In this work we introduce the free boundary value problem for the invasion phenomenon in biofilm reactors which takes into account the dynamics of the biofilm compartment as well as the bulk liquid phase in terms of both substrates and planktonic cells. We consider the biofilm as constituted by various particulate components growing in a liquid environment along with planktonic cells belonging to various microbial species that are able to move within the biofilm and the bulk liquid as well. The biofilm expansion depends on growth limiting nutrients which are dissolved in the liquid region or produced within the biofilm itself. The planktonic cells can diffuse and invade from the bulk liquid to the biofilm and switch their mode of growth from suspended to sessile when appropriate environmental conditions are found. The model is derived by coupling a reactor mass balance for planktonic cells and substrates with a full Wanner-Gujer type invasion model. The switch from motile to sessile state has been taken into account by introducing a growth rate term for sessile bacteria which depends on substrate and planktonic species concentrations within the biofilm. The movement of planktonic cells within the biofilm matrix has been modeled through the Fick's law of diffusion as we assumed a random character of motility. The model has been studied by both analytical methods and numerical simulations. In particular, the existence and uniqueness of the solutions has been proved in the class of continuous functions. Numerical simulations have been developed for a real engineering/ biological case which examines the invasion of specific microbial species, the Anammox bacteria, in a constituted wastewater biofilm. The invasion model has been adopted to illustrate the trends related to the establishment of such a multispecies community and to assess the effect of specific operational conditions on the biofilm colonization by Anammox bacteria. For all the cases analyzed, real data from existing literature is used to feed numerical simulations, which produce results in agreement with experimental findings.