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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications

Pages: 709 - 728, Volume 13, Issue 3, May 2010      doi:10.3934/dcdsb.2010.13.709

 
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Zhi-Xian Yu - School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China (email)
Rong Yuan - School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China (email)

Abstract: This paper is concerned with monotone traveling wave solutions of reaction-diffusion systems with spatio-temporal delay. Our approach is to use a new monotone iteration scheme based on a lower solution in the set of the profiles. The smoothness of upper and lower solutions is not required in this paper. We will apply our results to Nicholson's blowflies systems with non-monotone birth functions and show that the systems admit traveling wave solutions connecting two spatially homogeneous equilibria and the wave shape is monotone. Due to the biological realism, the positivity of the monotone traveling wave solutions can be directly obtained by the construction of suitable upper-lower solutions.

Keywords:  Traveling wave solutions; reaction-diffusion systems, upper-lower solutions, Nicholson's blowflies equations, monotone iteration.
Mathematics Subject Classification:  Primary: 35K57, 34K10.

Received: May 2009;      Revised: October 2009;      Available Online: February 2010.