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By reviewing our previous works on lifting dynamics in skew-product semi-flows and also the work of Johnson on almost periodic Floquet theory, we show several significant applications of the abstract theory of topological dynamics to the qualitative study of non-autonomous differential equations. The paper also contains some detailed discussions on a conjecture of Johnson.

MSC: 45C05 45G10 45M20 47G10 92D25 Keywords: Monostable equation Nonlocal dispersal Random dispersal Periodic habitat Spreading speed Spreading speed interval Traveling wave solution Principal eigenvalue Principal eigenfunction Variational principleThe current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established.

In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODEs) and in coupled map lattices (CMLs). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as time-discretizations of lattice ODEs. Finally, we show that these results apply to the discrete Nagumo equation. Academic Press

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