Estimates for the principal spectrum point for certain time-dependent parabolic operators

Non-autonomous parabolic equations are discussed. The periodic case is cons idered first and an estimate for the principal periodic-parabolic eigenvalu e is obtained by relating the original problem to the elliptic one obtained by time-averaging. It is then shown that an analogous bound may be obtaine d for the principal spectrum point in the almost periodic case. These resul ts have applications to the stability of nonlinear systems and hence, for e xample, to permanence for biological systems.