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

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.