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We investigate in this paper the dependence relation between the space-timeperiodic coefficients $A, q$ and $\mu$ of the reaction-diffusion equation$\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u =\mu(t, x) u(1 - u)$, and the spreading speed of the solutions of the Cauchyproblem associated with this equation and compactly supported initial data. Weprove in particular that (1) taking the spatial or temporal average of $\mu$decreases the minimal speed, (2) if the coefficients do not depend on $t$ and$q\not\equiv 0$, then increasing the amplitude of the diffusion matrix $A$increases the minimal speed, (3) if $A = I_N$, $\mu$ is a constant, then theintroduction of a space periodic drift term $q = \nabla Q$ increases theminimal speed. To prove these results, we use a variational characterization ofthe spreading speed that involves a family of periodic principal eigenvaluesassociated with the linearization of the equation near 0. We are thus back tothe investigation of the dependence relation between this family of eigenvaluesand the coefficients.

Some dependence results between the spreading speed and the coefficients of the space--time periodic Fisher--KPP equation