Abstract: Expanding populations are usually described by deterministic travelling wavefronts, solutions to some reaction-diffusion partial differential equations. These macroscopic models only provide an approximation of the real dynamics since biological systems are finite and thus undergo demographic fluctuations. In the deterministic framework, invasions are divided into two classes: pulled fronts, whose dynamics are driven by the small populations located at the leading edge of the fronts, and pushed fronts, driven by the individuals living in regions of intermediate density. This suggests that pushed fronts appear when the strength of the cooperation between individuals is large enough (compared to the intrinsic growth rate of the individuals) and that they are far less sensitive to fluctuations than pulled ones. However, recent studies from Birzu, Hallatschek and Korolev on fluctuating fronts reveal the existence of a subclass of pushed invasions, which turns out to be very sensitive to demographic fluctuations, namely the semi-pushed fronts.  In this work, we consider a microscopic toy model for what happens to the right of an invasion front to rigorously verify and make precise this classification.