Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity

Zhang, Liang; Wang, Zhi-Cheng; Zhao, Xiao-Qiang

doi:10.1090/tran/7709

Propagation dynamics of a time periodic and delayed reaction-diffusion model without quasi-monotonicity
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by Liang Zhang, Zhi-Cheng Wang and Xiao-Qiang Zhao PDF

Trans. Amer. Math. Soc. 372 (2019), 1751-1782 Request permission

Abstract:

In this paper, we consider a time periodic non-monotone and nonlocal delayed reaction-diffusion population model with stage structure. We first prove the existence of the asymptotic speed $c^*$ of spread by virtue of two auxiliary equations and comparison arguments. By the method of super- and sub-solutions and the fixed point theorem, as applied to the truncated problem on a finite interval, and the limiting arguments, we then establish the existence of time periodic traveling wave solutions of the model system with wave speed $c>c^*$. We further use the results of the asymptotic speed of spread to obtain the non-existence of traveling wave solutions for wave speed $c<c^*$. Finally, we prove the existence of the critical periodic traveling wave with wave speed $c=c^*$. It turns out that the asymptotic speed of spread coincides with the minimal wave speed for positive periodic traveling waves. These results are also applied to the model system with two prototypical birth functions.

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