Variational principle for spreading speeds and generalized propagating speeds in time almost periodic and space periodic KPP models

Abstract

Spatial spread and front propagation dynamics is one of the most important dynamical issues in KPP models. Such dynamics of KPP models in time independent or periodic media has been widely studied. Recently, the author of the current paper with Huang established some theoretical foundation for the study of spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. A notion of spreading speed intervals for such models was introduced in the above-mentioned paper and was shown to be the natural extension of the classical concept of the spreading speeds for time independent or periodic KPP models and that it could be used for more general time dependent KPP models. A notion of generalized propagating speed intervals of front solutions and a notion of traveling wave solutions to time almost periodic and space periodic KPP models were also introduced, which are the generalizations of wave speeds and traveling wave solutions in time independent or periodic KPP models. The aim of the current paper is to gain some further qualitative and quantitative understanding of the spatial spread and front propagation dynamics of KPP models in time almost periodic and space periodic media. By applying the principal Lyapunov exponent and the principal Floquet bundle theory for time almost periodic parabolic equations, we provide various useful estimates for spreading and generalized propagating speeds for such KPP models. Under the so-called linear determinacy condition, we show that the spreading speed interval in any given direction is a singleton (called the spreading speed). Moreover, in such a case we establish a variational principle for the spreading speed and prove that there is a front solution of speed $c$ in a given direction if and only if $c$ is greater than or equal to the spreading speed in that direction. Both the estimates and variational principle provide important and efficient tools for the spreading speeds analysis as well as the spreading speeds computation. Based on the variational principle, the influence of time and space variation of the media on the spreading speeds is also discussed in this paper. It is shown that the time and space variation cannot slow down the spatial spread and that it indeed speeds up the spatial spread except in certain degenerate cases, which provides deep insights into the understanding of the influence of the inhomogeneity of the underline media on the spatial spread in KPP models.