Mutual inclusion in a nonlocal competitive Lotka Volterra system

  • Xiaojie HouEmail author
  • Biao Wang
  • Zhengce Zhang
Original Paper Area 1


We investigate the traveling front solutions of a nonlocal Lotka Volterra system to illustrate the outcome of the competition between two species. The existence of the front solution is obtained through a new monotone iteration scheme, the uniqueness of the front solution corresponding to each propagation speed is proved by sliding domain method adapted to nonlocal systems, and the asymptotic decay rate of the fronts with critical and noncritical wave speeds is derived by a new method, which is different from the single equation case. The results demonstrate that in the long run, two weakly competing species can co-exist.


Nonlocal Lotka Volterra Traveling front Asymptotic behavior Uniqueness 

Mathematics Subject Classification (2000)

Primary 35B35 Secondary 91B18 35K57 35B40 


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  1. 1.
    Bates P., Chmaj A.: An integro-differential model for phase transitions: Stationary solutions in higher space dimensions. J. Stat. Phys. 95, 1119–1139 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bates P., Fife P., Ren X., Wang X.: Travelling waves in a convolution model for phase transitions. Arch. Rational Mech. Anal. 138, 105–136 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berestycki, H., Nirenberg, L.: On the method of moving planes and the sliding method. Bull. Braz. Math. Soc. 22(1), 1–37Google Scholar
  4. 4.
    Carr J., Chmaj A.: Uniqueness of travelling waves for nonlocal monostable equations. Proc. Am. Math. Soc. 132, 2433–2439 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen X.: Existence, uniqueness and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997)zbMATHGoogle Scholar
  6. 6.
    Cortazar C., Elgueta M., Rossi J.: A non-local diffusion equation whose solutions develop a free boundary. Ann. Henri Poincaré 6, 269–281 (2005)Google Scholar
  7. 7.
    Cortazar C., Elgueta M., Rossi J., Wolanski N.: Boundary fluxes for non-local diffusion. J. Differ. Equ. 234, 360–390 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ermrntrout B., Mcleod J.: Existence and uniqueness of traveling waves for a neural network. Proc. R. Soc. Edinburgh 123A, 1013–1022 (1994)Google Scholar
  9. 9.
    Coville, J.: Equations de réaction diffusion non-locale. Ph.D thesis, University of Paris (2003)Google Scholar
  10. 10.
    Coville J.: Maximum principles, sliding techniques and applications to nonlocal equations. Electron. J. Diff. Eqns. 68, 1–23 (2007)Google Scholar
  11. 11.
    Coville J., Dupaigne L.: Propagation speed of travelling fronts in nonlocal reaction diffusion equations. Nonl. Anal. TMA 60, 797–819 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Coville J., Dupaigne L.: On a nonlocal equation arising in population dynamics. Proc. R. Soc. Edinburgh 137, 1–29 (2007)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fang J.J., Zhao X.: Existence and uniqueness of traveling waves for non-monotone integral equations with applications. J. Differ. Equ. 248, 2199–2226 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fife P., Wang X.: A convolution model for interfacial motion: the generation and propagation of internal layers in higher space dimensions. Adv. Differ. Equ. 3, 85–110 (1998)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Hou, X.: On the minimal speed and asymptotics of the wave solutions for the lotka volterra system arXiv:1002.2882Google Scholar
  16. 16.
    Hou, X., Li, Y.: Competition exclusion—a case study of the front solutions to the nonlocal Lotka Volterra system, PreprintGoogle Scholar
  17. 17.
    Hou X., Feng W.: Traveling waves and their stability in a coupled reaction diffusion system. Comm. Pure Appl. Anal. 10, 141–160 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ignat L., Rossi J.: A nonlocal convection-diffusion equation. J. Funct. Anal. 251, 399–437 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lv G.: Asymptotic behavior of traveling wave fronts and entire solutions for a nonlocal monostable equation. Nonlinear Anal. 72, 3659–3668 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    de Masi A., Orlandi E., Presutti E.: Traveling fronts in nonlocal evolution equations. Arch. Rational Mech. Anal. 132(2), 143–205 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Mei M., Ou C., Zhao X.: Global stability of monostable traveling waves for nonlocal time-delayed reaction-difusion equations. SIAM J. Math. Anal. 42(6), 2762–2790 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Pan S., Li W., Lin G.: Traveling fronts in nonlocal delayed reaction-diffusion systems and applications. Z. Angew. Math. Phys. 60, 377–392 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Schumacher K.: Traveling-front solutions for integro-differential equations. I, Jurnal Fur Die Reine Und Angewandte Mathematik 316, 54–70 (1979)MathSciNetGoogle Scholar
  24. 24.
    Widder D.V.: The Laplace Transform. Princeton University Press, Princeton (1941)Google Scholar
  25. 25.
    Zhang L.: Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks. J. Differ. Eqn. 197, 162–196 (2004)CrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang L., Li B.: Traveling wave solutions in an integro-differential competition model. Discr. Continuous Dyn. Syst. Ser. B (DCDS-B) 17(1), 417–428 (2012)CrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North Carolina WilmingtonNCUSA
  2. 2.School of Mathematics and Statistics, Xi’an Jiaotong UniversityXi’anChina

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