Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats

Nar Rawal; Wenxian Shen; Aijun Zhang

doi:10.3934/dcds.2015.35.1609

Article Contents

2015, Volume 35, Issue 4: 1609-1640. Doi: 10.3934/dcds.2015.35.1609

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Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats

1.
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849
2.
Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849
3.
Department of Mathematics, Drexel University, Philadelphia, PA 19014

Received: June 30, 2013

Revised: May 31, 2014

Published: May 2017

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Abstract

This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution $u^*(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^*(\xi)$ in the direction of any given unit vector $\xi$. A variational characterization of $c^*(\xi)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting $u^*(\cdot,\cdot)$ and $0$ propagating in any given direction of $\xi$ with speed $c>c^*(\xi)$.

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Mathematics Subject Classification: 45C05, 45G10, 45M20, 47G20, 92D25.

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