American Institute of Mathematical Sciences

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May  2010, 13(3): 709-728. doi: 10.3934/dcdsb.2010.13.709

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

Zhi-Xian Yu 1, and  Rong Yuan 1,

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China, China

Received  May 2009 Revised  October 2009 Published  February 2010

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: upper-lower solutions, monotone iteration., Traveling wave solutions; reaction-diffusion systems, Nicholson's blowflies equations.
Mathematics Subject Classification: Primary: 35K57, 34K1.
Citation: Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
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