JOURNAL BROWSE
Search
Advanced SearchSearch Tips
TRAVELING WAVE SOLUTIONS IN NONLOCAL DISPERSAL MODELS WITH NONLOCAL DELAYS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
TRAVELING WAVE SOLUTIONS IN NONLOCAL DISPERSAL MODELS WITH NONLOCAL DELAYS
Pan, Shuxia;
  PDF(new window)
 Abstract
This paper is concerned with the traveling wave solutions of nonlocal dispersal models with nonlocal delays. The existence of traveling wave solutions is investigated by the upper and lower solutions, and the asymptotic behavior of traveling wave solutions is studied by the idea of contracting rectangles. To illustrate these results, a delayed competition model is considered by presenting the existence and nonexistence of traveling wave solutions, which completes and improves some known results. In particular, our conclusions can deal with the traveling wave solutions of evolutionary systems which admit large time delays reflecting intraspecific competition in population dynamics and leading to the failure of comparison principle in literature.
 Keywords
upper-lower solutions;asymptotic spreading;contracting rectangle;large delay;
 Language
English
 Cited by
 References
1.
P. W. Bates, On some nonlocal evolution equations arising in materials science, In: Nonlinear dynamics and evolution equations (Ed. by H. Brunner, X. Zhao and X. Zou), pp. 13-52, Fields Inst. Commun. 48, AMS, Providence, 2006.

2.
P. W. Bates, P. C. Fife, X. Ren, and X. Wang, Traveling waves in a convolution model for phase transitions, Arch. Rational Mech. Anal. 138 (1997), no. 2, 105-136. crossref(new window)

3.
J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2433-2439. crossref(new window)

4.
X. Chen, Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations 2 (1997), no. 1, 125-160.

5.
J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Anal. 60 (2005), no. 5, 797-819. crossref(new window)

6.
J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 4, 727-755. crossref(new window)

7.
S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, In: Nonlinear dynamics and evolution equations (Ed. by H. Brunner, X. Zhao and X. Zou), pp. 137-200, Fields Inst. Commun. 48, AMS, Providence, 2006.

8.
L. Hopf, Introduction to Differential Equations of Physics, Dover, New York, 1948.

9.
Y. Jin and X. Q. Zhao, Spatial dynamics of a periodic population model with dispersal, Nonlinearity 22 (2009), no. 5, 1167-1189. crossref(new window)

10.
W. T. Li, Y. Sun, and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl. 11 (2010), no. 4, 2302-2313. crossref(new window)

11.
X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math. 60 (2007), no. 1, 1-40. crossref(new window)

12.
G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterra competition-diffusion models with distributed delays, J. Dynam. Diff. Eqns., in press, DOI: 10.1007/s10884-014-9355-4. crossref(new window)

13.
J. D. Murray, Mathematical Biology, Springer, Berlin-Heidelberg-New York, 1993.

14.
S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl. 346 (2008), no. 2, 415-424. crossref(new window)

15.
S. Pan, W. T. Li, and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys. 60 (2009), no. 3, 377-392. crossref(new window)

16.
S. Pan, W. T. Li, and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay, Nonlinear Anal. 72 (2010), no. 6, 3150-3158. crossref(new window)

17.
W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differential Equations 249 (2010), no. 4, 747-795. crossref(new window)

18.
N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.

19.
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, AMS, Providence, RI, 1995.

20.
Y. Sun, W. T. Li, and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal. 74 (2011), no. 3, 814-826. crossref(new window)

21.
S. Wu and S. Liu, Traveling waves for delayed non-local diffusion equations with crossing-monostability, Appl. Math. Comput. 217 (2010), no. 4, 1435-1444. crossref(new window)

22.
J. Xia, Z. Yu, and R. Yuan, Traveling waves of a competitive Lotka-Volterra model with nonlocal diffusion and time delays, Acta Math. Appl. Sin. 34 (2011), no. 6, 1082-1093.

23.
Z. Xu and P. Weng, Traveling waves in a convolution model with infinite distributed delay and non-monotonicity, Nonlinear Anal. Real World Appl. 12 (2011), no. 1, 633-647. crossref(new window)

24.
Z. Yu and R. Yuan, Travelling wave solutions in nonlocal reaction-diffusion systems with delays and applications, ANZIAM J. 51 (2009), no. 1, 49-66. crossref(new window)

25.
Z. Yu and R. Yuan, Travelling wave solutions in non-local convolution diffusive competitive-cooperative systems, IMA J. Appl. Math. 76 (2011), no. 4, 493-513. crossref(new window)

26.
G. Zhang, W. T. Li, and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling 49 (2009), no. 5-6, 1021-1029. crossref(new window)

27.
G. Zhang, W. T. Li, and Z. C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations 252 (2012), no. 9, 5096-5124. crossref(new window)