Density dependence plays an important role in population regulation, and has a long history in ecology as a mechanism that can induce local density fluctuations. Yet much less is known about how these endogenous processes affect spatial population dynamics. Biological invasions occur through the combined action of population growth (demography), and movement (dispersal), making them relevant for understanding how density dependence regulates spatial spread. While classical ecological theory suggests that many invasions move at a constant speed, empirical work is illuminating the highly variable nature of biological invasions, which can lead to non-constant spreading speeds. Here, we explore endogenous density dependence as a mechanism for inducing variability in biological invasions. We constructed a set of integrodifference population models that incorporate classic population fluctuation mechanisms to determine how density dependence in demography, including Allee effects, and in dispersal affects the speed of biological invasions. We show that density dependence is a key factor in producing fluctuations in spreading speed when Allee effects are acting on population densities that fluctuate locally. We show that the necessary density fluctuations can arise from either a nonmonotone population growth function where densities fluctuate locally (e.g., overcompensatory population growth), or from density-dependent dispersal when the population growth function results in constant local densities. As density dependence in both demography and dispersal are common, this mechanism of variability may influence many invading organisms.