Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats

Authors:
Wenxian Shen and Aijun Zhang

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1681-1696

MSC (2010):
Primary 45C05, 45G10, 45M20, 47G10, 92D25

Published electronically:
September 2, 2011

MathSciNet review:
2869152

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors' earlier work for the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend our earlier results to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

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Additional Information

**Wenxian Shen**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
wenxish@auburn.edu

**Aijun Zhang**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
zhangai@auburn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11011-6

Keywords:
Monostable equation,
nonlocal dispersal,
random dispersal,
periodic habitat,
spreading speed,
principal eigenvalue,
principal eigenfunction,
variational principle.

Received by editor(s):
September 17, 2010

Received by editor(s) in revised form:
January 17, 2011

Published electronically:
September 2, 2011

Additional Notes:
This work was partially supported by NSF grant DMS–0907752

Communicated by:
Yingfei Yi

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.