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## Spatial Spread Dynamics of Monostable Equations in Spatially Locally Inhomogeneous Media with Temporal Periodicity

##### Date

2013-05-31##### Author

Kong, Liang

##### Type of Degree

dissertation##### Department

Mathematics and Statistics##### Metadata

Show full item record##### Abstract

This dissertation is devoted to the study of semilinear dispersal evolution equations.
These type of equations are called as Monostable or KPP type equations, which arise
in modeling the population dynamics of many species which exhibit local, nonlocal and
discrete internal interactions and live in locally spatially inhomogeneous media with temporal
periodicity. The following main results are proved in the dissertation.
Firstly, it is proved that Liouville type property holds for such equations, that is, time
periodic strictly positive solutions are unique. It is proved that if time periodic strictly
positive solutions (if exists) are globally stable with respect to strictly positive perturbations.
Moreover, it is proved that if the trivial solution u = 0 of the limit equation of such an
equation is linearly unstable, then the equation has a time periodic strictly positive solution.
Secondly, spatial spreading speeds of such equations is investigated. It is also proved
that if u 0 is a linearly unstable solution to the time and space periodic limit equation of
such an equation, then the original equation has a spatial spreading speed in every direction.
Moreover, it is proved that the localized spatial inhomogeneity neither slows down nor speeds
up the spatial spreading speeds. In addition, in the time dependent case, various spreading
features of the spreading speeds are obtained. Finally, the e ects of temporal and spatial variations on the uniform persistence and
spatial spreading speeds of such equations are considered. As in the periodic media case, it
is shown that temporal and spatial variations favor the population's persistence and do not
reduce the spatial spreading speeds.

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