Fast varying active transmitter sets are a key feature of wireless communication networks with very short transmissions arising in machine-to-machine communications. A consequence is that the interference is dynamic, leading to non-Gaussian statistics. In this paper, we study the behavior of large-scale communication networks in the presence of isotropic α-stable interference, which forms a model for dynamic interference. We first characterize the achievable rate of each link by considering a non-Gaussian input distribution, which is shown to outperform a Gaussian input. Moreover, we analyze the area spectral efficiency, which is the total rate per square meter. Our analysis suggests that analogously to the common model of slowly varying active transmitter sets, dense networks maximize the area spectral efficiency.