Monotonicity of the principal eigenvalue for a linear time-periodic parabolic operator

We investigate the effect of frequency on the principal eigenvalue of a time-periodic parabolic operator with Dirichlet, Robin or Neumann boundary conditions. The monotonicity and asymptotic behaviors of the principal eigenvalue with respect to the frequency parameter are established. Our results prove a conjecture raised by Hutson, Michaikow and Pol'{a}\v{c}ik [2001 J. Math. Biol.].