Spectral theory for general nonautonomous/random dispersal evolution operators

We investigate the spectral theory of the following general nonautonomous evolution equation where D is a bounded subset of which can be a smooth domain or a discrete set, is a general linear dispersal operator (for example a Laplacian operator, an integral operator with positive kernel or a cooperative discrete operator) and h(t,x) is a smooth function on . We first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal Lyapunov exponent of a time-dependent dispersal equation is always greater than or equal to that of the time-averaged one. Several results about the principal eigenvalue of time-periodic parabolic equations are extended to general time-periodic dispersal ones. Finally, the investigation is generalized to random time-dependent dispersal equations