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This paper is concerned with the periodic principal eigenvalue$k_\lambda(\mu)$ associated with the operator $- {d^2\over dx^2} - 2\lambda{d\over dx} - \mu(x) - \lambda^2$ , (1) where $\lambda\in \mathbb{R}$ and $\mu$is continuous and periodic in $x\in\mathbb{R}$. Our main result is that$k_\lambda(\mu^*) \le k_\lambda(\mu)$, where $\mu^*$ is the Schwarzrearrangement of the function $\mu$. From a population dynamics point of view,using reaction-diffusion modeling, this result means that the fragmentation ofthe habitat of an invading population slows down the invasion. We prove thatthis property does not hold in higher dimension, if $\mu^*$ is the Steinersymmetrization of $\mu$. For heterogeneous diffusion and advection, we provethat increasing the period of the coefficients decreases $k_\lambda$ and wecompute the limit of $k_\lambda$ when the period of the coefficients goes to 0.Lastly, we prove that, in dimension 1, rearranging the diffusion term decreases$k_\lambda$. These results rely on some new formula for the periodic principaleigenvalue of a nonsymmetric operator.

The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator