Spectral theory for general nonautonomous/random dispersal evolution operators

We investigate the spectral theory of the following general nonautonomous evolution equation∂tu(t,x)=A(u(t,⋅))(x)+h(t,x)u(t,x),x∈D, where D   is a bounded subset of RNRN which can be a smooth domain or a discrete set, AA 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)h(t,x) is a smooth function on R×D¯. 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.