Methods and Applications of Analysis

Volume 18 (2011)

Number 4

Positive stationary solutions and spreading speeds of KPP equations in locally spatially in homogeneous media

Pages: 427 – 456

DOI: https://dx.doi.org/10.4310/MAA.2011.v18.n4.a5

Authors

Liang Kong (Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, U.S.A.)

Wenxian Shen (Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, U.S.A.)

Abstract

The current paper is concerned with positive stationary solutions and spatial spreading speeds of KPP type evolution equations with local (i.e. the standard Laplacian) or nonlocal or discrete dispersal in locally spatially inhomogeneous media. It is shown that such an equation has a unique globally stable positive stationary solution and has a spreading speed in every direction. Moreover, it is shown that the localized spatial inhomogeneity of the medium neither slows down nor speeds up the spatial spreading in all the directions.

Keywords

KPP equations, random dispersal, nonlocal dispersal, discrete dispersal, localized spatial inhomogeneity, spreading speed, positive stationary solution, principal eigenvalue, subsolution, super-solution, comparison principle

2010 Mathematics Subject Classification

35K57, 45G10, 58D20, 92D25

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Published 23 July 2012