).
    [13] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system revisited, Discrete Contin. Dyn. Syst.-B, 15 (2011), 171–196.
    [14] M. Mei, C. Ou and X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal timedelayed reaction-diffusion equations, SIAM J. Math. Anal., 42 (2010), 2762–2790; Erratum, SIAM J. Math. Anal., 44 (2012), 538–540.
    [15] D. Sattinger, On the stability of traveling waves, Adv. Math., 22 (1976), 312–355.
    [16] M. Bramson, Convergence of solutions of the Kolmogorov equations to traveling waves, Mem. Amer. Math. Soc., 44 (1983), 285.
    [17] A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling wave solutions of parabolic systems, Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, RI, 1994.
    [18] J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161–230.
    [19] G.-S. Chen, S.-L. Wu and C.-H. Hsu, Stability of traveling wavefronts for a discrete diffusive competition system with three species, J. Math. Anal. Appl., 474 (2019), 909–930.
    [20] C.-H. Hsu, T.-S. Yang and Z. X. Yu, Existence and exponential stability of traveling waves for delayed reaction-diffusion systems, Nonlinearity, 32 (2019), 1206–1236.
    [21] M. Mei, C. K. Lin, C. T. Lin, et al., Traveling wavefronts for time-delayed reaction-diffusion equation:(I) local nonlinearity, J. Differ. Equations, 247 (2009), 495–510.
    [22] M. Mei, C. K. Lin, C. T. Lin, et al., Traveling wavefronts for time-delayed reaction-diffusion equation:(II)local nonlinearity, J. Differ. Equations, 247 (2009), 511–529.
    [23] K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philos. Soc., 81 (1977), 431–433.
    [24] M. A. Lewis, B. Li and H. F.Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219–233.
    [25] R. Martin and H. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1–44.
    [26] M. Mei, J.W.-H. So, M. Y. Li, et al., Asymptotic stability of traveling waves for Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh, 134A (2004), 579–594.
    [27] W.-T. Li, L. Zhang and G.-B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531–1560.
    [28] L.-C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Japan J. Indust. Appl. Math., 29 (2012), 237–251.
    [29] W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dynam. Differ. Equations, 22 (2010), 285–297.
    [30] N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka- Volterra system, Nonlinear Anal. Real World Appl., 4 (2003) 503–524.
    [31] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, New York Berlin, 1981.
    [32] C.-H. Hsu, J.-J. Lin and T.-S. Yang, Traveling wave solutions for delayed lattice reaction-diffusion systems, IMA J. Appl. Math., 80 (2015), 302–323.
    [33] C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behavior of traveling waves for a epidemic model, Nonlinearity, 26 (2013), 121–139. Corrigendum: 26 (2013), 2925–2928.
    [34] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Anal.-Theor., 44 (2001), 239–246.
    [35] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal.-Theor., 28 (1997), 145–164.
    [36] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differ. Equations, 171 (2001), 294–314.
    [37] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257–270.
    [38] M. Rodrigo and M. Mimura, Exact solutions of reaction-diffusion systems and nonlinear wave equations, Japan J. Indust. Appl. Math., 18 (2001), 657–696.
    [39] Q. Ye, Z. Li, M. X.Wang, et al., Introduction to Reaction-Diffusion Equations, 2nd edition, Science Press, Beijing, 2011.
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(3961) PDF downloads(512) Cited by(3)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog