
Mathematical Biosciences and Engineering, 2019, 16(5): 41514181. doi: 10.3934/mbe.2019207.
Research article Special Issues
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Existence and stability of traveling wavefronts for discrete three species competitivecooperative 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: , Accepted: , Published:
Special Issues: Spatial dynamics for epidemic models with dispersal of organisms and heterogenity of environment
Keywords: traveling wavefronts; monotone system; supersolution; subsolution; weighted energy estimate
Citation: ChengHsiung Hsu, JianJhong Lin, ShiLiang Wu. Existence and stability of traveling wavefronts for discrete three species competitivecooperative systems. Mathematical Biosciences and Engineering, 2019, 16(5): 41514181. doi: 10.3934/mbe.2019207
References:
 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 threespecies competitiondiffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2653–2669.
 3. C. C. Chen, L, C, Hung, M. Mimura, et al., Semiexact equilibrium solutions for threespecies competitiondiffusion systems, Hiroshima Math. J., 43 (2013), 179–206.
 4. M. Mimura and M. Tohma, Dynamic coexistence in a threespecies competitiondiffusion system, Ecol. Complex., 21 (2015), 215–232.
 5. H. Ikeda, Travelling wave solutions of threecomponent systems with competition and diffusion, Toyama Math. J., 24 (2001), 37–66.
 6. H. Ikeda, Dynamics of weakly interacting front and back waves in threecomponent systems, Toyama Math. J., 30 (2007), 1–34.
 7. Y. Kanon and M. Mimura, Singular perturbation approach to a 3component reactiondiffusion system arising in population dynamics, SIAM J. Math. Anal., 29 (1998), 1519–1536.
 8. P. D. Miller, Nonmonotone waves in a three species reactiondiffusion model, Methods and Applications of Analysis, 4 (1997), 261–282.
 9. M. Mimura and P. C. Fife, A 3component 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 competitivecooperative LotkaVolterra 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 competitivecooperative system of three species, preprint, (2018).
 13. A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for LotkaVolterra 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 reactiondiffusion 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 reactiondiffusion systems, Nonlinearity, 32 (2019), 1206–1236.
 21. M. Mei, C. K. Lin, C. T. Lin, et al., Traveling wavefronts for timedelayed reactiondiffusion 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 timedelayed reactiondiffusion 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 twospecies competition models, J. Math. Biol., 45 (2002), 219–233.
 25. R. Martin and H. Smith, Abstract functional differential equations and reactiondiffusion 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 LotkaVolterra systems of two competing species, Japan J. Indust. Appl. Math., 29 (2012), 237–251.
 29. W. Huang, Problem on minimum wave speed for a LotkaVolterra reactiondiffusion 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, SpringerVerlag, New York Berlin, 1981.
 32. C.H. Hsu, J.J. Lin and T.S. Yang, Traveling wave solutions for delayed lattice reactiondiffusion 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. Kanon, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Anal.Theor., 44 (2001), 239–246.
 35. Y. Kanon, Fisher wave fronts for the LotkaVolterra competition model with diffusion, Nonlinear Anal.Theor., 28 (1997), 145–164.
 36. S. Ma, Traveling wavefronts for delayed reactiondiffusion systems via a fixed point theorem, J. Differ. Equations, 171 (2001), 294–314.
 37. M. Rodrigo and M. Mimura, Exact solutions of a competitiondiffusion system, Hiroshima Math. J., 30 (2000), 257–270.
 38. M. Rodrigo and M. Mimura, Exact solutions of reactiondiffusion 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 ReactionDiffusion Equations, 2_{nd} 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 Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *