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We study the smallest positive eigenvalue λ1(M) of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface of genus g ≥2. We show the existence of a constant C > 0 only depending on g so that λ1(M) ≢ [C-1/vol(M)², C log vol(M)/vol(M)²2g-2/(²2g-2-1)] and that this estimate is essentially sharp. We show that if M is typical or random, then we have λ1(M)≢ [C-1/vol(M)², C/vol(M)2]. This rests on a result of independent interest about reccurence properties of axes of random pseudo-Anosov elements. [ABSTRACT FROM AUTHOR]