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

Abstract

We investigate in this paper the dependence relation between the space-time periodic coefficients $A, q$ and $μ$ of the reaction-diffusion equation $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = μ(t, x) u(1 - u)$, and the spreading speed of the solutions of the Cauchy problem associated with this equation and compactly supported initial data. We prove in particular that (1) taking the spatial or temporal average of $μ$ 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 $A = I_N$, $μ$ is a constant, then the introduction of a space periodic drift term $q = \nabla Q$ increases the minimal speed. To prove these results, we use a variational characterization of the spreading speed that involves a family of periodic principal eigenvalues associated with the linearization of the equation near 0. We are thus back to the investigation of the dependence relation between this family of eigenvalues and the coefficients.