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Criteria for the existence of principal eigenvalues of time periodic cooperative linear systems with nonlocal dispersal


Authors: Xiongxiong Bao and Wenxian Shen
Journal: Proc. Amer. Math. Soc. 145 (2017), 2881-2894
MSC (2010): Primary 35K55, 45C05, 45M15, 45G15, 47G20
DOI: https://doi.org/10.1090/proc/13602
Published electronically: February 21, 2017
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Abstract: The current paper establishes criteria for the existence of principal eigenvalues of time periodic cooperative linear nonlocal dispersal systems with Dirichlet type, Neumann type or periodic type boundary conditions. It is shown that such a nonlocal dispersal system has a principal eigenvalue in the following cases: the nonlocal dispersal distance is sufficiently small; the spatial inhomogeneity satisfies a so-called vanishing condition; or the spatial inhomogeneity is nearly globally homogeneous. Moreover, it is shown that the principal eigenvalue of a time periodic cooperative linear nonlocal dispersal system (if it exists) is algebraically simple. A linear nonlocal dispersal system may not have a principal eigenvalue. The results established in the current paper extend those in literature for time independent or periodic nonlocal dispersal equations to time periodic cooperative nonlocal dispersal systems and will serve as a basic tool for the study of cooperative nonlinear systems with nonlocal dispersal.


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Additional Information

Xiongxiong Bao
Affiliation: School of Science, Chang’an University, Xi’an, Shaanxi 710064, People’s Republic of China
Email: baoxx2016@chd.edu.cn

Wenxian Shen
Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn University, Alabama 36849
Email: wenxish@auburn.edu

DOI: https://doi.org/10.1090/proc/13602
Keywords: Nonlocal dispersal, cooperative system, principal spectrum point, principal eigenvalue
Received by editor(s): May 6, 2016
Received by editor(s) in revised form: May 17, 2016
Published electronically: February 21, 2017
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society