Propagation of monostable traveling fronts in discrete periodic media with delay

Shi-Liang Wu; Cheng-Hsiung Hsu

doi:10.3934/dcds.2018128

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2018, Volume 38, Issue 6: 2987-3022. Doi: 10.3934/dcds.2018128

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Propagation of monostable traveling fronts in discrete periodic media with delay

Shi-Liang Wu^1, , and
Cheng-Hsiung Hsu^2,

1.
School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China
2.
Department of Mathematics, National Central University, Chungli 32001, Taiwan

E-mail address: slwu@xidian.edu.cn(S.-L. Wu)(Corresponding author)
E-mail address: slwu@xidian.edu.cn(S.-L. Wu)(Corresponding author)

Received: August 31, 2017

Published: May 2018

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Abstract

This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 37L60, 92D25.

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