Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion

1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

This paper is concerned with invasion entire solutions of a monostable time periodic Lotka-Volterra competition-diffusion system. We first give the asymptotic behaviors of time periodic traveling wave solutions at infinity by a dynamical approach coupled with the two-sided Laplace transform. According to these asymptotic behaviors, we then obtain some key estimates which are crucial for the construction of an appropriate pair of sub-super solutions. Finally, using the sub-super solutions method and comparison principle, we establish the existence of invasion entire solutions which behave as two periodic traveling fronts with different speeds propagating from both sides of x-axis. In other words, we formulate a new invasion way of the superior species to the inferior one in a time periodic environment.

  Figure/Table
  Supplementary
  Article Metrics

Keywords Time periodic traveling waves; asymptotic behavior; comparison principle; invasion entire solution

Citation: Li-Jun Du, Wan-Tong Li, Jia-Bing Wang. Invasion entire solutions in a time periodic Lotka-Volterra competition system with diffusion. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1187-1213. doi: 10.3934/mbe.2017061

References

  • [1] N. D. Alikakos,P. W. Bates,X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999): 2777-2805.
  • [2] X. Bao,W. T. Li,Z. C. Wang, Time periodic traveling curved fronts in the periodic Lotka-Volterra competition-diffusion system, J. Dynam. Differential Equations, null (2015): 1-36.
  • [3] X. Bao,Z. C. Wang, Existence and stability of time periodic traveling waves for a periodic bistable Lotka-Volterra competition system, J. Differential Equations, 255 (2013): 2402-2435.
  • [4] P. W. Bates,F. Chen, Periodic traveling waves for a nonlocal integro-differential model, Electron. J. Differential Equations, 1999 (1999): 1-19.
  • [5] H. Berestycki,F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002): 949-1032.
  • [6] Z. H. Bu,Z. C. Wang,N. W. Liu, Asymptotic behavior of pulsating fronts and entire solutions of reaction-advection-diffusion equations in periodic media, Nonlinear Anal. Real World Appl., 28 (2016): 48-71.
  • [7] X. Chen,J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005): 62-84.
  • [8] C. Conley,R. Gardner, An application of the generalized morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984): 319-343.
  • [9] J. Foldes,P. Poláčik, On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry, Discrete Cont. Dynam. Syst. Ser. A., 25 (2009): 133-157.
  • [10] Y. Fukao,Y. Morita,H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004): 15-32.
  • [11] R. A. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations., 44 (1982): 343-364.
  • [12] J. S. Guo,Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst. Ser. A., 12 (2005): 193-212.
  • [13] J. S. Guo,C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku Math. J., 62 (2010): 17-28.
  • [14] F. Hamel, Qualitative properties of monostable pulsating fronts: Exponential decayed monotonicity, J. Math. Pures Appl., 89 (2008): 355-399.
  • [15] F. Hamel,N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999): 1255-1276.
  • [16] F. Hamel,N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb R^N$, Arch. Ration. Mech. Anal., 157 (2001): 91-163.
  • [17] Y. Hosono, Singular perturbation analysis of travelling waves of diffusive Lotka-Volterra competition models, Numerical and Applied Mathematics, Part Ⅱ (Paris 1988), null (1989): 687-692.
  • [18] X. Hou,A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008): 2196-2213.
  • [19] Y. Kan-On, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995): 340-363.
  • [20] Y. Kan-On, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997): 145-164.
  • [21] W. T. Li,Y. J. Sun,Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010): 2302-2313.
  • [22] W. T. Li,Z. C. Wang,J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008): 102-129.
  • [23] W. T. Li,J. B. Wang,L. Zhang, Entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats, J. Differential Equations, 261 (2016): 2472-2501.
  • [24] W. T. Li,L. Zhang,G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst. Ser. A., 35 (2015): 1531-1560.
  • [25] N. W. Liu,W. T. Li,Z. C. Wang, Pulsating type entire solutions of monostable reaction-advection-diffusion equations in periodic excitable media, Nonlinear Anal., 75 (2012): 1869-1880.
  • [26] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser, Boston, 1995.
  • [27] G. Lv,M. Wang, Traveling wave front in diffusive and competitive Lotka-Volterra system with delays, Nonlinear Anal. Real World Appl., 11 (2010): 1323-1329.
  • [28] Y. Morita,H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006): 841-861.
  • [29] Y. Morita,K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009): 2217-2240.
  • [30] G. Nadin, Traveling fronts in space-time periodic media, J. Math. Pures Appl., 92 (2009): 232-262.
  • [31] G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion equation, J. Differential Equations, 249 (2010): 1288-1304.
  • [32] J. Nolen,M. Rudd,J. Xin, Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds, Dyn. Partial Differ. Equ., 2 (2005): 1-24.
  • [33] W. Shen, Traveling waves in time periodic lattice differential equations, Nonlinear Anal., 54 (2003): 319-339.
  • [34] W. J. Sheng and J. B. Wang, Entire solutions of time periodic bistable reaction-advection-diffusion equations in infinite cylinders J. Math. Phys., 56 (2015), 081501, 17 pp.
  • [35] Y. J. Sun,W. T. Li,Z. C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity, J. Differential Equations, 251 (2011): 551-581.
  • [36] M. M. Tang,P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal., 73 (1980): 69-77.
  • [37] J. H. Vuuren, The existence of traveling plane waves in a general class of competition-diffusion systems, SIMA J. Appl. Math., 55 (1995): 135-148.
  • [38] M. Wang,G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delays, Nonlinearity, 23 (2010): 1609-1630.
  • [39] Z. C. Wang,W. T. Li,S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009): 2047-2084.
  • [40] Z. C. Wang,W. T. Li,J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009): 2392-2420.
  • [41] H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003): 117-164.
  • [42] L. Zhang,W. T. Li,S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016): 189-224.
  • [43] G. Zhao,S. Ruan, Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion, J. Math. Pures Appl., 95 (2011): 627-671.
  • [44] G. Zhao,S. Ruan, Time periodic traveling wave solutions for periodic advection-reaction-diffusion systems, J. Differential Equations, 257 (2014): 1078-1147.
  • [45] X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.

 

This article has been cited by

  • 1. Li-Jun Du, Wan-Tong Li, Jia-Bing Wang, Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system, Journal of Differential Equations, 2018, 10.1016/j.jde.2018.07.024
  • 2. Wei-Jian Bo, Guo Lin, Yuanwei Qi, Propagation dynamics of a time periodic diffusion equation with degenerate nonlinearity, Nonlinear Analysis: Real World Applications, 2019, 45, 376, 10.1016/j.nonrwa.2018.07.010
  • 3. Li-Jun Du, Wan-Tong Li, Shi-Liang Wu, Pulsating fronts and front-like entire solutions for a reaction–advection–diffusion competition model in a periodic habitat, Journal of Differential Equations, 2019, 10.1016/j.jde.2018.12.029
  • 4. Li-Jun Du, Wan-Tong Li, Shi-Liang Wu, Propagation phenomena for a bistable Lotka–Volterra competition system with advection in a periodic habitat, Zeitschrift für angewandte Mathematik und Physik, 2020, 71, 1, 10.1007/s00033-019-1236-6

Reader Comments

your name: *   your email: *  

Copyright Info: 2017, Wan-Tong Li, et al., 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)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved